Toniolo and Linder, Equation (13)

Percentage Accurate: 50.2% → 64.5%

Time: 28.0s

Alternatives: 18

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.2% 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: 64.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := l\_m \cdot \frac{l\_m}{Om}\\ t_2 := {\left(\frac{l\_m}{Om}\right)}^{2}\\ t_3 := \left(n \cdot t\_2\right) \cdot \left(U* - U\right)\\ t_4 := \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, t\_1, n \cdot \left(t\_2 \cdot \left(-U*\right)\right)\right)\right)\right)}\\ t_5 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_3\right)\\ \mathbf{if}\;t\_5 \leq 10^{-252}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_5 \leq 10^{+295}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_3 - 2 \cdot t\_1\right)\right)}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - \frac{2}{Om}\right)\right)} \cdot \left(l\_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* l_m (/ l_m Om)))
        (t_2 (pow (/ l_m Om) 2.0))
        (t_3 (* (* n t_2) (- U* U)))
        (t_4
         (sqrt (* (* 2.0 n) (* U (- t (fma 2.0 t_1 (* n (* t_2 (- U*)))))))))
        (t_5 (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_3))))
   (if (<= t_5 1e-252)
     t_4
     (if (<= t_5 1e+295)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_3 (* 2.0 t_1)))))
       (if (<= t_5 INFINITY)
         t_4
         (*
          (sqrt (* U (* n (- (/ (* n (- U* U)) (pow Om 2.0)) (/ 2.0 Om)))))
          (* l_m (sqrt 2.0))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = l_m * (l_m / Om);
	double t_2 = pow((l_m / Om), 2.0);
	double t_3 = (n * t_2) * (U_42_ - U);
	double t_4 = sqrt(((2.0 * n) * (U * (t - fma(2.0, t_1, (n * (t_2 * -U_42_)))))));
	double t_5 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_3);
	double tmp;
	if (t_5 <= 1e-252) {
		tmp = t_4;
	} else if (t_5 <= 1e+295) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_3 - (2.0 * t_1)))));
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = sqrt((U * (n * (((n * (U_42_ - U)) / pow(Om, 2.0)) - (2.0 / 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(l_m * Float64(l_m / Om))
	t_2 = Float64(l_m / Om) ^ 2.0
	t_3 = Float64(Float64(n * t_2) * Float64(U_42_ - U))
	t_4 = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - fma(2.0, t_1, Float64(n * Float64(t_2 * Float64(-U_42_))))))))
	t_5 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_3))
	tmp = 0.0
	if (t_5 <= 1e-252)
		tmp = t_4;
	elseif (t_5 <= 1e+295)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(t_3 - Float64(2.0 * t_1)))));
	elseif (t_5 <= Inf)
		tmp = t_4;
	else
		tmp = Float64(sqrt(Float64(U * Float64(n * Float64(Float64(Float64(n * Float64(U_42_ - U)) / (Om ^ 2.0)) - Float64(2.0 / 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[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(n * t$95$2), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * t$95$1 + N[(n * N[(t$95$2 * (-U$42$)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = 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$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 1e-252], t$95$4, If[LessEqual[t$95$5, 1e+295], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(t$95$3 - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$5, Infinity], t$95$4, 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 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.99999999999999943e-253 or 9.9999999999999998e294 < (*.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 27.0%

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

   3. Simplified 45.3%

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

   5. Taylor expanded in U around 0 36.1%

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

      1. mul-1-neg 36.1%

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

      2. associate-/l* 36.1%

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

      3. unpow2 36.1%

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

      4. unpow2 36.1%

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

      5. times-frac 45.5%

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

      6. unpow2 45.5%

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

   7. Simplified 45.5%

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

   if 9.99999999999999943e-253 < (*.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*)))) < 9.9999999999999998e294

