Toniolo and Linder, Equation (13)

Percentage Accurate: 49.6% → 62.7%

Time: 22.8s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 49.6% 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: 62.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_1\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{l\_m}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_1 - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(n \cdot \frac{U* - U}{{Om}^{2}} - \frac{2}{Om}\right)\right)} \cdot \left(l\_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_2 (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_1))))
   (if (<= t_2 0.0)
     (pow (* (* 2.0 n) (* U (- t (* 2.0 (/ (pow l_m 2.0) Om))))) 0.5)
     (if (<= t_2 5e+303)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_1 (* 2.0 (* l_m (/ l_m Om)))))))
       (*
        (sqrt (* U (* n (- (* n (/ (- U* U) (pow Om 2.0))) (/ 2.0 Om)))))
        (* l_m (sqrt 2.0)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = pow(((2.0 * n) * (U * (t - (2.0 * (pow(l_m, 2.0) / Om))))), 0.5);
	} else if (t_2 <= 5e+303) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = sqrt((U * (n * ((n * ((U_42_ - U) / pow(Om, 2.0))) - (2.0 / Om))))) * (l_m * sqrt(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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (n * ((l_m / om) ** 2.0d0)) * (u_42 - u)
    t_2 = ((2.0d0 * n) * u) * ((t - (2.0d0 * ((l_m * l_m) / om))) + t_1)
    if (t_2 <= 0.0d0) then
        tmp = ((2.0d0 * n) * (u * (t - (2.0d0 * ((l_m ** 2.0d0) / om))))) ** 0.5d0
    else if (t_2 <= 5d+303) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + (t_1 - (2.0d0 * (l_m * (l_m / om)))))))
    else
        tmp = sqrt((u * (n * ((n * ((u_42 - u) / (om ** 2.0d0))) - (2.0d0 / om))))) * (l_m * sqrt(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 t_1 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = Math.pow(((2.0 * n) * (U * (t - (2.0 * (Math.pow(l_m, 2.0) / Om))))), 0.5);
	} else if (t_2 <= 5e+303) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = Math.sqrt((U * (n * ((n * ((U_42_ - U) / Math.pow(Om, 2.0))) - (2.0 / Om))))) * (l_m * Math.sqrt(2.0));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)
	tmp = 0
	if t_2 <= 0.0:
		tmp = math.pow(((2.0 * n) * (U * (t - (2.0 * (math.pow(l_m, 2.0) / Om))))), 0.5)
	elif t_2 <= 5e+303:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))))
	else:
		tmp = math.sqrt((U * (n * ((n * ((U_42_ - U) / math.pow(Om, 2.0))) - (2.0 / Om))))) * (l_m * math.sqrt(2.0))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_2 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_1))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64((l_m ^ 2.0) / Om))))) ^ 0.5;
	elseif (t_2 <= 5e+303)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(t_1 - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))))));
	else
		tmp = Float64(sqrt(Float64(U * Float64(n * Float64(Float64(n * Float64(Float64(U_42_ - U) / (Om ^ 2.0))) - Float64(2.0 / Om))))) * Float64(l_m * sqrt(2.0)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = ((2.0 * n) * (U * (t - (2.0 * ((l_m ^ 2.0) / Om))))) ^ 0.5;
	elseif (t_2 <= 5e+303)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	else
		tmp = sqrt((U * (n * ((n * ((U_42_ - U) / (Om ^ 2.0))) - (2.0 / Om))))) * (l_m * sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[t$95$2, 5e+303], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(t$95$1 - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * N[(n * N[(N[(n * N[(N[(U$42$ - U), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 15.9%

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

   3. Simplified 42.2%

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

      1. *-commutative 42.2%

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

      2. sqrt-prod 33.8%

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

      3. associate-*r/ 33.8%

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

      4. pow2 33.8%

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

   6. Applied egg-rr 33.8%

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

      1. fma-define 33.8%

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

      2. +-commutative 33.8%

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

      3. fma-define 33.8%

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

      4. associate-*r/ 33.8%

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

   8. Simplified 33.8%

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

   9. Taylor expanded in n around 0 36.1%

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

       1. sqrt-unprod 48.9%

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

       2. pow1/2 49.1%

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

   11. Applied egg-rr 49.1%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.9999999999999997e303

   1. Initial program 98.3%

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

   3. Simplified 98.4%

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

   if 4.9999999999999997e303 < (*.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 19.7%

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

   3. Simplified 29.2%

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

   5. Taylor expanded in l around inf 28.6%

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

      1. associate-/l* 28.8%

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

      2. associate-*r/ 28.8%

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

      3. metadata-eval 28.8%

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

   7. Simplified 28.8%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(n \cdot \frac{U* - U}{{Om}^{2}} - \frac{2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 60.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_1\right)}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_1 - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{l\_m}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_2
         (sqrt (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_1)))))
   (if (<= t_2 0.0)
     (sqrt (* (* 2.0 n) (* U t)))
     (if (<= t_2 INFINITY)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_1 (* 2.0 (* l_m (/ l_m Om)))))))
       (pow (* (* 2.0 n) (* U (- t (* 2.0 (/ (pow l_m 2.0) Om))))) 0.5)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt(((2.0 * n) * (U * t)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = pow(((2.0 * n) * (U * (t - (2.0 * (pow(l_m, 2.0) / Om))))), 0.5);
	}
	return tmp;
}

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = Math.sqrt(((2.0 * n) * (U * t)));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = Math.pow(((2.0 * n) * (U * (t - (2.0 * (Math.pow(l_m, 2.0) / Om))))), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_2 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)))
	tmp = 0
	if t_2 <= 0.0:
		tmp = math.sqrt(((2.0 * n) * (U * t)))
	elif t_2 <= math.inf:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))))
	else:
		tmp = math.pow(((2.0 * n) * (U * (t - (2.0 * (math.pow(l_m, 2.0) / Om))))), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_1)))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t)));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(t_1 - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))))));
	else
		tmp = Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64((l_m ^ 2.0) / Om))))) ^ 0.5;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = sqrt(((2.0 * n) * (U * t)));
	elseif (t_2 <= Inf)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	else
		tmp = ((2.0 * n) * (U * (t - (2.0 * ((l_m ^ 2.0) / Om))))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(t$95$1 - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 17.9%

