Toniolo and Linder, Equation (13)

Percentage Accurate: 50.6% → 65.8%

Time: 27.3s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.6% 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: 65.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_1\right)\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-295}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\mathsf{fma}\left(-2, {l\_m}^{2}, U* \cdot \frac{n \cdot {l\_m}^{2}}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_1 - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + 2 \cdot \frac{-1}{Om}\right)\right)} \cdot \left(l\_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

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 5e-295)
     (sqrt
      (*
       (* 2.0 n)
       (*
        U
        (+
         t
         (/ (fma -2.0 (pow l_m 2.0) (* U* (/ (* n (pow l_m 2.0)) Om))) Om)))))
     (if (<= t_2 INFINITY)
       (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 (/ -1.0 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 <= 5e-295) {
		tmp = sqrt(((2.0 * n) * (U * (t + (fma(-2.0, pow(l_m, 2.0), (U_42_ * ((n * pow(l_m, 2.0)) / Om))) / Om)))));
	} else if (t_2 <= ((double) INFINITY)) {
		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 * (-1.0 / Om)))))) * (l_m * 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 <= 5e-295)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(fma(-2.0, (l_m ^ 2.0), Float64(U_42_ * Float64(Float64(n * (l_m ^ 2.0)) / Om))) / Om)))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(t_1 - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))))));
	else
		tmp = Float64(sqrt(Float64(U * Float64(n * Float64(Float64(Float64(n * Float64(U_42_ - U)) / (Om ^ 2.0)) + Float64(2.0 * Float64(-1.0 / Om)))))) * Float64(l_m * sqrt(2.0)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * 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, 5e-295], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(-2.0 * N[Power[l$95$m, 2.0], $MachinePrecision] + N[(U$42$ * N[(N[(n * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(t$95$1 - N[(2.0 * N[(l$95$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]), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(-1.0 / Om), $MachinePrecision]), $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*)))) < 5.00000000000000008e-295

   1. Initial program 18.0%

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

   3. Simplified 39.5%

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

   5. Taylor expanded in U around 0 34.7%

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

      1. mul-1-neg 34.7%

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

      2. associate-/l* 36.9%

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

      3. unpow2 36.9%

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

      4. unpow2 36.9%

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

      5. times-frac 39.5%

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

      6. unpow2 39.5%

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

   7. Simplified 39.5%

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

   8. Taylor expanded in Om around -inf 43.5%

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

      1. mul-1-neg 43.5%

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

      2. fma-define 43.5%

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

      3. associate-/l* 45.7%

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

   10. Simplified 45.7%

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

   if 5.00000000000000008e-295 < (*.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 64.5%

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

   3. Simplified 70.0%

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

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

   1. Initial program 0.0%

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

   3. Simplified 3.9%

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

   5. Taylor expanded in l around inf 27.8%

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 5 \cdot 10^{-295}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{\mathsf{fma}\left(-2, {\ell}^{2}, U* \cdot \frac{n \cdot {\ell}^{2}}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + 2 \cdot \frac{-1}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_1\right)}\\ \mathbf{if}\;t\_2 \leq 10^{-147}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(-2 \cdot \frac{U \cdot \left(l\_m \cdot l\_m\right)}{Om} + U \cdot t\right)\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_1 - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + 2 \cdot \frac{-1}{Om}\right)\right)} \cdot \left(l\_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

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
         (sqrt (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_1)))))
   (if (<= t_2 1e-147)
     (sqrt (* 2.0 (* n (+ (* -2.0 (/ (* U (* l_m l_m)) Om)) (* U t)))))
     (if (<= t_2 INFINITY)
       (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 (/ -1.0 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 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	double tmp;
	if (t_2 <= 1e-147) {
		tmp = sqrt((2.0 * (n * ((-2.0 * ((U * (l_m * l_m)) / Om)) + (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 = sqrt((U * (n * (((n * (U_42_ - U)) / pow(Om, 2.0)) + (2.0 * (-1.0 / Om)))))) * (l_m * sqrt(2.0));
	}
	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 <= 1e-147) {
		tmp = Math.sqrt((2.0 * (n * ((-2.0 * ((U * (l_m * l_m)) / Om)) + (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.sqrt((U * (n * (((n * (U_42_ - U)) / Math.pow(Om, 2.0)) + (2.0 * (-1.0 / 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 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)))
	tmp = 0
	if t_2 <= 1e-147:
		tmp = math.sqrt((2.0 * (n * ((-2.0 * ((U * (l_m * l_m)) / Om)) + (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.sqrt((U * (n * (((n * (U_42_ - U)) / math.pow(Om, 2.0)) + (2.0 * (-1.0 / 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 = 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 <= 1e-147)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(Float64(-2.0 * Float64(Float64(U * Float64(l_m * l_m)) / Om)) + 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(sqrt(Float64(U * Float64(n * Float64(Float64(Float64(n * Float64(U_42_ - U)) / (Om ^ 2.0)) + Float64(2.0 * Float64(-1.0 / 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 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	tmp = 0.0;
	if (t_2 <= 1e-147)
		tmp = sqrt((2.0 * (n * ((-2.0 * ((U * (l_m * l_m)) / Om)) + (U * t)))));
	elseif (t_2 <= Inf)
		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 * (-1.0 / 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[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, 1e-147], N[Sqrt[N[(2.0 * N[(n * N[(N[(-2.0 * N[(N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(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]), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(-1.0 / Om), $MachinePrecision]), $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 (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 9.9999999999999997e-148

   1. Initial program 21.9%

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

   3. Simplified 44.9%

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

   5. Taylor expanded in Om around inf 47.5%

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

      1. unpow2 47.5%

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

   7. Applied egg-rr 47.5%

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

   if 9.9999999999999997e-148 < (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 64.5%

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

   3. Simplified 70.0%

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

   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 5.8%

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

   5. Taylor expanded in l around inf 27.4%

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 10^{-147}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(-2 \cdot \frac{U \cdot \left(\ell \cdot \ell\right)}{Om} + U \cdot t\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} + 2 \cdot \frac{-1}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.5% accurate, 0.3× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_2
         (sqrt (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_1)))))
   (if (<= t_2 1e-147)
     (sqrt (* 2.0 (* n (+ (* -2.0 (/ (* U (* l_m l_m)) Om)) (* U t)))))
     (if (<= t_2 INFINITY)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_1 (* 2.0 (* l_m (/ l_m Om)))))))
       (*
        (* l_m (sqrt 2.0))
        (sqrt (* U (* n (- (/ (* n U*) (pow Om 2.0)) (/ 2.0 Om))))))))))

C

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

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	double tmp;
	if (t_2 <= 1e-147) {
		tmp = Math.sqrt((2.0 * (n * ((-2.0 * ((U * (l_m * l_m)) / Om)) + (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 = (l_m * Math.sqrt(2.0)) * Math.sqrt((U * (n * (((n * U_42_) / Math.pow(Om, 2.0)) - (2.0 / Om)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_2 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)))
	tmp = 0
	if t_2 <= 1e-147:
		tmp = math.sqrt((2.0 * (n * ((-2.0 * ((U * (l_m * l_m)) / Om)) + (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 = (l_m * math.sqrt(2.0)) * math.sqrt((U * (n * (((n * U_42_) / math.pow(Om, 2.0)) - (2.0 / Om)))))
	return tmp

Julia

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

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	tmp = 0.0;
	if (t_2 <= 1e-147)
		tmp = sqrt((2.0 * (n * ((-2.0 * ((U * (l_m * l_m)) / Om)) + (U * t)))));
	elseif (t_2 <= Inf)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	else
		tmp = (l_m * sqrt(2.0)) * sqrt((U * (n * (((n * U_42_) / (Om ^ 2.0)) - (2.0 / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 9.9999999999999997e-148

   1. Initial program 21.9%

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

   3. Simplified 44.9%

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

   5. Taylor expanded in Om around inf 47.5%

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

      1. unpow2 47.5%

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

   7. Applied egg-rr 47.5%

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

   if 9.9999999999999997e-148 < (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 64.5%