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

   3. Simplified 45.0%

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

   5. Taylor expanded in t around inf 45.0%

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

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

   1. Initial program 69.9%

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

   3. Simplified 75.5%

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

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

   1. Initial program 0.0%

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

   3. Simplified 10.8%

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

      1. *-commutative 10.8%

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

      2. sqrt-prod 2.3%

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

      3. associate-*r/ 0.3%

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

      4. pow2 0.3%

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

   6. Applied egg-rr 0.3%

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

      1. fma-define 0.3%

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

      2. +-commutative 0.3%

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

      3. fma-define 0.5%

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

      4. associate-*r/ 0.5%

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

   8. Simplified 0.5%

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

   9. Taylor expanded in n around 0 3.2%

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

       1. sqrt-unprod 10.3%

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

       2. pow1/2 38.9%

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

   11. Applied egg-rr 38.9%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\\ t_2 := \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot U*\right) - t\_1\right)\right)}\\ \mathbf{if}\;n \leq -1.75 \cdot 10^{+29}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;n \leq -2 \cdot 10^{-310}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{l\_m}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;n \leq 1.35 \cdot 10^{-97}:\\ \;\;\;\;\sqrt{U \cdot \left(t - t\_1\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* l_m (/ l_m Om))))
        (t_2
         (sqrt
          (*
           (* 2.0 (* n U))
           (+ t (- (* n (* (pow (/ l_m Om) 2.0) U*)) t_1))))))
   (if (<= n -1.75e+29)
     t_2
     (if (<= n -2e-310)
       (pow (* (* 2.0 n) (* U (- t (* 2.0 (/ (pow l_m 2.0) Om))))) 0.5)
       (if (<= n 1.35e-97) (* (sqrt (* U (- t t_1))) (sqrt (* 2.0 n))) t_2)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = 2.0 * (l_m * (l_m / Om));
	double t_2 = sqrt(((2.0 * (n * U)) * (t + ((n * (pow((l_m / Om), 2.0) * U_42_)) - t_1))));
	double tmp;
	if (n <= -1.75e+29) {
		tmp = t_2;
	} else if (n <= -2e-310) {
		tmp = pow(((2.0 * n) * (U * (t - (2.0 * (pow(l_m, 2.0) / Om))))), 0.5);
	} else if (n <= 1.35e-97) {
		tmp = sqrt((U * (t - t_1))) * sqrt((2.0 * n));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 2.0d0 * (l_m * (l_m / om))
    t_2 = sqrt(((2.0d0 * (n * u)) * (t + ((n * (((l_m / om) ** 2.0d0) * u_42)) - t_1))))
    if (n <= (-1.75d+29)) then
        tmp = t_2
    else if (n <= (-2d-310)) then
        tmp = ((2.0d0 * n) * (u * (t - (2.0d0 * ((l_m ** 2.0d0) / om))))) ** 0.5d0
    else if (n <= 1.35d-97) then
        tmp = sqrt((u * (t - t_1))) * sqrt((2.0d0 * n))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = 2.0 * (l_m * (l_m / Om));
	double t_2 = Math.sqrt(((2.0 * (n * U)) * (t + ((n * (Math.pow((l_m / Om), 2.0) * U_42_)) - t_1))));
	double tmp;
	if (n <= -1.75e+29) {
		tmp = t_2;
	} else if (n <= -2e-310) {
		tmp = Math.pow(((2.0 * n) * (U * (t - (2.0 * (Math.pow(l_m, 2.0) / Om))))), 0.5);
	} else if (n <= 1.35e-97) {
		tmp = Math.sqrt((U * (t - t_1))) * Math.sqrt((2.0 * n));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = 2.0 * (l_m * (l_m / Om))
	t_2 = math.sqrt(((2.0 * (n * U)) * (t + ((n * (math.pow((l_m / Om), 2.0) * U_42_)) - t_1))))
	tmp = 0
	if n <= -1.75e+29:
		tmp = t_2
	elif n <= -2e-310:
		tmp = math.pow(((2.0 * n) * (U * (t - (2.0 * (math.pow(l_m, 2.0) / Om))))), 0.5)
	elif n <= 1.35e-97:
		tmp = math.sqrt((U * (t - t_1))) * math.sqrt((2.0 * n))
	else:
		tmp = t_2
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(2.0 * Float64(l_m * Float64(l_m / Om)))
	t_2 = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(n * Float64((Float64(l_m / Om) ^ 2.0) * U_42_)) - t_1))))
	tmp = 0.0
	if (n <= -1.75e+29)
		tmp = t_2;
	elseif (n <= -2e-310)
		tmp = Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64((l_m ^ 2.0) / Om))))) ^ 0.5;
	elseif (n <= 1.35e-97)
		tmp = Float64(sqrt(Float64(U * Float64(t - t_1))) * sqrt(Float64(2.0 * n)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = 2.0 * (l_m * (l_m / Om));
	t_2 = sqrt(((2.0 * (n * U)) * (t + ((n * (((l_m / Om) ^ 2.0) * U_42_)) - t_1))));
	tmp = 0.0;
	if (n <= -1.75e+29)
		tmp = t_2;
	elseif (n <= -2e-310)
		tmp = ((2.0 * n) * (U * (t - (2.0 * ((l_m ^ 2.0) / Om))))) ^ 0.5;
	elseif (n <= 1.35e-97)
		tmp = sqrt((U * (t - t_1))) * sqrt((2.0 * n));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(n * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * U$42$), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -1.75e+29], t$95$2, If[LessEqual[n, -2e-310], N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[n, 1.35e-97], N[(N[Sqrt[N[(U * N[(t - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.74999999999999989e29 or 1.34999999999999993e-97 < n