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

   3. Simplified 70.0%

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

   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 5.7%

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

   5. Taylor expanded in U around 0 0.9%

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

      1. mul-1-neg 0.9%

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

      2. associate-/l* 0.9%

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

      3. unpow2 0.9%

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

      4. unpow2 0.9%

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

      5. times-frac 5.9%

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

      6. unpow2 5.9%

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

   7. Simplified 5.9%

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

      1. pow1/2 6.1%

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

      2. *-commutative 6.1%

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

      3. unpow-prod-down 5.4%

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

   9. Applied egg-rr 0.4%

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

   10. Taylor expanded in l around inf 27.4%

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

       1. associate-*r/ 27.4%

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

       2. metadata-eval 27.4%

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

   12. Simplified 27.4%

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

4. Final simplification 59.2%

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

Alternative 4: 61.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_1\right)\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-295}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(-2 \cdot \frac{U \cdot \left(l\_m \cdot l\_m\right)}{Om} + U \cdot t\right)\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_1 - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot {\left(\left(n \cdot U\right) \cdot \left(t - \frac{2 \cdot {l\_m}^{2}}{Om}\right)\right)}^{0.5}\\ \end{array} \end{array} \]

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 (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_1))))
   (if (<= t_2 5e-295)
     (sqrt (* 2.0 (* n (+ (* -2.0 (/ (* U (* l_m l_m)) Om)) (* U t)))))
     (if (<= t_2 INFINITY)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_1 (* 2.0 (* l_m (/ l_m Om)))))))
       (*
        (sqrt 2.0)
        (pow (* (* 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 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	double tmp;
	if (t_2 <= 5e-295) {
		tmp = sqrt((2.0 * (n * ((-2.0 * ((U * (l_m * l_m)) / Om)) + (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 = sqrt(2.0) * pow(((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 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	double tmp;
	if (t_2 <= 5e-295) {
		tmp = Math.sqrt((2.0 * (n * ((-2.0 * ((U * (l_m * l_m)) / Om)) + (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.sqrt(2.0) * Math.pow(((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 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)
	tmp = 0
	if t_2 <= 5e-295:
		tmp = math.sqrt((2.0 * (n * ((-2.0 * ((U * (l_m * l_m)) / Om)) + (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.sqrt(2.0) * math.pow(((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 = 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 <= 5e-295)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(Float64(-2.0 * Float64(Float64(U * Float64(l_m * l_m)) / Om)) + 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(sqrt(2.0) * (Float64(Float64(n * U) * Float64(t - Float64(Float64(2.0 * (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 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	tmp = 0.0;
	if (t_2 <= 5e-295)
		tmp = sqrt((2.0 * (n * ((-2.0 * ((U * (l_m * l_m)) / Om)) + (U * t)))));
	elseif (t_2 <= Inf)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l_m * (l_m / Om)))))));
	else
		tmp = sqrt(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[(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, 5e-295], N[Sqrt[N[(2.0 * N[(n * N[(N[(-2.0 * N[(N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(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[2.0], $MachinePrecision] * N[Power[N[(N[(n * U), $MachinePrecision] * N[(t - N[(N[(2.0 * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $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*)))) < 5.00000000000000008e-295

   1. Initial program 18.0%

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

   3. Simplified 39.5%

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

   5. Taylor expanded in Om around inf 43.5%

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

      1. unpow2 43.5%

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

   7. Applied egg-rr 43.5%

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

   if 5.00000000000000008e-295 < (*.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 64.5%

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

   3. Simplified 70.0%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.6%

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

      1. associate-*r* 0.6%

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

      2. fma-define 12.0%

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

      3. associate-*r* 12.0%

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

   6. Applied egg-rr 12.0%

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

      1. pow1/2 12.5%

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

      2. associate-*l* 12.5%

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

      3. unpow-prod-down 12.5%

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

      4. pow1/2 12.5%

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

   8. Applied egg-rr 12.5%

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

   9. Taylor expanded in Om around inf 29.4%

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

       1. metadata-eval 29.4%

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

       2. cancel-sign-sub-inv 29.4%

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

       3. associate-*r/ 29.4%

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

   11. Simplified 29.4%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 5 \cdot 10^{-295}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(-2 \cdot \frac{U \cdot \left(\ell \cdot \ell\right)}{Om} + U \cdot t\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot {\left(\left(n \cdot U\right) \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\\ t_2 := t\_1 \cdot \left(U* - U\right)\\ t_3 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_2\right)\\ \mathbf{if}\;t\_3 \leq 5 \cdot 10^{-295}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(-2 \cdot \frac{U \cdot \left(l\_m \cdot l\_m\right)}{Om} + U \cdot t\right)\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_2 - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(t\_1 \cdot \left(U \cdot U*\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* n (pow (/ l_m Om) 2.0)))
        (t_2 (* t_1 (- U* U)))
        (t_3 (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_2))))
   (if (<= t_3 5e-295)
     (sqrt (* 2.0 (* n (+ (* -2.0 (/ (* U (* l_m l_m)) Om)) (* U t)))))
     (if (<= t_3 INFINITY)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_2 (* 2.0 (* l_m (/ l_m Om)))))))
       (sqrt (* (* 2.0 n) (* t_1 (* U U*))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = n * pow((l_m / Om), 2.0);
	double t_2 = t_1 * (U_42_ - U);
	double t_3 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2);
	double tmp;
	if (t_3 <= 5e-295) {
		tmp = sqrt((2.0 * (n * ((-2.0 * ((U * (l_m * l_m)) / Om)) + (U * t)))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_2 - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = sqrt(((2.0 * n) * (t_1 * (U * U_42_))));
	}
	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);
	double t_2 = t_1 * (U_42_ - U);
	double t_3 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2);
	double tmp;
	if (t_3 <= 5e-295) {
		tmp = Math.sqrt((2.0 * (n * ((-2.0 * ((U * (l_m * l_m)) / Om)) + (U * t)))));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + (t_2 - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (t_1 * (U * U_42_))));
	}
	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)
	t_2 = t_1 * (U_42_ - U)
	t_3 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2)
	tmp = 0
	if t_3 <= 5e-295:
		tmp = math.sqrt((2.0 * (n * ((-2.0 * ((U * (l_m * l_m)) / Om)) + (U * t)))))
	elif t_3 <= math.inf:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (t_2 - (2.0 * (l_m * (l_m / Om)))))))
	else:
		tmp = math.sqrt(((2.0 * n) * (t_1 * (U * U_42_))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(n * (Float64(l_m / Om) ^ 2.0))
	t_2 = Float64(t_1 * Float64(U_42_ - U))
	t_3 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_2))
	tmp = 0.0
	if (t_3 <= 5e-295)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(Float64(-2.0 * Float64(Float64(U * Float64(l_m * l_m)) / Om)) + Float64(U * t)))));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(t_2 - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(t_1 * 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_)
	t_1 = n * ((l_m / Om) ^ 2.0);
	t_2 = t_1 * (U_42_ - U);
	t_3 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_2);
	tmp = 0.0;
	if (t_3 <= 5e-295)
		tmp = sqrt((2.0 * (n * ((-2.0 * ((U * (l_m * l_m)) / Om)) + (U * t)))));
	elseif (t_3 <= Inf)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_2 - (2.0 * (l_m * (l_m / Om)))))));
	else
		tmp = sqrt(((2.0 * n) * (t_1 * (U * U_42_))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 5e-295], N[Sqrt[N[(2.0 * N[(n * N[(N[(-2.0 * N[(N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(t$95$2 - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(t$95$1 * N[(U * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.00000000000000008e-295

   1. Initial program 18.0%

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

   3. Simplified 39.5%

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

   5. Taylor expanded in Om around inf 43.5%

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

      1. unpow2 43.5%

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

   7. Applied egg-rr 43.5%

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

   if 5.00000000000000008e-295 < (*.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 64.5%