   1. Initial program 56.6%

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

   3. Simplified 62.0%

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

      1. associate-*r* 63.5%

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

      2. pow1 63.5%

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

   6. Applied egg-rr 63.5%

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

      1. unpow1 63.5%

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

   8. Simplified 63.5%

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

   9. Taylor expanded in U around 0 49.6%

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

       1. mul-1-neg 49.6%

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

       2. associate-/l* 49.7%

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

       3. unpow2 49.7%

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

       4. unpow2 49.7%

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

       5. times-frac 63.5%

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

       6. unpow2 63.5%

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

   11. Simplified 63.5%

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

   if -1.74999999999999989e29 < n < -1.999999999999994e-310

   1. Initial program 40.3%

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

   3. Simplified 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. Step-by-step derivation

      1. *-commutative 47.3%

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

      2. sqrt-prod 0.0%

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

      3. associate-*r/ 0.0%

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

      4. pow2 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. fma-define 0.0%

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

      2. +-commutative 0.0%

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

      3. fma-define 0.0%

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

      4. associate-*r/ 0.0%

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

   8. Simplified 0.0%

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

   9. Taylor expanded in n around 0 0.0%

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

       1. sqrt-unprod 46.4%

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

       2. pow1/2 49.4%

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

   11. Applied egg-rr 49.4%

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

   if -1.999999999999994e-310 < n < 1.34999999999999993e-97

   1. Initial program 43.0%

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

   3. Simplified 51.4%

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

      1. *-commutative 51.4%

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

      2. sqrt-prod 66.3%

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

      3. associate-*r/ 64.2%

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

      4. pow2 64.2%

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

   6. Applied egg-rr 64.2%

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

      1. fma-define 64.2%

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

      2. +-commutative 64.2%

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

      3. fma-define 64.2%

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

      4. associate-*r/ 64.2%

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

   8. Simplified 64.2%

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

   9. Taylor expanded in n around 0 66.9%

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

       1. unpow2 66.9%

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

       2. associate-*r/ 69.0%

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

       3. *-commutative 69.0%

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

   11. Applied egg-rr 69.0%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.75 \cdot 10^{+29}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot U*\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;n \leq -2 \cdot 10^{-310}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;n \leq 1.35 \cdot 10^{-97}:\\ \;\;\;\;\sqrt{U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot U*\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;Om \leq -8.2 \cdot 10^{+113} \lor \neg \left(Om \leq 4.7 \cdot 10^{+68}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot U*\right) - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t + \frac{{l\_m}^{2} \cdot \left(\frac{U \cdot \left(n \cdot \left(U* - U\right)\right)}{Om} - 2 \cdot U\right)}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (or (<= Om -8.2e+113) (not (<= Om 4.7e+68)))
   (sqrt
    (*
     (* 2.0 (* n U))
     (+ t (- (* n (* (pow (/ l_m Om) 2.0) U*)) (* 2.0 (* l_m (/ l_m Om)))))))
   (sqrt
    (*
     2.0
     (*
      n
      (+
       (* U t)
       (/ (* (pow l_m 2.0) (- (/ (* U (* n (- U* U))) Om) (* 2.0 U))) 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 ((Om <= -8.2e+113) || !(Om <= 4.7e+68)) {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * (pow((l_m / Om), 2.0) * U_42_)) - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = sqrt((2.0 * (n * ((U * t) + ((pow(l_m, 2.0) * (((U * (n * (U_42_ - U))) / Om) - (2.0 * U))) / 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 ((om <= (-8.2d+113)) .or. (.not. (om <= 4.7d+68))) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + ((n * (((l_m / om) ** 2.0d0) * u_42)) - (2.0d0 * (l_m * (l_m / om)))))))
    else
        tmp = sqrt((2.0d0 * (n * ((u * t) + (((l_m ** 2.0d0) * (((u * (n * (u_42 - u))) / om) - (2.0d0 * u))) / 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 ((Om <= -8.2e+113) || !(Om <= 4.7e+68)) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((n * (Math.pow((l_m / Om), 2.0) * U_42_)) - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = Math.sqrt((2.0 * (n * ((U * t) + ((Math.pow(l_m, 2.0) * (((U * (n * (U_42_ - U))) / Om) - (2.0 * U))) / Om)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if (Om <= -8.2e+113) or not (Om <= 4.7e+68):
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((n * (math.pow((l_m / Om), 2.0) * U_42_)) - (2.0 * (l_m * (l_m / Om)))))))
	else:
		tmp = math.sqrt((2.0 * (n * ((U * t) + ((math.pow(l_m, 2.0) * (((U * (n * (U_42_ - U))) / Om) - (2.0 * U))) / Om)))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if ((Om <= -8.2e+113) || !(Om <= 4.7e+68))
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * 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 = sqrt(Float64(2.0 * Float64(n * Float64(Float64(U * t) + Float64(Float64((l_m ^ 2.0) * Float64(Float64(Float64(U * Float64(n * Float64(U_42_ - U))) / Om) - Float64(2.0 * U))) / 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 ((Om <= -8.2e+113) || ~((Om <= 4.7e+68)))
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * (((l_m / Om) ^ 2.0) * U_42_)) - (2.0 * (l_m * (l_m / Om)))))));
	else
		tmp = sqrt((2.0 * (n * ((U * t) + (((l_m ^ 2.0) * (((U * (n * (U_42_ - U))) / Om) - (2.0 * U))) / 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[Om, -8.2e+113], N[Not[LessEqual[Om, 4.7e+68]], $MachinePrecision]], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * 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[Sqrt[N[(2.0 * N[(n * N[(N[(U * t), $MachinePrecision] + N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[(N[(U * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -8.19999999999999985e113 or 4.6999999999999996e68 < Om

   1. Initial program 55.4%

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

   3. Simplified 66.8%

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

      1. associate-*r* 66.8%

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

      2. pow1 66.8%

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

   6. Applied egg-rr 66.8%

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

      1. unpow1 66.8%

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

   8. Simplified 66.8%

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

   9. Taylor expanded in U around 0 50.8%

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

       1. mul-1-neg 50.8%

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

       2. associate-/l* 54.7%

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

       3. unpow2 54.7%

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

       4. unpow2 54.7%

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

       5. times-frac 66.8%

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

       6. unpow2 66.8%

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

   11. Simplified 66.8%

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

   if -8.19999999999999985e113 < Om < 4.6999999999999996e68

   1. Initial program 44.5%

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

   3. Simplified 48.3%

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

   5. Taylor expanded in Om around -inf 46.3%

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

      1. +-commutative 46.3%

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

      2. mul-1-neg 46.3%

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

      3. unsub-neg 46.3%

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

      4. +-commutative 46.3%

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

      5. mul-1-neg 46.3%

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

      6. unsub-neg 46.3%

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

      7. associate-*r* 46.3%

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

      8. associate-/l* 48.9%

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

      9. associate-/l* 47.5%

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

   7. Simplified 47.5%

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

   8. Taylor expanded in l around 0 55.9%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -8.2 \cdot 10^{+113} \lor \neg \left(Om \leq 4.7 \cdot 10^{+68}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot U*\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t + \frac{{\ell}^{2} \cdot \left(\frac{U \cdot \left(n \cdot \left(U* - U\right)\right)}{Om} - 2 \cdot U\right)}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 3.2 \cdot 10^{-197}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right|}\\ \mathbf{elif}\;l\_m \leq 4.6 \cdot 10^{+213}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot \left(n \cdot U*\right) - \frac{l\_m}{Om} \cdot \left(2 \cdot l\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(U \cdot -4\right) \cdot \frac{n \cdot {l\_m}^{2}}{Om}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 3.2e-197)
   (sqrt (fabs (* 2.0 (* n (* U t)))))
   (if (<= l_m 4.6e+213)
     (sqrt
      (*
       (* 2.0 (* n U))
       (+ t (- (* (pow (/ l_m Om) 2.0) (* n U*)) (* (/ l_m Om) (* 2.0 l_m))))))
     (pow (* (* U -4.0) (/ (* n (pow l_m 2.0)) Om)) 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 (l_m <= 3.2e-197) {
		tmp = sqrt(fabs((2.0 * (n * (U * t)))));
	} else if (l_m <= 4.6e+213) {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((pow((l_m / Om), 2.0) * (n * U_42_)) - ((l_m / Om) * (2.0 * l_m))))));
	} else {
		tmp = pow(((U * -4.0) * ((n * pow(l_m, 2.0)) / Om)), 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 (l_m <= 3.2d-197) then
        tmp = sqrt(abs((2.0d0 * (n * (u * t)))))
    else if (l_m <= 4.6d+213) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + ((((l_m / om) ** 2.0d0) * (n * u_42)) - ((l_m / om) * (2.0d0 * l_m))))))
    else
        tmp = ((u * (-4.0d0)) * ((n * (l_m ** 2.0d0)) / om)) ** 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 (l_m <= 3.2e-197) {
		tmp = Math.sqrt(Math.abs((2.0 * (n * (U * t)))));
	} else if (l_m <= 4.6e+213) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((Math.pow((l_m / Om), 2.0) * (n * U_42_)) - ((l_m / Om) * (2.0 * l_m))))));
	} else {
		tmp = Math.pow(((U * -4.0) * ((n * Math.pow(l_m, 2.0)) / Om)), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 3.2e-197:
		tmp = math.sqrt(math.fabs((2.0 * (n * (U * t)))))
	elif l_m <= 4.6e+213:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((math.pow((l_m / Om), 2.0) * (n * U_42_)) - ((l_m / Om) * (2.0 * l_m))))))
	else:
		tmp = math.pow(((U * -4.0) * ((n * math.pow(l_m, 2.0)) / Om)), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 3.2e-197)
		tmp = sqrt(abs(Float64(2.0 * Float64(n * Float64(U * t)))));
	elseif (l_m <= 4.6e+213)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64((Float64(l_m / Om) ^ 2.0) * Float64(n * U_42_)) - Float64(Float64(l_m / Om) * Float64(2.0 * l_m))))));
	else
		tmp = Float64(Float64(U * -4.0) * Float64(Float64(n * (l_m ^ 2.0)) / Om)) ^ 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 (l_m <= 3.2e-197)
		tmp = sqrt(abs((2.0 * (n * (U * t)))));
	elseif (l_m <= 4.6e+213)
		tmp = sqrt(((2.0 * (n * U)) * (t + ((((l_m / Om) ^ 2.0) * (n * U_42_)) - ((l_m / Om) * (2.0 * l_m))))));
	else
		tmp = ((U * -4.0) * ((n * (l_m ^ 2.0)) / Om)) ^ 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[l$95$m, 3.2e-197], N[Sqrt[N[Abs[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 4.6e+213], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(n * U$42$), $MachinePrecision]), $MachinePrecision] - N[(N[(l$95$m / Om), $MachinePrecision] * N[(2.0 * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(U * -4.0), $MachinePrecision] * N[(N[(n * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.1999999999999997e-197

   1. Initial program 48.4%

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

   3. Simplified 54.4%

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

   5. Taylor expanded in t around inf 41.8%

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

      1. add-sqr-sqrt 41.8%

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

      2. pow1/2 41.8%

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

      3. pow1/2 41.8%

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

      4. pow-prod-down 27.3%

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

      5. pow2 27.3%

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

      6. associate-*l* 27.3%

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

   7. Applied egg-rr 27.3%

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

      1. unpow1/2 27.3%

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

      2. unpow2 27.3%

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

      3. rem-sqrt-square 42.1%

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

   9. Simplified 42.1%

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

   if 3.1999999999999997e-197 < l < 4.59999999999999996e213

   1. Initial program 58.3%

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

   3. Simplified 63.8%

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

      1. associate-*r* 63.8%

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

      2. pow1 63.8%

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

   6. Applied egg-rr 63.8%

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

      1. unpow1 63.8%

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

   8. Simplified 63.8%

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

   9. Taylor expanded in U around 0 50.8%

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

       1. mul-1-neg 50.8%

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

       2. associate-/l* 54.1%

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

       3. unpow2 54.1%

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

       4. unpow2 54.1%

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

       5. times-frac 63.8%

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

       6. unpow2 63.8%

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

   11. Simplified 63.8%

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

       1. associate-*r* 63.8%

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

       2. fma-define 67.2%

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

       3. distribute-lft-neg-in 67.2%

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

   13. Applied egg-rr 67.2%

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

       1. distribute-lft-neg-out 67.2%

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

       2. distribute-rgt-neg-in 67.2%

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

       3. fmm-undef 63.8%

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

       4. *-commutative 63.8%

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

       5. associate-*r* 61.6%

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

       6. *-commutative 61.6%

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

   15. Simplified 61.6%

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

   if 4.59999999999999996e213 < l

   1. Initial program 6.7%

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

   3. Simplified 12.3%

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

   5. Taylor expanded in Om around inf 9.0%

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

      1. add-sqr-sqrt 9.0%

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

      2. pow2 9.0%

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

      3. fma-define 9.0%

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

      4. associate-/l* 9.0%

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

   7. Applied egg-rr 9.0%

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

   8. Taylor expanded in l around inf 9.0%

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

      1. associate-*r/ 9.0%

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

      2. associate-*r* 9.0%

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

   10. Simplified 9.0%

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

       1. pow1/2 26.5%

          \[\leadsto \color{blue}{{\left(\frac{\left(-4 \cdot U\right) \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}^{0.5}} \]