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

   3. Simplified 70.0%

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

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

   1. Initial program 0.0%

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

   3. Simplified 3.8%

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

      1. *-commutative 3.8%

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

      2. sqrt-prod 5.9%

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

   6. Applied egg-rr 0.5%

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

   7. Taylor expanded in U* around inf 19.6%

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

      1. associate-*r* 19.5%

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

   9. Simplified 19.5%

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

       1. pow1 19.5%

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

       2. sqrt-unprod 28.6%

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

       3. associate-/l* 28.7%

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

       4. *-commutative 28.7%

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

   11. Applied egg-rr 28.7%

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

       1. unpow1 28.7%

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

       2. *-commutative 28.7%

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

       3. *-commutative 28.7%

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

       4. associate-/l* 28.6%

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

       5. unpow2 28.6%

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

       6. unpow2 28.6%

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

       7. times-frac 29.1%

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

       8. unpow2 29.1%

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

   13. Simplified 29.1%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 5 \cdot 10^{-295}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(-2 \cdot \frac{U \cdot \left(\ell \cdot \ell\right)}{Om} + U \cdot t\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U \cdot U*\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 44.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{U \cdot U*}\\ \mathbf{if}\;Om \leq -3.9 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{l\_m}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq -3.2 \cdot 10^{-217}:\\ \;\;\;\;t\_1 \cdot \left(l\_m \cdot \left(n \cdot \frac{\sqrt{2}}{-Om}\right)\right)\\ \mathbf{elif}\;Om \leq 1.2 \cdot 10^{-247}:\\ \;\;\;\;\frac{t\_1 \cdot \left(l\_m \cdot \left(n \cdot \sqrt{2}\right)\right)}{Om}\\ \mathbf{elif}\;Om \leq 1.05 \cdot 10^{-55}:\\ \;\;\;\;\frac{l\_m}{Om} \cdot \sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(n \cdot U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \frac{-2 \cdot {l\_m}^{2}}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* U U*))))
   (if (<= Om -3.9e-112)
     (sqrt (* (* 2.0 n) (* U (- t (* 2.0 (/ (pow l_m 2.0) Om))))))
     (if (<= Om -3.2e-217)
       (* t_1 (* l_m (* n (/ (sqrt 2.0) (- Om)))))
       (if (<= Om 1.2e-247)
         (/ (* t_1 (* l_m (* n (sqrt 2.0)))) Om)
         (if (<= Om 1.05e-55)
           (* (/ l_m Om) (sqrt (* U (* (* 2.0 n) (* n U*)))))
           (sqrt (* (* 2.0 U) (* n (+ t (/ (* -2.0 (pow l_m 2.0)) Om)))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt((U * U_42_));
	double tmp;
	if (Om <= -3.9e-112) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (pow(l_m, 2.0) / Om))))));
	} else if (Om <= -3.2e-217) {
		tmp = t_1 * (l_m * (n * (sqrt(2.0) / -Om)));
	} else if (Om <= 1.2e-247) {
		tmp = (t_1 * (l_m * (n * sqrt(2.0)))) / Om;
	} else if (Om <= 1.05e-55) {
		tmp = (l_m / Om) * sqrt((U * ((2.0 * n) * (n * U_42_))));
	} else {
		tmp = sqrt(((2.0 * U) * (n * (t + ((-2.0 * pow(l_m, 2.0)) / Om)))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((u * u_42))
    if (om <= (-3.9d-112)) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * ((l_m ** 2.0d0) / om))))))
    else if (om <= (-3.2d-217)) then
        tmp = t_1 * (l_m * (n * (sqrt(2.0d0) / -om)))
    else if (om <= 1.2d-247) then
        tmp = (t_1 * (l_m * (n * sqrt(2.0d0)))) / om
    else if (om <= 1.05d-55) then
        tmp = (l_m / om) * sqrt((u * ((2.0d0 * n) * (n * u_42))))
    else
        tmp = sqrt(((2.0d0 * u) * (n * (t + (((-2.0d0) * (l_m ** 2.0d0)) / om)))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = Math.sqrt((U * U_42_));
	double tmp;
	if (Om <= -3.9e-112) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * (Math.pow(l_m, 2.0) / Om))))));
	} else if (Om <= -3.2e-217) {
		tmp = t_1 * (l_m * (n * (Math.sqrt(2.0) / -Om)));
	} else if (Om <= 1.2e-247) {
		tmp = (t_1 * (l_m * (n * Math.sqrt(2.0)))) / Om;
	} else if (Om <= 1.05e-55) {
		tmp = (l_m / Om) * Math.sqrt((U * ((2.0 * n) * (n * U_42_))));
	} else {
		tmp = Math.sqrt(((2.0 * U) * (n * (t + ((-2.0 * Math.pow(l_m, 2.0)) / Om)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = math.sqrt((U * U_42_))
	tmp = 0
	if Om <= -3.9e-112:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * (math.pow(l_m, 2.0) / Om))))))
	elif Om <= -3.2e-217:
		tmp = t_1 * (l_m * (n * (math.sqrt(2.0) / -Om)))
	elif Om <= 1.2e-247:
		tmp = (t_1 * (l_m * (n * math.sqrt(2.0)))) / Om
	elif Om <= 1.05e-55:
		tmp = (l_m / Om) * math.sqrt((U * ((2.0 * n) * (n * U_42_))))
	else:
		tmp = math.sqrt(((2.0 * U) * (n * (t + ((-2.0 * math.pow(l_m, 2.0)) / Om)))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(U * U_42_))
	tmp = 0.0
	if (Om <= -3.9e-112)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64((l_m ^ 2.0) / Om))))));
	elseif (Om <= -3.2e-217)
		tmp = Float64(t_1 * Float64(l_m * Float64(n * Float64(sqrt(2.0) / Float64(-Om)))));
	elseif (Om <= 1.2e-247)
		tmp = Float64(Float64(t_1 * Float64(l_m * Float64(n * sqrt(2.0)))) / Om);
	elseif (Om <= 1.05e-55)
		tmp = Float64(Float64(l_m / Om) * sqrt(Float64(U * Float64(Float64(2.0 * n) * Float64(n * U_42_)))));
	else
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * Float64(t + Float64(Float64(-2.0 * (l_m ^ 2.0)) / Om)))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt((U * U_42_));
	tmp = 0.0;
	if (Om <= -3.9e-112)
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * ((l_m ^ 2.0) / Om))))));
	elseif (Om <= -3.2e-217)
		tmp = t_1 * (l_m * (n * (sqrt(2.0) / -Om)));
	elseif (Om <= 1.2e-247)
		tmp = (t_1 * (l_m * (n * sqrt(2.0)))) / Om;
	elseif (Om <= 1.05e-55)
		tmp = (l_m / Om) * sqrt((U * ((2.0 * n) * (n * U_42_))));
	else
		tmp = sqrt(((2.0 * U) * (n * (t + ((-2.0 * (l_m ^ 2.0)) / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -3.9e-112], N[Sqrt[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]], $MachinePrecision], If[LessEqual[Om, -3.2e-217], N[(t$95$1 * N[(l$95$m * N[(n * N[(N[Sqrt[2.0], $MachinePrecision] / (-Om)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Om, 1.2e-247], N[(N[(t$95$1 * N[(l$95$m * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision], If[LessEqual[Om, 1.05e-55], N[(N[(l$95$m / Om), $MachinePrecision] * N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * N[(n * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(t + N[(N[(-2.0 * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if Om < -3.9000000000000001e-112

   1. Initial program 53.7%

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

   3. Simplified 59.5%

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

   5. Taylor expanded in Om around inf 53.3%

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

   if -3.9000000000000001e-112 < Om < -3.2000000000000001e-217

   1. Initial program 38.6%

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

   3. Simplified 38.6%

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

      1. *-commutative 38.6%

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

      2. sqrt-prod 47.1%

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

   6. Applied egg-rr 47.1%

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

   7. Taylor expanded in U* around inf 37.5%

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

      1. associate-*r* 37.5%

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

   9. Simplified 37.5%

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

   10. Taylor expanded in l around -inf 38.2%

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

       1. mul-1-neg 38.2%

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

       2. associate-*r/ 46.6%

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

       3. *-commutative 46.6%

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

       4. distribute-rgt-neg-in 46.6%

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

       5. associate-/l* 46.7%

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

   12. Simplified 46.7%

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

   if -3.2000000000000001e-217 < Om < 1.20000000000000005e-247

   1. Initial program 13.7%

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

   3. Simplified 13.7%

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

      1. associate-*r* 13.7%

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

      2. fma-define 19.5%

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

      3. associate-*r* 25.2%

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

   6. Applied egg-rr 25.2%

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

      1. pow1/2 25.3%

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

      2. associate-*l* 25.3%

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

      3. unpow-prod-down 25.3%

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

      4. pow1/2 25.3%

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

   8. Applied egg-rr 25.3%

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

   9. Taylor expanded in U* around inf 13.5%

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

       1. associate-*l/ 13.6%

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

   11. Simplified 13.6%

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

   if 1.20000000000000005e-247 < Om < 1.0500000000000001e-55

   1. Initial program 37.0%

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

   3. Simplified 38.3%

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

      1. *-commutative 38.3%

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

      2. sqrt-prod 12.1%

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

   6. Applied egg-rr 12.1%

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

   7. Taylor expanded in U* around inf 11.6%

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

      1. pow1 11.6%

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

      2. associate-*l* 11.5%

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

      3. pow1/2 11.6%

         \[\leadsto {\left(\frac{\ell}{Om} \cdot \left(\color{blue}{{\left(U \cdot \left(U* \cdot n\right)\right)}^{0.5}} \cdot \sqrt{2 \cdot n}\right)\right)}^{1} \]