       2. associate-/l* 26.5%

          \[\leadsto {\color{blue}{\left(\left(-4 \cdot U\right) \cdot \frac{{\ell}^{2} \cdot n}{Om}\right)}}^{0.5} \]

       3. *-commutative 26.5%

          \[\leadsto {\left(\color{blue}{\left(U \cdot -4\right)} \cdot \frac{{\ell}^{2} \cdot n}{Om}\right)}^{0.5} \]

       4. *-commutative 26.5%

          \[\leadsto {\left(\left(U \cdot -4\right) \cdot \frac{\color{blue}{n \cdot {\ell}^{2}}}{Om}\right)}^{0.5} \]

   12. Applied egg-rr 26.5%

       \[\leadsto \color{blue}{{\left(\left(U \cdot -4\right) \cdot \frac{n \cdot {\ell}^{2}}{Om}\right)}^{0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.2 \cdot 10^{-197}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right|}\\ \mathbf{elif}\;\ell \leq 4.6 \cdot 10^{+213}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot U*\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(U \cdot -4\right) \cdot \frac{n \cdot {\ell}^{2}}{Om}\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{-310}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{l\_m}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(t - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)} \cdot \sqrt{2 \cdot n}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n -2e-310)
   (pow (* (* 2.0 n) (* U (- t (* 2.0 (/ (pow l_m 2.0) Om))))) 0.5)
   (* (sqrt (* U (- t (* 2.0 (* l_m (/ l_m Om)))))) (sqrt (* 2.0 n)))))

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 <= -2e-310) {
		tmp = pow(((2.0 * n) * (U * (t - (2.0 * (pow(l_m, 2.0) / Om))))), 0.5);
	} else {
		tmp = sqrt((U * (t - (2.0 * (l_m * (l_m / Om)))))) * sqrt((2.0 * n));
	}
	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 <= (-2d-310)) then
        tmp = ((2.0d0 * n) * (u * (t - (2.0d0 * ((l_m ** 2.0d0) / om))))) ** 0.5d0
    else
        tmp = sqrt((u * (t - (2.0d0 * (l_m * (l_m / om)))))) * sqrt((2.0d0 * n))
    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 <= -2e-310) {
		tmp = Math.pow(((2.0 * n) * (U * (t - (2.0 * (Math.pow(l_m, 2.0) / Om))))), 0.5);
	} else {
		tmp = Math.sqrt((U * (t - (2.0 * (l_m * (l_m / Om)))))) * Math.sqrt((2.0 * n));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if n <= -2e-310:
		tmp = math.pow(((2.0 * n) * (U * (t - (2.0 * (math.pow(l_m, 2.0) / Om))))), 0.5)
	else:
		tmp = math.sqrt((U * (t - (2.0 * (l_m * (l_m / Om)))))) * math.sqrt((2.0 * n))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= -2e-310)
		tmp = Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64((l_m ^ 2.0) / Om))))) ^ 0.5;
	else
		tmp = Float64(sqrt(Float64(U * Float64(t - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))))) * sqrt(Float64(2.0 * n)));
	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 <= -2e-310)
		tmp = ((2.0 * n) * (U * (t - (2.0 * ((l_m ^ 2.0) / Om))))) ^ 0.5;
	else
		tmp = sqrt((U * (t - (2.0 * (l_m * (l_m / Om)))))) * sqrt((2.0 * n));
	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, -2e-310], N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[(N[Sqrt[N[(U * N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.999999999999994e-310