      4. pow1/2 11.6%

         \[\leadsto {\left(\frac{\ell}{Om} \cdot \left({\left(U \cdot \left(U* \cdot n\right)\right)}^{0.5} \cdot \color{blue}{{\left(2 \cdot n\right)}^{0.5}}\right)\right)}^{1} \]

      5. pow-prod-down 22.4%

         \[\leadsto {\left(\frac{\ell}{Om} \cdot \color{blue}{{\left(\left(U \cdot \left(U* \cdot n\right)\right) \cdot \left(2 \cdot n\right)\right)}^{0.5}}\right)}^{1} \]

      6. *-commutative 22.4%

         \[\leadsto {\left(\frac{\ell}{Om} \cdot {\left(\left(U \cdot \color{blue}{\left(n \cdot U*\right)}\right) \cdot \left(2 \cdot n\right)\right)}^{0.5}\right)}^{1} \]

   9. Applied egg-rr 22.4%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{Om} \cdot {\left(\left(U \cdot \left(n \cdot U*\right)\right) \cdot \left(2 \cdot n\right)\right)}^{0.5}\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 22.4%

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

       2. unpow1/2 22.2%

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

       3. associate-*l* 22.3%

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

   11. Simplified 22.3%

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

   if 1.0500000000000001e-55 < Om

   1. Initial program 52.0%

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

   3. Simplified 56.2%

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

   5. Taylor expanded in n around 0 49.5%

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

      1. associate-*r* 49.5%

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

      2. metadata-eval 49.5%

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

      3. cancel-sign-sub-inv 49.5%

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

      4. cancel-sign-sub-inv 49.5%

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

      5. metadata-eval 49.5%

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

      6. associate-*r/ 49.5%

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

   7. Simplified 49.5%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \frac{-2 \cdot {\ell}^{2}}{Om}\right)\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -3.9 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq -3.2 \cdot 10^{-217}:\\ \;\;\;\;\sqrt{U \cdot U*} \cdot \left(\ell \cdot \left(n \cdot \frac{\sqrt{2}}{-Om}\right)\right)\\ \mathbf{elif}\;Om \leq 1.2 \cdot 10^{-247}:\\ \;\;\;\;\frac{\sqrt{U \cdot U*} \cdot \left(\ell \cdot \left(n \cdot \sqrt{2}\right)\right)}{Om}\\ \mathbf{elif}\;Om \leq 1.05 \cdot 10^{-55}:\\ \;\;\;\;\frac{\ell}{Om} \cdot \sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(n \cdot U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \frac{-2 \cdot {\ell}^{2}}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{U \cdot U*}\\ \mathbf{if}\;Om \leq -3.2 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(-2 \cdot \frac{U \cdot \left(l\_m \cdot l\_m\right)}{Om} + U \cdot t\right)\right)}\\ \mathbf{elif}\;Om \leq -3.1 \cdot 10^{-217}:\\ \;\;\;\;t\_1 \cdot \left(l\_m \cdot \left(n \cdot \frac{\sqrt{2}}{-Om}\right)\right)\\ \mathbf{elif}\;Om \leq 1.22 \cdot 10^{-247}:\\ \;\;\;\;\frac{t\_1 \cdot \left(l\_m \cdot \left(n \cdot \sqrt{2}\right)\right)}{Om}\\ \mathbf{elif}\;Om \leq 5.4 \cdot 10^{-49}:\\ \;\;\;\;\frac{l\_m}{Om} \cdot \sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(n \cdot U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \frac{-2 \cdot {l\_m}^{2}}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* U U*))))
   (if (<= Om -3.2e-114)
     (sqrt (* 2.0 (* n (+ (* -2.0 (/ (* U (* l_m l_m)) Om)) (* U t)))))
     (if (<= Om -3.1e-217)
       (* t_1 (* l_m (* n (/ (sqrt 2.0) (- Om)))))
       (if (<= Om 1.22e-247)
         (/ (* t_1 (* l_m (* n (sqrt 2.0)))) Om)
         (if (<= Om 5.4e-49)
           (* (/ l_m Om) (sqrt (* U (* (* 2.0 n) (* n U*)))))
           (sqrt (* (* 2.0 U) (* n (+ t (/ (* -2.0 (pow l_m 2.0)) Om)))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt((U * U_42_));
	double tmp;
	if (Om <= -3.2e-114) {
		tmp = sqrt((2.0 * (n * ((-2.0 * ((U * (l_m * l_m)) / Om)) + (U * t)))));
	} else if (Om <= -3.1e-217) {
		tmp = t_1 * (l_m * (n * (sqrt(2.0) / -Om)));
	} else if (Om <= 1.22e-247) {
		tmp = (t_1 * (l_m * (n * sqrt(2.0)))) / Om;
	} else if (Om <= 5.4e-49) {
		tmp = (l_m / Om) * sqrt((U * ((2.0 * n) * (n * U_42_))));
	} else {
		tmp = sqrt(((2.0 * U) * (n * (t + ((-2.0 * pow(l_m, 2.0)) / Om)))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((u * u_42))
    if (om <= (-3.2d-114)) then
        tmp = sqrt((2.0d0 * (n * (((-2.0d0) * ((u * (l_m * l_m)) / om)) + (u * t)))))
    else if (om <= (-3.1d-217)) then
        tmp = t_1 * (l_m * (n * (sqrt(2.0d0) / -om)))
    else if (om <= 1.22d-247) then
        tmp = (t_1 * (l_m * (n * sqrt(2.0d0)))) / om
    else if (om <= 5.4d-49) then
        tmp = (l_m / om) * sqrt((u * ((2.0d0 * n) * (n * u_42))))
    else
        tmp = sqrt(((2.0d0 * u) * (n * (t + (((-2.0d0) * (l_m ** 2.0d0)) / om)))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = Math.sqrt((U * U_42_));
	double tmp;
	if (Om <= -3.2e-114) {
		tmp = Math.sqrt((2.0 * (n * ((-2.0 * ((U * (l_m * l_m)) / Om)) + (U * t)))));
	} else if (Om <= -3.1e-217) {
		tmp = t_1 * (l_m * (n * (Math.sqrt(2.0) / -Om)));
	} else if (Om <= 1.22e-247) {
		tmp = (t_1 * (l_m * (n * Math.sqrt(2.0)))) / Om;
	} else if (Om <= 5.4e-49) {
		tmp = (l_m / Om) * Math.sqrt((U * ((2.0 * n) * (n * U_42_))));
	} else {
		tmp = Math.sqrt(((2.0 * U) * (n * (t + ((-2.0 * Math.pow(l_m, 2.0)) / Om)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = math.sqrt((U * U_42_))
	tmp = 0
	if Om <= -3.2e-114:
		tmp = math.sqrt((2.0 * (n * ((-2.0 * ((U * (l_m * l_m)) / Om)) + (U * t)))))
	elif Om <= -3.1e-217:
		tmp = t_1 * (l_m * (n * (math.sqrt(2.0) / -Om)))
	elif Om <= 1.22e-247:
		tmp = (t_1 * (l_m * (n * math.sqrt(2.0)))) / Om
	elif Om <= 5.4e-49:
		tmp = (l_m / Om) * math.sqrt((U * ((2.0 * n) * (n * U_42_))))
	else:
		tmp = math.sqrt(((2.0 * U) * (n * (t + ((-2.0 * math.pow(l_m, 2.0)) / Om)))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(U * U_42_))
	tmp = 0.0
	if (Om <= -3.2e-114)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(Float64(-2.0 * Float64(Float64(U * Float64(l_m * l_m)) / Om)) + Float64(U * t)))));
	elseif (Om <= -3.1e-217)
		tmp = Float64(t_1 * Float64(l_m * Float64(n * Float64(sqrt(2.0) / Float64(-Om)))));
	elseif (Om <= 1.22e-247)
		tmp = Float64(Float64(t_1 * Float64(l_m * Float64(n * sqrt(2.0)))) / Om);
	elseif (Om <= 5.4e-49)
		tmp = Float64(Float64(l_m / Om) * sqrt(Float64(U * Float64(Float64(2.0 * n) * Float64(n * U_42_)))));
	else
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * Float64(t + Float64(Float64(-2.0 * (l_m ^ 2.0)) / Om)))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt((U * U_42_));
	tmp = 0.0;
	if (Om <= -3.2e-114)
		tmp = sqrt((2.0 * (n * ((-2.0 * ((U * (l_m * l_m)) / Om)) + (U * t)))));
	elseif (Om <= -3.1e-217)
		tmp = t_1 * (l_m * (n * (sqrt(2.0) / -Om)));
	elseif (Om <= 1.22e-247)
		tmp = (t_1 * (l_m * (n * sqrt(2.0)))) / Om;
	elseif (Om <= 5.4e-49)
		tmp = (l_m / Om) * sqrt((U * ((2.0 * n) * (n * U_42_))));
	else
		tmp = sqrt(((2.0 * U) * (n * (t + ((-2.0 * (l_m ^ 2.0)) / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -3.2e-114], N[Sqrt[N[(2.0 * N[(n * N[(N[(-2.0 * N[(N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, -3.1e-217], N[(t$95$1 * N[(l$95$m * N[(n * N[(N[Sqrt[2.0], $MachinePrecision] / (-Om)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Om, 1.22e-247], N[(N[(t$95$1 * N[(l$95$m * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision], If[LessEqual[Om, 5.4e-49], N[(N[(l$95$m / Om), $MachinePrecision] * N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * N[(n * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(t + N[(N[(-2.0 * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if Om < -3.2000000000000002e-114

   1. Initial program 53.7%

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

   3. Simplified 59.6%

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

   5. Taylor expanded in Om around inf 52.4%

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

      1. unpow2 52.4%

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

   7. Applied egg-rr 52.4%

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

   if -3.2000000000000002e-114 < Om < -3.0999999999999999e-217

   1. Initial program 38.6%

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

   3. Simplified 38.6%

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

      1. *-commutative 38.6%

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

      2. sqrt-prod 47.1%

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

   6. Applied egg-rr 47.1%

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

   7. Taylor expanded in U* around inf 37.5%

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

      1. associate-*r* 37.5%

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

   9. Simplified 37.5%

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

   10. Taylor expanded in l around -inf 38.2%

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

       1. mul-1-neg 38.2%

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

       2. associate-*r/ 46.6%

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

       3. *-commutative 46.6%

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

       4. distribute-rgt-neg-in 46.6%

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

       5. associate-/l* 46.7%

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

   12. Simplified 46.7%

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

   if -3.0999999999999999e-217 < Om < 1.22e-247

   1. Initial program 13.7%

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

   3. Simplified 13.7%

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

      1. associate-*r* 13.7%

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

      2. fma-define 19.5%

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

      3. associate-*r* 25.2%

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

   6. Applied egg-rr 25.2%

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

      1. pow1/2 25.3%

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

      2. associate-*l* 25.3%

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

      3. unpow-prod-down 25.3%

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

      4. pow1/2 25.3%

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

   8. Applied egg-rr 25.3%

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

   9. Taylor expanded in U* around inf 13.5%

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

       1. associate-*l/ 13.6%

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

   11. Simplified 13.6%

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

   if 1.22e-247 < Om < 5.3999999999999999e-49

   1. Initial program 37.0%

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

   3. Simplified 38.3%

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

      1. *-commutative 38.3%

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

      2. sqrt-prod 12.1%

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

   6. Applied egg-rr 12.1%

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

   7. Taylor expanded in U* around inf 11.6%

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

      1. pow1 11.6%

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

      2. associate-*l* 11.5%

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

      3. pow1/2 11.6%

         \[\leadsto {\left(\frac{\ell}{Om} \cdot \left(\color{blue}{{\left(U \cdot \left(U* \cdot n\right)\right)}^{0.5}} \cdot \sqrt{2 \cdot n}\right)\right)}^{1} \]

      4. pow1/2 11.6%

         \[\leadsto {\left(\frac{\ell}{Om} \cdot \left({\left(U \cdot \left(U* \cdot n\right)\right)}^{0.5} \cdot \color{blue}{{\left(2 \cdot n\right)}^{0.5}}\right)\right)}^{1} \]

      5. pow-prod-down 22.4%

         \[\leadsto {\left(\frac{\ell}{Om} \cdot \color{blue}{{\left(\left(U \cdot \left(U* \cdot n\right)\right) \cdot \left(2 \cdot n\right)\right)}^{0.5}}\right)}^{1} \]

      6. *-commutative 22.4%

         \[\leadsto {\left(\frac{\ell}{Om} \cdot {\left(\left(U \cdot \color{blue}{\left(n \cdot U*\right)}\right) \cdot \left(2 \cdot n\right)\right)}^{0.5}\right)}^{1} \]

   9. Applied egg-rr 22.4%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{Om} \cdot {\left(\left(U \cdot \left(n \cdot U*\right)\right) \cdot \left(2 \cdot n\right)\right)}^{0.5}\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 22.4%

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

       2. unpow1/2 22.2%

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

       3. associate-*l* 22.3%

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

   11. Simplified 22.3%

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

   if 5.3999999999999999e-49 < Om

   1. Initial program 52.0%

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

   3. Simplified 56.2%

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

   5. Taylor expanded in n around 0 49.5%

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

      1. associate-*r* 49.5%

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

      2. metadata-eval 49.5%

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

      3. cancel-sign-sub-inv 49.5%

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

      4. cancel-sign-sub-inv 49.5%

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

      5. metadata-eval 49.5%

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

      6. associate-*r/ 49.5%

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

   7. Simplified 49.5%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \frac{-2 \cdot {\ell}^{2}}{Om}\right)\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -3.2 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(-2 \cdot \frac{U \cdot \left(\ell \cdot \ell\right)}{Om} + U \cdot t\right)\right)}\\ \mathbf{elif}\;Om \leq -3.1 \cdot 10^{-217}:\\ \;\;\;\;\sqrt{U \cdot U*} \cdot \left(\ell \cdot \left(n \cdot \frac{\sqrt{2}}{-Om}\right)\right)\\ \mathbf{elif}\;Om \leq 1.22 \cdot 10^{-247}:\\ \;\;\;\;\frac{\sqrt{U \cdot U*} \cdot \left(\ell \cdot \left(n \cdot \sqrt{2}\right)\right)}{Om}\\ \mathbf{elif}\;Om \leq 5.4 \cdot 10^{-49}:\\ \;\;\;\;\frac{\ell}{Om} \cdot \sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(n \cdot U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + \frac{-2 \cdot {\ell}^{2}}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := 2 \cdot \left(n \cdot U\right)\\ t_2 := {\left(\frac{l\_m}{Om}\right)}^{2}\\ \mathbf{if}\;U* \leq -9 \cdot 10^{+209} \lor \neg \left(U* \leq 5.4 \cdot 10^{+163}\right):\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t + \left(n \cdot t\_2\right) \cdot U*\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t + \left(n \cdot \left(t\_2 \cdot U*\right) - \frac{l\_m}{Om} \cdot \left(2 \cdot l\_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* n U))) (t_2 (pow (/ l_m Om) 2.0)))
   (if (or (<= U* -9e+209) (not (<= U* 5.4e+163)))
     (sqrt (* t_1 (+ t (* (* n t_2) U*))))
     (sqrt (* t_1 (+ t (- (* n (* t_2 U*)) (* (/ l_m Om) (* 2.0 l_m)))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double t_2 = pow((l_m / Om), 2.0);
	double tmp;
	if ((U_42_ <= -9e+209) || !(U_42_ <= 5.4e+163)) {
		tmp = sqrt((t_1 * (t + ((n * t_2) * U_42_))));
	} else {
		tmp = sqrt((t_1 * (t + ((n * (t_2 * U_42_)) - ((l_m / Om) * (2.0 * l_m))))));
	}
	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 * (n * u)
    t_2 = (l_m / om) ** 2.0d0
    if ((u_42 <= (-9d+209)) .or. (.not. (u_42 <= 5.4d+163))) then
        tmp = sqrt((t_1 * (t + ((n * t_2) * u_42))))
    else
        tmp = sqrt((t_1 * (t + ((n * (t_2 * u_42)) - ((l_m / om) * (2.0d0 * l_m))))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double t_2 = Math.pow((l_m / Om), 2.0);
	double tmp;
	if ((U_42_ <= -9e+209) || !(U_42_ <= 5.4e+163)) {
		tmp = Math.sqrt((t_1 * (t + ((n * t_2) * U_42_))));
	} else {
		tmp = Math.sqrt((t_1 * (t + ((n * (t_2 * U_42_)) - ((l_m / Om) * (2.0 * l_m))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = 2.0 * (n * U)
	t_2 = math.pow((l_m / Om), 2.0)
	tmp = 0
	if (U_42_ <= -9e+209) or not (U_42_ <= 5.4e+163):
		tmp = math.sqrt((t_1 * (t + ((n * t_2) * U_42_))))
	else:
		tmp = math.sqrt((t_1 * (t + ((n * (t_2 * U_42_)) - ((l_m / Om) * (2.0 * l_m))))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(2.0 * Float64(n * U))
	t_2 = Float64(l_m / Om) ^ 2.0
	tmp = 0.0
	if ((U_42_ <= -9e+209) || !(U_42_ <= 5.4e+163))
		tmp = sqrt(Float64(t_1 * Float64(t + Float64(Float64(n * t_2) * U_42_))));
	else
		tmp = sqrt(Float64(t_1 * Float64(t + Float64(Float64(n * Float64(t_2 * U_42_)) - Float64(Float64(l_m / Om) * Float64(2.0 * l_m))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = 2.0 * (n * U);
	t_2 = (l_m / Om) ^ 2.0;
	tmp = 0.0;
	if ((U_42_ <= -9e+209) || ~((U_42_ <= 5.4e+163)))
		tmp = sqrt((t_1 * (t + ((n * t_2) * U_42_))));
	else
		tmp = sqrt((t_1 * (t + ((n * (t_2 * U_42_)) - ((l_m / Om) * (2.0 * l_m))))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]}, If[Or[LessEqual[U$42$, -9e+209], N[Not[LessEqual[U$42$, 5.4e+163]], $MachinePrecision]], N[Sqrt[N[(t$95$1 * N[(t + N[(N[(n * t$95$2), $MachinePrecision] * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t$95$1 * N[(t + N[(N[(n * N[(t$95$2 * U$42$), $MachinePrecision]), $MachinePrecision] - N[(N[(l$95$m / Om), $MachinePrecision] * N[(2.0 * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if U* < -9.0000000000000007e209 or 5.39999999999999998e163 < U*