   1. Initial program 46.0%

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

   3. Simplified 54.3%

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

      1. *-commutative 54.3%

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

      2. sqrt-prod 0.0%

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

      3. associate-*r/ 0.0%

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

      4. pow2 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. fma-define 0.0%

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

      2. +-commutative 0.0%

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

      3. fma-define 0.0%

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

      4. associate-*r/ 0.0%

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

   8. Simplified 0.0%

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

   9. Taylor expanded in n around 0 0.0%

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

       1. sqrt-unprod 42.5%

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

       2. pow1/2 50.0%

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

   11. Applied egg-rr 50.0%

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

   if -1.999999999999994e-310 < n

   1. Initial program 52.3%

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

   3. Simplified 56.9%

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

      1. *-commutative 56.9%

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

      2. sqrt-prod 65.3%

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

      3. associate-*r/ 62.0%

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

      4. pow2 62.0%

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

   6. Applied egg-rr 62.0%

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

      1. fma-define 62.0%

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

      2. +-commutative 62.0%

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

      3. fma-define 62.1%

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

      4. associate-*r/ 62.1%

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

   8. Simplified 62.1%

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

   9. Taylor expanded in n around 0 53.1%

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

       1. unpow2 53.1%

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

       2. associate-*r/ 56.4%

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

       3. *-commutative 56.4%

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

   11. Applied egg-rr 56.4%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{-310}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)} \cdot \sqrt{2 \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 46.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.1 \cdot 10^{+105}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t + -2 \cdot \frac{U \cdot \left(l\_m \cdot l\_m\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(U \cdot -4\right) \cdot \frac{n \cdot {l\_m}^{2}}{Om}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.1e+105)
   (sqrt (* 2.0 (* n (+ (* U t) (* -2.0 (/ (* U (* l_m l_m)) Om))))))
   (pow (* (* U -4.0) (/ (* n (pow l_m 2.0)) Om)) 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 (l_m <= 2.1e+105) {
		tmp = sqrt((2.0 * (n * ((U * t) + (-2.0 * ((U * (l_m * l_m)) / Om))))));
	} else {
		tmp = pow(((U * -4.0) * ((n * pow(l_m, 2.0)) / Om)), 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 (l_m <= 2.1d+105) then
        tmp = sqrt((2.0d0 * (n * ((u * t) + ((-2.0d0) * ((u * (l_m * l_m)) / om))))))
    else
        tmp = ((u * (-4.0d0)) * ((n * (l_m ** 2.0d0)) / om)) ** 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 (l_m <= 2.1e+105) {
		tmp = Math.sqrt((2.0 * (n * ((U * t) + (-2.0 * ((U * (l_m * l_m)) / Om))))));
	} else {
		tmp = Math.pow(((U * -4.0) * ((n * Math.pow(l_m, 2.0)) / Om)), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 2.1e+105:
		tmp = math.sqrt((2.0 * (n * ((U * t) + (-2.0 * ((U * (l_m * l_m)) / Om))))))
	else:
		tmp = math.pow(((U * -4.0) * ((n * math.pow(l_m, 2.0)) / Om)), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2.1e+105)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(Float64(U * t) + Float64(-2.0 * Float64(Float64(U * Float64(l_m * l_m)) / Om))))));
	else
		tmp = Float64(Float64(U * -4.0) * Float64(Float64(n * (l_m ^ 2.0)) / Om)) ^ 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 (l_m <= 2.1e+105)
		tmp = sqrt((2.0 * (n * ((U * t) + (-2.0 * ((U * (l_m * l_m)) / Om))))));
	else
		tmp = ((U * -4.0) * ((n * (l_m ^ 2.0)) / Om)) ^ 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[l$95$m, 2.1e+105], N[Sqrt[N[(2.0 * N[(n * N[(N[(U * t), $MachinePrecision] + N[(-2.0 * N[(N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(U * -4.0), $MachinePrecision] * N[(N[(n * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.1000000000000001e105

   1. Initial program 54.2%

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

   3. Simplified 59.8%

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

   5. Taylor expanded in Om around inf 48.4%

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

      1. unpow2 48.4%

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

   7. Applied egg-rr 48.4%

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

   if 2.1000000000000001e105 < l

   1. Initial program 20.8%

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

   3. Simplified 32.9%

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

   5. Taylor expanded in Om around inf 17.2%

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

      1. add-sqr-sqrt 17.2%

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

      2. pow2 17.2%

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

      3. fma-define 17.2%

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

      4. associate-/l* 19.5%

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

   7. Applied egg-rr 19.5%

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

   8. Taylor expanded in l around inf 17.2%

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

      1. associate-*r/ 17.2%

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

      2. associate-*r* 17.2%

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

   10. Simplified 17.2%

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

       1. pow1/2 34.8%

          \[\leadsto \color{blue}{{\left(\frac{\left(-4 \cdot U\right) \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}^{0.5}} \]

       2. associate-/l* 34.9%

          \[\leadsto {\color{blue}{\left(\left(-4 \cdot U\right) \cdot \frac{{\ell}^{2} \cdot n}{Om}\right)}}^{0.5} \]

       3. *-commutative 34.9%

          \[\leadsto {\left(\color{blue}{\left(U \cdot -4\right)} \cdot \frac{{\ell}^{2} \cdot n}{Om}\right)}^{0.5} \]

       4. *-commutative 34.9%

          \[\leadsto {\left(\left(U \cdot -4\right) \cdot \frac{\color{blue}{n \cdot {\ell}^{2}}}{Om}\right)}^{0.5} \]

   12. Applied egg-rr 34.9%

       \[\leadsto \color{blue}{{\left(\left(U \cdot -4\right) \cdot \frac{n \cdot {\ell}^{2}}{Om}\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.1 \cdot 10^{+105}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t + -2 \cdot \frac{U \cdot \left(\ell \cdot \ell\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(U \cdot -4\right) \cdot \frac{n \cdot {\ell}^{2}}{Om}\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2 \cdot 10^{+113}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t + -2 \cdot \frac{U \cdot \left(l\_m \cdot l\_m\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \left(\left(n \cdot \frac{\sqrt{2}}{Om}\right) \cdot \sqrt{U \cdot U*}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2e+113)
   (sqrt (* 2.0 (* n (+ (* U t) (* -2.0 (/ (* U (* l_m l_m)) Om))))))
   (* l_m (* (* n (/ (sqrt 2.0) Om)) (sqrt (* U U*))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2e+113) {
		tmp = sqrt((2.0 * (n * ((U * t) + (-2.0 * ((U * (l_m * l_m)) / Om))))));
	} else {
		tmp = l_m * ((n * (sqrt(2.0) / Om)) * sqrt((U * U_42_)));
	}
	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 <= 2d+113) then
        tmp = sqrt((2.0d0 * (n * ((u * t) + ((-2.0d0) * ((u * (l_m * l_m)) / om))))))
    else
        tmp = l_m * ((n * (sqrt(2.0d0) / om)) * sqrt((u * u_42)))
    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 <= 2e+113) {
		tmp = Math.sqrt((2.0 * (n * ((U * t) + (-2.0 * ((U * (l_m * l_m)) / Om))))));
	} else {
		tmp = l_m * ((n * (Math.sqrt(2.0) / Om)) * Math.sqrt((U * U_42_)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 2e+113:
		tmp = math.sqrt((2.0 * (n * ((U * t) + (-2.0 * ((U * (l_m * l_m)) / Om))))))
	else:
		tmp = l_m * ((n * (math.sqrt(2.0) / Om)) * math.sqrt((U * U_42_)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2e+113)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(Float64(U * t) + Float64(-2.0 * Float64(Float64(U * Float64(l_m * l_m)) / Om))))));
	else
		tmp = Float64(l_m * Float64(Float64(n * Float64(sqrt(2.0) / Om)) * sqrt(Float64(U * U_42_))));
	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 <= 2e+113)
		tmp = sqrt((2.0 * (n * ((U * t) + (-2.0 * ((U * (l_m * l_m)) / Om))))));
	else
		tmp = l_m * ((n * (sqrt(2.0) / Om)) * sqrt((U * U_42_)));
	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, 2e+113], N[Sqrt[N[(2.0 * N[(n * N[(N[(U * t), $MachinePrecision] + N[(-2.0 * N[(N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[(N[(n * N[(N[Sqrt[2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2e113