   1. Initial program 40.2%

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

   3. Simplified 41.9%

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

      1. associate-*r* 41.9%

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

      2. fma-define 48.4%

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

      3. associate-*r* 45.3%

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

   6. Applied egg-rr 45.3%

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

      1. fma-undefine 38.7%

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

   8. Applied egg-rr 38.7%

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

   9. Taylor expanded in U* around inf 46.6%

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

       1. mul-1-neg 46.6%

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

       2. associate-/l* 51.4%

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

       3. distribute-rgt-neg-in 51.4%

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

       4. *-commutative 51.4%

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

       5. associate-/l* 51.5%

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

       6. distribute-lft-neg-in 51.5%

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

       7. unpow2 51.5%

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

       8. unpow2 51.5%

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

       9. times-frac 56.5%

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

       10. unpow2 56.5%

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

   11. Simplified 56.5%

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

   if -9.0000000000000007e209 < U* < 5.39999999999999998e163

   1. Initial program 49.2%

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

   3. Simplified 54.7%

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

      1. associate-*r* 54.7%

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

      2. fma-define 54.8%

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

      3. associate-*r* 54.8%

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

   6. Applied egg-rr 54.8%

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

      1. fma-undefine 54.7%

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

   8. Applied egg-rr 54.7%

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

   9. Taylor expanded in U around 0 44.5%

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

       1. mul-1-neg 43.1%

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

       2. associate-/l* 43.7%

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

       3. unpow2 43.7%

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

       4. unpow2 43.7%

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

       5. times-frac 54.4%

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

       6. unpow2 54.4%

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

   11. Simplified 55.0%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq -9 \cdot 10^{+209} \lor \neg \left(U* \leq 5.4 \cdot 10^{+163}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot U*\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot U*\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{-23} \lor \neg \left(n \leq 1.1 \cdot 10^{-148}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot U*\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{l\_m}^{2}}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (or (<= n -2e-23) (not (<= n 1.1e-148)))
   (sqrt (* (* 2.0 (* n U)) (+ t (* (* n (pow (/ l_m Om) 2.0)) U*))))
   (sqrt (* (* 2.0 n) (* U (- t (* 2.0 (/ (pow l_m 2.0) Om))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((n <= -2e-23) || !(n <= 1.1e-148)) {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * pow((l_m / Om), 2.0)) * U_42_))));
	} else {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (pow(l_m, 2.0) / Om))))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((n <= (-2d-23)) .or. (.not. (n <= 1.1d-148))) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + ((n * ((l_m / om) ** 2.0d0)) * u_42))))
    else
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * ((l_m ** 2.0d0) / om))))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((n <= -2e-23) || !(n <= 1.1e-148)) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((n * Math.pow((l_m / Om), 2.0)) * U_42_))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * (Math.pow(l_m, 2.0) / Om))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if (n <= -2e-23) or not (n <= 1.1e-148):
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((n * math.pow((l_m / Om), 2.0)) * U_42_))))
	else:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * (math.pow(l_m, 2.0) / Om))))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if ((n <= -2e-23) || !(n <= 1.1e-148))
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * U_42_))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64((l_m ^ 2.0) / Om))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if ((n <= -2e-23) || ~((n <= 1.1e-148)))
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * ((l_m / Om) ^ 2.0)) * U_42_))));
	else
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * ((l_m ^ 2.0) / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[Or[LessEqual[n, -2e-23], N[Not[LessEqual[n, 1.1e-148]], $MachinePrecision]], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[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]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.99999999999999992e-23 or 1.10000000000000009e-148 < n

   1. Initial program 47.4%

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

   3. Simplified 52.4%

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

      1. associate-*r* 52.4%

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

      2. fma-define 55.0%

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

      3. associate-*r* 55.6%

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

   6. Applied egg-rr 55.6%

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

      1. fma-undefine 53.0%

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

   8. Applied egg-rr 53.0%

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

   9. Taylor expanded in U* around inf 44.5%

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

       1. mul-1-neg 44.5%

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

       2. associate-/l* 47.0%

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

       3. distribute-rgt-neg-in 47.0%

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

       4. *-commutative 47.0%

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

       5. associate-/l* 46.6%

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

       6. distribute-lft-neg-in 46.6%

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

       7. unpow2 46.6%

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

       8. unpow2 46.6%

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

       9. times-frac 52.8%

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

       10. unpow2 52.8%

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

   11. Simplified 52.8%

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

   if -1.99999999999999992e-23 < n < 1.10000000000000009e-148

   1. Initial program 46.4%

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

   3. Simplified 50.5%

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

   5. Taylor expanded in Om around inf 53.8%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{-23} \lor \neg \left(n \leq 1.1 \cdot 10^{-148}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot U*\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 3 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{l\_m}^{2}}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot U*\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 3e+151)
   (sqrt (* (* 2.0 n) (* U (- t (* 2.0 (/ (pow l_m 2.0) Om))))))
   (sqrt (* (* 2.0 (* n U)) (* (* n (pow (/ l_m Om) 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 <= 3e+151) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (pow(l_m, 2.0) / Om))))));
	} else {
		tmp = sqrt(((2.0 * (n * U)) * ((n * pow((l_m / Om), 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 <= 3d+151) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * ((l_m ** 2.0d0) / om))))))
    else
        tmp = sqrt(((2.0d0 * (n * u)) * ((n * ((l_m / om) ** 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 <= 3e+151) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * (Math.pow(l_m, 2.0) / Om))))));
	} else {
		tmp = Math.sqrt(((2.0 * (n * U)) * ((n * Math.pow((l_m / Om), 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 <= 3e+151:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * (math.pow(l_m, 2.0) / Om))))))
	else:
		tmp = math.sqrt(((2.0 * (n * U)) * ((n * math.pow((l_m / Om), 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 <= 3e+151)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64((l_m ^ 2.0) / Om))))));
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(Float64(n * (Float64(l_m / Om) ^ 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 <= 3e+151)
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * ((l_m ^ 2.0) / Om))))));
	else
		tmp = sqrt(((2.0 * (n * U)) * ((n * ((l_m / Om) ^ 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, 3e+151], N[Sqrt[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]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * U$42$), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.9999999999999999e151