   1. Initial program 54.4%

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

   3. Simplified 59.9%

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

   5. Taylor expanded in Om around inf 48.3%

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

      1. unpow2 48.3%

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

   7. Applied egg-rr 48.3%

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

   if 2e113 < l

   1. Initial program 17.2%

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

   3. Simplified 30.2%

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

   5. Taylor expanded in U* around inf 27.5%

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

      1. associate-/l* 30.0%

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

   7. Simplified 30.0%

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

   8. Taylor expanded in l around 0 27.5%

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

      1. associate-/l* 30.0%

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

      2. associate-*r/ 30.0%

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

      3. *-commutative 30.0%

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

      4. associate-*l* 30.0%

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

   10. Simplified 30.0%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{+113}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t + -2 \cdot \frac{U \cdot \left(\ell \cdot \ell\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\left(n \cdot \frac{\sqrt{2}}{Om}\right) \cdot \sqrt{U \cdot U*}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.45 \cdot 10^{+113}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t + -2 \cdot \frac{U \cdot \left(l\_m \cdot l\_m\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \frac{n}{Om}\right) \cdot \sqrt{U \cdot \left(2 \cdot U*\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.45e+113)
   (sqrt (* 2.0 (* n (+ (* U t) (* -2.0 (/ (* U (* l_m l_m)) Om))))))
   (* (* l_m (/ n Om)) (sqrt (* U (* 2.0 U*))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2.45e+113) {
		tmp = sqrt((2.0 * (n * ((U * t) + (-2.0 * ((U * (l_m * l_m)) / Om))))));
	} else {
		tmp = (l_m * (n / Om)) * sqrt((U * (2.0 * U_42_)));
	}
	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 <= 2.45d+113) then
        tmp = sqrt((2.0d0 * (n * ((u * t) + ((-2.0d0) * ((u * (l_m * l_m)) / om))))))
    else
        tmp = (l_m * (n / om)) * sqrt((u * (2.0d0 * u_42)))
    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 <= 2.45e+113) {
		tmp = Math.sqrt((2.0 * (n * ((U * t) + (-2.0 * ((U * (l_m * l_m)) / Om))))));
	} else {
		tmp = (l_m * (n / Om)) * Math.sqrt((U * (2.0 * U_42_)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 2.45e+113:
		tmp = math.sqrt((2.0 * (n * ((U * t) + (-2.0 * ((U * (l_m * l_m)) / Om))))))
	else:
		tmp = (l_m * (n / Om)) * math.sqrt((U * (2.0 * U_42_)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2.45e+113)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(Float64(U * t) + Float64(-2.0 * Float64(Float64(U * Float64(l_m * l_m)) / Om))))));
	else
		tmp = Float64(Float64(l_m * Float64(n / Om)) * sqrt(Float64(U * Float64(2.0 * U_42_))));
	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 <= 2.45e+113)
		tmp = sqrt((2.0 * (n * ((U * t) + (-2.0 * ((U * (l_m * l_m)) / Om))))));
	else
		tmp = (l_m * (n / Om)) * sqrt((U * (2.0 * U_42_)));
	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, 2.45e+113], N[Sqrt[N[(2.0 * N[(n * N[(N[(U * t), $MachinePrecision] + N[(-2.0 * N[(N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[(n / Om), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U * N[(2.0 * U$42$), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.45000000000000011e113

   1. Initial program 54.4%

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

   3. Simplified 59.9%

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

   5. Taylor expanded in Om around inf 48.3%

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

      1. unpow2 48.3%

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

   7. Applied egg-rr 48.3%

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

   if 2.45000000000000011e113 < l

   1. Initial program 17.2%

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

   3. Simplified 32.5%

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

      1. associate-*r* 32.5%

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

      2. pow1 32.5%

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

   6. Applied egg-rr 32.5%

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

      1. unpow1 32.5%

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

   8. Simplified 32.5%

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

   9. Taylor expanded in U around 0 16.4%

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

       1. mul-1-neg 16.4%

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

       2. associate-/l* 16.4%

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

       3. unpow2 16.4%

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

       4. unpow2 16.4%

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

       5. times-frac 32.7%

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

       6. unpow2 32.7%

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

   11. Simplified 32.7%

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

       1. add-cube-cbrt 32.6%

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

       2. pow3 32.6%

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

   13. Applied egg-rr 32.6%

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

   14. Taylor expanded in n around inf 27.4%

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

       1. associate-/l* 29.8%

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

       2. rem-cube-cbrt 30.0%

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

       3. *-commutative 30.0%

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

   16. Simplified 30.0%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.45 \cdot 10^{+113}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t + -2 \cdot \frac{U \cdot \left(\ell \cdot \ell\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{n}{Om}\right) \cdot \sqrt{U \cdot \left(2 \cdot U*\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 38.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.7 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;l\_m \leq 1.26 \cdot 10^{+113}:\\ \;\;\;\;\sqrt{\frac{\left(U \cdot -4\right) \cdot \left(n \cdot \left(l\_m \cdot l\_m\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \frac{n}{Om}\right) \cdot \sqrt{U \cdot \left(2 \cdot U*\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.7e+42)
   (sqrt (* (* 2.0 n) (* U t)))
   (if (<= l_m 1.26e+113)
     (sqrt (/ (* (* U -4.0) (* n (* l_m l_m))) Om))
     (* (* l_m (/ n Om)) (sqrt (* U (* 2.0 U*)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2.7e+42) {
		tmp = sqrt(((2.0 * n) * (U * t)));
	} else if (l_m <= 1.26e+113) {
		tmp = sqrt((((U * -4.0) * (n * (l_m * l_m))) / Om));
	} else {
		tmp = (l_m * (n / Om)) * sqrt((U * (2.0 * U_42_)));
	}
	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 <= 2.7d+42) then
        tmp = sqrt(((2.0d0 * n) * (u * t)))
    else if (l_m <= 1.26d+113) then
        tmp = sqrt((((u * (-4.0d0)) * (n * (l_m * l_m))) / om))
    else
        tmp = (l_m * (n / om)) * sqrt((u * (2.0d0 * u_42)))
    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 <= 2.7e+42) {
		tmp = Math.sqrt(((2.0 * n) * (U * t)));
	} else if (l_m <= 1.26e+113) {
		tmp = Math.sqrt((((U * -4.0) * (n * (l_m * l_m))) / Om));
	} else {
		tmp = (l_m * (n / Om)) * Math.sqrt((U * (2.0 * U_42_)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 2.7e+42:
		tmp = math.sqrt(((2.0 * n) * (U * t)))
	elif l_m <= 1.26e+113:
		tmp = math.sqrt((((U * -4.0) * (n * (l_m * l_m))) / Om))
	else:
		tmp = (l_m * (n / Om)) * math.sqrt((U * (2.0 * U_42_)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2.7e+42)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t)));
	elseif (l_m <= 1.26e+113)
		tmp = sqrt(Float64(Float64(Float64(U * -4.0) * Float64(n * Float64(l_m * l_m))) / Om));
	else
		tmp = Float64(Float64(l_m * Float64(n / Om)) * sqrt(Float64(U * Float64(2.0 * U_42_))));
	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 <= 2.7e+42)
		tmp = sqrt(((2.0 * n) * (U * t)));
	elseif (l_m <= 1.26e+113)
		tmp = sqrt((((U * -4.0) * (n * (l_m * l_m))) / Om));
	else
		tmp = (l_m * (n / Om)) * sqrt((U * (2.0 * U_42_)));
	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, 2.7e+42], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 1.26e+113], N[Sqrt[N[(N[(N[(U * -4.0), $MachinePrecision] * N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[(n / Om), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U * N[(2.0 * U$42$), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.7000000000000001e42

   1. Initial program 52.3%

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

   3. Simplified 58.3%

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

   5. Taylor expanded in t around inf 42.2%

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

   if 2.7000000000000001e42 < l < 1.2599999999999999e113

   1. Initial program 75.3%

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

   3. Simplified 75.7%

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

   5. Taylor expanded in Om around inf 66.3%

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

      1. add-sqr-sqrt 65.9%

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

      2. pow2 65.9%

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

      3. fma-define 65.9%

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

      4. associate-/l* 70.8%

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

   7. Applied egg-rr 70.8%

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

   8. Taylor expanded in l around inf 57.2%

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

      1. associate-*r/ 57.2%

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

      2. associate-*r* 57.2%

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

   10. Simplified 57.2%

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

       1. unpow2 66.3%

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

   12. Applied egg-rr 57.2%

       \[\leadsto \sqrt{\frac{\left(-4 \cdot U\right) \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)}{Om}} \]

   if 1.2599999999999999e113 < l

   1. Initial program 17.2%

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

   3. Simplified 32.5%

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

      1. associate-*r* 32.5%

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

      2. pow1 32.5%

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

   6. Applied egg-rr 32.5%

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

      1. unpow1 32.5%

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

   8. Simplified 32.5%

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

   9. Taylor expanded in U around 0 16.4%

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

       1. mul-1-neg 16.4%

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

       2. associate-/l* 16.4%

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

       3. unpow2 16.4%

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

       4. unpow2 16.4%

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

       5. times-frac 32.7%

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

       6. unpow2 32.7%

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

   11. Simplified 32.7%

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

       1. add-cube-cbrt 32.6%

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

       2. pow3 32.6%

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

   13. Applied egg-rr 32.6%

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

   14. Taylor expanded in n around inf 27.4%

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

       1. associate-/l* 29.8%

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

       2. rem-cube-cbrt 30.0%

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

       3. *-commutative 30.0%

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

   16. Simplified 30.0%

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

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.7 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;\ell \leq 1.26 \cdot 10^{+113}:\\ \;\;\;\;\sqrt{\frac{\left(U \cdot -4\right) \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{n}{Om}\right) \cdot \sqrt{U \cdot \left(2 \cdot U*\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 39.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 5.4 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(U \cdot -4\right) \cdot \left(n \cdot \left(l\_m \cdot l\_m\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 5.4e+38)
   (sqrt (* (* 2.0 n) (* U t)))
   (sqrt (/ (* (* U -4.0) (* n (* l_m l_m))) 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 <= 5.4e+38) {
		tmp = sqrt(((2.0 * n) * (U * t)));
	} else {
		tmp = sqrt((((U * -4.0) * (n * (l_m * l_m))) / 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 <= 5.4d+38) then
        tmp = sqrt(((2.0d0 * n) * (u * t)))
    else
        tmp = sqrt((((u * (-4.0d0)) * (n * (l_m * l_m))) / 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 <= 5.4e+38) {
		tmp = Math.sqrt(((2.0 * n) * (U * t)));
	} else {
		tmp = Math.sqrt((((U * -4.0) * (n * (l_m * l_m))) / Om));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 5.4e+38:
		tmp = math.sqrt(((2.0 * n) * (U * t)))
	else:
		tmp = math.sqrt((((U * -4.0) * (n * (l_m * l_m))) / Om))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 5.4e+38)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t)));
	else
		tmp = sqrt(Float64(Float64(Float64(U * -4.0) * Float64(n * Float64(l_m * l_m))) / 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 <= 5.4e+38)
		tmp = sqrt(((2.0 * n) * (U * t)));
	else
		tmp = sqrt((((U * -4.0) * (n * (l_m * l_m))) / 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, 5.4e+38], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(U * -4.0), $MachinePrecision] * N[(n * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.39999999999999992e38

   1. Initial program 52.3%

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

   3. Simplified 58.3%

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

   5. Taylor expanded in t around inf 42.2%

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

   if 5.39999999999999992e38 < l

   1. Initial program 37.2%

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

   3. Simplified 45.9%

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

   5. Taylor expanded in Om around inf 33.2%

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

      1. add-sqr-sqrt 33.0%

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

      2. pow2 33.0%

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

      3. fma-define 33.0%

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

      4. associate-/l* 34.7%

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

   7. Applied egg-rr 34.7%

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

   8. Taylor expanded in l around inf 28.4%

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

      1. associate-*r/ 28.4%

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

      2. associate-*r* 28.4%

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

   10. Simplified 28.4%

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

       1. unpow2 33.2%

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

   12. Applied egg-rr 28.4%

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

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.4 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(U \cdot -4\right) \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 35.1% accurate, 2.1× speedup

Math

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

FPCore

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.9%

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

3. Simplified 55.5%

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

5. Taylor expanded in t around inf 35.8%

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

Alternative 13: 35.5% accurate, 2.1× speedup

Math

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

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(Float64(2.0 * 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[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 48.9%

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

3. Simplified 55.5%

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

5. Taylor expanded in l around 0 32.6%

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

   1. associate-*r* 32.6%

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

7. Simplified 32.6%

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

Reproduce

herbie shell --seed 2024180 
(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*))))))