   1. Initial program 51.2%

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

   3. Simplified 53.4%

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

   5. Taylor expanded in Om around inf 48.4%

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

   if 2.9999999999999999e151 < l

   1. Initial program 12.5%

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

   3. Simplified 33.6%

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

      1. associate-*r* 33.6%

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

      2. fma-define 37.5%

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

      3. associate-*r* 34.1%

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

   6. Applied egg-rr 34.1%

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

      1. fma-undefine 30.2%

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

   8. Applied egg-rr 30.2%

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

   9. Taylor expanded in U* around inf 19.4%

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

       1. associate-/l* 19.4%

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

       2. *-commutative 19.4%

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

       3. associate-/l* 19.5%

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

       4. unpow2 19.5%

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

       5. unpow2 19.5%

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

       6. times-frac 34.3%

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

       7. unpow2 34.3%

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

   11. Simplified 34.3%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot U*\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U \leq -9.5 \cdot 10^{+108} \lor \neg \left(U \leq 1.95 \cdot 10^{+27}\right):\\ \;\;\;\;\sqrt{\left|\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(-2 \cdot \frac{U \cdot \left(l\_m \cdot l\_m\right)}{Om} + U \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (or (<= U -9.5e+108) (not (<= U 1.95e+27)))
   (sqrt (fabs (* (* (* 2.0 n) U) t)))
   (sqrt (* 2.0 (* n (+ (* -2.0 (/ (* U (* l_m l_m)) Om)) (* U t)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((U <= -9.5e+108) || !(U <= 1.95e+27)) {
		tmp = sqrt(fabs((((2.0 * n) * U) * t)));
	} else {
		tmp = sqrt((2.0 * (n * ((-2.0 * ((U * (l_m * l_m)) / Om)) + (U * t)))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((u <= (-9.5d+108)) .or. (.not. (u <= 1.95d+27))) then
        tmp = sqrt(abs((((2.0d0 * n) * u) * t)))
    else
        tmp = sqrt((2.0d0 * (n * (((-2.0d0) * ((u * (l_m * l_m)) / om)) + (u * t)))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((U <= -9.5e+108) || !(U <= 1.95e+27)) {
		tmp = Math.sqrt(Math.abs((((2.0 * n) * U) * t)));
	} else {
		tmp = Math.sqrt((2.0 * (n * ((-2.0 * ((U * (l_m * l_m)) / Om)) + (U * t)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if (U <= -9.5e+108) or not (U <= 1.95e+27):
		tmp = math.sqrt(math.fabs((((2.0 * n) * U) * t)))
	else:
		tmp = math.sqrt((2.0 * (n * ((-2.0 * ((U * (l_m * l_m)) / Om)) + (U * t)))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if ((U <= -9.5e+108) || !(U <= 1.95e+27))
		tmp = sqrt(abs(Float64(Float64(Float64(2.0 * n) * U) * t)));
	else
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(Float64(-2.0 * Float64(Float64(U * Float64(l_m * l_m)) / Om)) + Float64(U * t)))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if ((U <= -9.5e+108) || ~((U <= 1.95e+27)))
		tmp = sqrt(abs((((2.0 * n) * U) * t)));
	else
		tmp = sqrt((2.0 * (n * ((-2.0 * ((U * (l_m * l_m)) / Om)) + (U * t)))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[Or[LessEqual[U, -9.5e+108], N[Not[LessEqual[U, 1.95e+27]], $MachinePrecision]], N[Sqrt[N[Abs[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(n * N[(N[(-2.0 * N[(N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -9.50000000000000097e108 or 1.9499999999999999e27 < U

   1. Initial program 62.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in t around inf 53.0%

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

      1. add-sqr-sqrt 53.0%

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

      2. pow1/2 53.0%

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

      3. pow1/2 57.5%

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

      4. pow-prod-down 36.6%

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

      5. pow2 36.6%

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

      6. associate-*r* 36.6%

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

   7. Applied egg-rr 36.6%

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

      1. unpow1/2 36.6%

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

      2. unpow2 36.6%

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

      3. rem-sqrt-square 58.2%

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

      4. *-commutative 58.2%

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

      5. *-commutative 58.2%

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

      6. *-commutative 58.2%

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

   9. Simplified 58.2%

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

   if -9.50000000000000097e108 < U < 1.9499999999999999e27

   1. Initial program 41.3%

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

   3. Simplified 50.7%

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

   5. Taylor expanded in Om around inf 44.8%

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

      1. unpow2 44.8%

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

   7. Applied egg-rr 44.8%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -9.5 \cdot 10^{+108} \lor \neg \left(U \leq 1.95 \cdot 10^{+27}\right):\\ \;\;\;\;\sqrt{\left|\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(-2 \cdot \frac{U \cdot \left(\ell \cdot \ell\right)}{Om} + U \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U \leq -1.08 \cdot 10^{+108} \lor \neg \left(U \leq 5.2 \cdot 10^{+24}\right):\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(-2 \cdot \frac{U \cdot \left(l\_m \cdot l\_m\right)}{Om} + U \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (or (<= U -1.08e+108) (not (<= U 5.2e+24)))
   (pow (* (* (* 2.0 n) U) t) 0.5)
   (sqrt (* 2.0 (* n (+ (* -2.0 (/ (* U (* l_m l_m)) Om)) (* U t)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((U <= -1.08e+108) || !(U <= 5.2e+24)) {
		tmp = pow((((2.0 * n) * U) * t), 0.5);
	} else {
		tmp = sqrt((2.0 * (n * ((-2.0 * ((U * (l_m * l_m)) / Om)) + (U * t)))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((u <= (-1.08d+108)) .or. (.not. (u <= 5.2d+24))) then
        tmp = (((2.0d0 * n) * u) * t) ** 0.5d0
    else
        tmp = sqrt((2.0d0 * (n * (((-2.0d0) * ((u * (l_m * l_m)) / om)) + (u * t)))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((U <= -1.08e+108) || !(U <= 5.2e+24)) {
		tmp = Math.pow((((2.0 * n) * U) * t), 0.5);
	} else {
		tmp = Math.sqrt((2.0 * (n * ((-2.0 * ((U * (l_m * l_m)) / Om)) + (U * t)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if (U <= -1.08e+108) or not (U <= 5.2e+24):
		tmp = math.pow((((2.0 * n) * U) * t), 0.5)
	else:
		tmp = math.sqrt((2.0 * (n * ((-2.0 * ((U * (l_m * l_m)) / Om)) + (U * t)))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if ((U <= -1.08e+108) || !(U <= 5.2e+24))
		tmp = Float64(Float64(Float64(2.0 * n) * U) * t) ^ 0.5;
	else
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(Float64(-2.0 * Float64(Float64(U * Float64(l_m * l_m)) / Om)) + Float64(U * t)))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if ((U <= -1.08e+108) || ~((U <= 5.2e+24)))
		tmp = (((2.0 * n) * U) * t) ^ 0.5;
	else
		tmp = sqrt((2.0 * (n * ((-2.0 * ((U * (l_m * l_m)) / Om)) + (U * t)))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[Or[LessEqual[U, -1.08e+108], N[Not[LessEqual[U, 5.2e+24]], $MachinePrecision]], N[Power[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(2.0 * N[(n * N[(N[(-2.0 * N[(N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -1.0800000000000001e108 or 5.1999999999999997e24 < U

   1. Initial program 61.6%

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

   3. Simplified 64.5%

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

   5. Taylor expanded in t around inf 52.3%

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

      1. pow1/2 56.7%

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

      2. associate-*r* 56.7%

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

   7. Applied egg-rr 56.7%

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

   if -1.0800000000000001e108 < U < 5.1999999999999997e24

   1. Initial program 41.5%

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

   3. Simplified 51.0%

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

   5. Taylor expanded in Om around inf 45.0%

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

      1. unpow2 45.0%

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

   7. Applied egg-rr 45.0%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -1.08 \cdot 10^{+108} \lor \neg \left(U \leq 5.2 \cdot 10^{+24}\right):\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(-2 \cdot \frac{U \cdot \left(\ell \cdot \ell\right)}{Om} + U \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 39.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;Om \leq 1.9 \cdot 10^{-279}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5}\\ \mathbf{elif}\;Om \leq 60:\\ \;\;\;\;\frac{l\_m}{Om} \cdot \sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(n \cdot U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= Om 1.9e-279)
   (pow (* (* 2.0 n) (* U t)) 0.5)
   (if (<= Om 60.0)
     (* (/ l_m Om) (sqrt (* U (* (* 2.0 n) (* n U*)))))
     (pow (* (* (* 2.0 n) U) t) 0.5))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (Om <= 1.9e-279) {
		tmp = pow(((2.0 * n) * (U * t)), 0.5);
	} else if (Om <= 60.0) {
		tmp = (l_m / Om) * sqrt((U * ((2.0 * n) * (n * U_42_))));
	} else {
		tmp = pow((((2.0 * n) * U) * t), 0.5);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (om <= 1.9d-279) then
        tmp = ((2.0d0 * n) * (u * t)) ** 0.5d0
    else if (om <= 60.0d0) then
        tmp = (l_m / om) * sqrt((u * ((2.0d0 * n) * (n * u_42))))
    else
        tmp = (((2.0d0 * n) * u) * t) ** 0.5d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (Om <= 1.9e-279) {
		tmp = Math.pow(((2.0 * n) * (U * t)), 0.5);
	} else if (Om <= 60.0) {
		tmp = (l_m / Om) * Math.sqrt((U * ((2.0 * n) * (n * U_42_))));
	} else {
		tmp = Math.pow((((2.0 * n) * U) * t), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if Om <= 1.9e-279:
		tmp = math.pow(((2.0 * n) * (U * t)), 0.5)
	elif Om <= 60.0:
		tmp = (l_m / Om) * math.sqrt((U * ((2.0 * n) * (n * U_42_))))
	else:
		tmp = math.pow((((2.0 * n) * U) * t), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (Om <= 1.9e-279)
		tmp = Float64(Float64(2.0 * n) * Float64(U * t)) ^ 0.5;
	elseif (Om <= 60.0)
		tmp = Float64(Float64(l_m / Om) * sqrt(Float64(U * Float64(Float64(2.0 * n) * Float64(n * U_42_)))));
	else
		tmp = Float64(Float64(Float64(2.0 * n) * U) * t) ^ 0.5;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (Om <= 1.9e-279)
		tmp = ((2.0 * n) * (U * t)) ^ 0.5;
	elseif (Om <= 60.0)
		tmp = (l_m / Om) * sqrt((U * ((2.0 * n) * (n * U_42_))));
	else
		tmp = (((2.0 * n) * U) * t) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[Om, 1.9e-279], N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[Om, 60.0], N[(N[(l$95$m / Om), $MachinePrecision] * N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * N[(n * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision], 0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if Om < 1.90000000000000016e-279

   1. Initial program 49.0%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in l around 0 38.2%

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

      1. pow1/2 39.8%

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

      2. associate-*r* 39.8%

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

   7. Applied egg-rr 39.8%

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

   if 1.90000000000000016e-279 < Om < 60

   1. Initial program 33.3%

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

   3. Simplified 34.0%

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

      1. *-commutative 34.0%

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

      2. sqrt-prod 19.3%

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

   6. Applied egg-rr 19.4%

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

   7. Taylor expanded in U* around inf 12.3%

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

      1. pow1 12.3%

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

      2. associate-*l* 12.2%

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

      3. pow1/2 12.3%

         \[\leadsto {\left(\frac{\ell}{Om} \cdot \left(\color{blue}{{\left(U \cdot \left(U* \cdot n\right)\right)}^{0.5}} \cdot \sqrt{2 \cdot n}\right)\right)}^{1} \]

      4. pow1/2 12.3%

         \[\leadsto {\left(\frac{\ell}{Om} \cdot \left({\left(U \cdot \left(U* \cdot n\right)\right)}^{0.5} \cdot \color{blue}{{\left(2 \cdot n\right)}^{0.5}}\right)\right)}^{1} \]

      5. pow-prod-down 22.3%

         \[\leadsto {\left(\frac{\ell}{Om} \cdot \color{blue}{{\left(\left(U \cdot \left(U* \cdot n\right)\right) \cdot \left(2 \cdot n\right)\right)}^{0.5}}\right)}^{1} \]

      6. *-commutative 22.3%

         \[\leadsto {\left(\frac{\ell}{Om} \cdot {\left(\left(U \cdot \color{blue}{\left(n \cdot U*\right)}\right) \cdot \left(2 \cdot n\right)\right)}^{0.5}\right)}^{1} \]

   9. Applied egg-rr 22.3%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{Om} \cdot {\left(\left(U \cdot \left(n \cdot U*\right)\right) \cdot \left(2 \cdot n\right)\right)}^{0.5}\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 22.3%

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

       2. unpow1/2 22.2%

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

       3. associate-*l* 22.3%

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

   11. Simplified 22.3%

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

   if 60 < Om

   1. Initial program 54.8%

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

   3. Simplified 65.7%

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

   5. Taylor expanded in t around inf 44.6%

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

      1. pow1/2 44.6%

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

      2. associate-*r* 44.7%

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

   7. Applied egg-rr 44.7%

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

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 1.9 \cdot 10^{-279}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot t\right)\right)}^{0.5}\\ \mathbf{elif}\;Om \leq 60:\\ \;\;\;\;\frac{\ell}{Om} \cdot \sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(n \cdot U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 37.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;Om \leq -3.9 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= Om -3.9e+68)
   (sqrt (* 2.0 (* n (* U t))))
   (pow (* 2.0 (* U (* n t))) 0.5)))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (Om <= -3.9e+68) {
		tmp = sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (om <= (-3.9d+68)) then
        tmp = sqrt((2.0d0 * (n * (u * t))))
    else
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (Om <= -3.9e+68) {
		tmp = Math.sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if Om <= -3.9e+68:
		tmp = math.sqrt((2.0 * (n * (U * t))))
	else:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (Om <= -3.9e+68)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * t))));
	else
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (Om <= -3.9e+68)
		tmp = sqrt((2.0 * (n * (U * t))));
	else
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[Om, -3.9e+68], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -3.90000000000000019e68

   1. Initial program 62.4%

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

   3. Simplified 72.5%

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

   5. Taylor expanded in l around 0 59.0%

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

   if -3.90000000000000019e68 < Om

   1. Initial program 42.4%

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

   3. Simplified 45.0%

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

   5. Taylor expanded in l around 0 29.3%

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

      1. associate-*r* 29.3%

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

   7. Simplified 29.3%

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

      1. pow1/2 31.4%

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

      2. associate-*l* 31.4%

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

   9. Applied egg-rr 31.4%

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

Alternative 15: 35.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;Om \leq -1.5 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= Om -1.5e+67)
   (sqrt (* 2.0 (* n (* U t))))
   (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_) {
	double tmp;
	if (Om <= -1.5e+67) {
		tmp = sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = sqrt(((2.0 * U) * (n * t)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (om <= (-1.5d+67)) then
        tmp = sqrt((2.0d0 * (n * (u * t))))
    else
        tmp = sqrt(((2.0d0 * u) * (n * t)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (Om <= -1.5e+67) {
		tmp = Math.sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = Math.sqrt(((2.0 * U) * (n * t)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if Om <= -1.5e+67:
		tmp = math.sqrt((2.0 * (n * (U * t))))
	else:
		tmp = math.sqrt(((2.0 * U) * (n * t)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (Om <= -1.5e+67)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * t))));
	else
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (Om <= -1.5e+67)
		tmp = sqrt((2.0 * (n * (U * t))));
	else
		tmp = sqrt(((2.0 * U) * (n * t)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[Om, -1.5e+67], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -1.50000000000000005e67

   1. Initial program 62.4%

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

   3. Simplified 72.5%

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

   5. Taylor expanded in l around 0 59.0%

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

   if -1.50000000000000005e67 < Om

   1. Initial program 42.4%

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

   3. Simplified 45.0%

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

   5. Taylor expanded in l around 0 29.3%

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

      1. associate-*r* 29.3%

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

   7. Simplified 29.3%

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

Alternative 16: 35.1% accurate, 2.1× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* 2.0 (* 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(2.0 * Float64(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[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 47.0%

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

3. Simplified 51.3%

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

5. Taylor expanded in l around 0 33.1%

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

Reproduce

herbie shell --seed 2024167 
(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*))))))