   1. Initial program 97.2%

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

   3. Simplified 97.2%

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

   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 7.2%

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

      1. add-sqr-sqrt 0.4%

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

      2. pow2 0.4%

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

      3. sqrt-prod 0.4%

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

      4. sqrt-pow1 0.6%

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

      5. metadata-eval 0.6%

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

      6. pow1 0.6%

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

   6. Applied egg-rr 0.6%

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

   7. Taylor expanded in l around inf 28.8%

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

      1. associate-*r/ 28.8%

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

      2. metadata-eval 28.8%

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

   9. Simplified 28.8%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{-252}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(-U*\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{+295}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(-U*\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - \frac{2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 60.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_1\right)}\\ \mathbf{if}\;t\_2 \leq 10^{-126}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t + -2 \cdot \frac{U \cdot \left(l\_m \cdot l\_m\right)}{Om}\right)\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_1 - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\left(U \cdot {l\_m}^{2}\right) \cdot \left(n \cdot \frac{U* - U}{Om}\right)}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 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-126)
     (sqrt (* 2.0 (* n (+ (* U t) (* -2.0 (/ (* U (* l_m l_m)) Om))))))
     (if (<= t_2 INFINITY)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_1 (* 2.0 (* l_m (/ l_m Om)))))))
       (sqrt
        (*
         (* 2.0 n)
         (+ (* U t) (/ (* (* U (pow l_m 2.0)) (* n (/ (- U* U) Om))) Om))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	double tmp;
	if (t_2 <= 1e-126) {
		tmp = sqrt((2.0 * (n * ((U * t) + (-2.0 * ((U * (l_m * l_m)) / 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(((2.0 * n) * ((U * t) + (((U * pow(l_m, 2.0)) * (n * ((U_42_ - U) / Om))) / Om))));
	}
	return tmp;
}

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	double tmp;
	if (t_2 <= 1e-126) {
		tmp = Math.sqrt((2.0 * (n * ((U * t) + (-2.0 * ((U * (l_m * l_m)) / Om))))));
	} 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 * n) * ((U * t) + (((U * Math.pow(l_m, 2.0)) * (n * ((U_42_ - U) / Om))) / Om))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_2 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)))
	tmp = 0
	if t_2 <= 1e-126:
		tmp = math.sqrt((2.0 * (n * ((U * t) + (-2.0 * ((U * (l_m * l_m)) / Om))))))
	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 * n) * ((U * t) + (((U * math.pow(l_m, 2.0)) * (n * ((U_42_ - U) / Om))) / Om))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_1)))
	tmp = 0.0
	if (t_2 <= 1e-126)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(Float64(U * t) + Float64(-2.0 * Float64(Float64(U * Float64(l_m * l_m)) / 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 = sqrt(Float64(Float64(2.0 * n) * Float64(Float64(U * t) + Float64(Float64(Float64(U * (l_m ^ 2.0)) * Float64(n * Float64(Float64(U_42_ - U) / Om))) / Om))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)));
	tmp = 0.0;
	if (t_2 <= 1e-126)
		tmp = sqrt((2.0 * (n * ((U * t) + (-2.0 * ((U * (l_m * l_m)) / Om))))));
	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) + (((U * (l_m ^ 2.0)) * (n * ((U_42_ - U) / Om))) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 1e-126], N[Sqrt[N[(2.0 * N[(n * N[(N[(U * t), $MachinePrecision] + N[(-2.0 * N[(N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 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[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(N[(U * t), $MachinePrecision] + N[(N[(N[(U * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(n * N[(N[(U$42$ - U), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 9.9999999999999995e-127

   1. Initial program 26.0%

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

   3. Simplified 55.0%

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

   5. Taylor expanded in Om around inf 58.2%

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

      1. unpow2 58.2%

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

   7. Applied egg-rr 58.2%

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

   if 9.9999999999999995e-127 < (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 66.0%

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

   3. Simplified 68.9%

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

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

   1. Initial program 0.0%

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

   3. Simplified 11.8%

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

   5. Taylor expanded in Om around -inf 17.2%

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

   6. Taylor expanded in Om around inf 17.2%

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

      1. fma-define 17.2%

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

      2. mul-1-neg 17.2%

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

      3. fmm-undef 17.2%

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

      4. associate-*r* 17.2%

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

      5. associate-/l* 18.9%

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

      6. associate-/l* 18.9%

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

   8. Simplified 18.9%

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

   9. Taylor expanded in n around inf 29.7%

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

       1. mul-1-neg 29.7%

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

       2. associate-/l* 31.5%

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

       3. associate-/l* 31.5%

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

       4. associate-*r/ 31.5%

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

       5. associate-*l* 35.0%

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

       6. distribute-rgt-neg-in 35.0%

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

       7. distribute-rgt-neg-in 35.0%

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

   11. Simplified 35.0%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 10^{-126}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t + -2 \cdot \frac{U \cdot \left(\ell \cdot \ell\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\left(U \cdot {\ell}^{2}\right) \cdot \left(n \cdot \frac{U* - U}{Om}\right)}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := l\_m \cdot \frac{l\_m}{Om}\\ t_2 := {\left(\frac{l\_m}{Om}\right)}^{2}\\ t_3 := \left(n \cdot t\_2\right) \cdot \left(U* - U\right)\\ t_4 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_3\right)\\ \mathbf{if}\;t\_4 \leq 10^{-252}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, t\_1, n \cdot \left(U \cdot t\_2\right)\right)\right)\right)}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_3 - 2 \cdot t\_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - \frac{2}{Om}\right)\right)} \cdot \left(l\_m \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* l_m (/ l_m Om)))
        (t_2 (pow (/ l_m Om) 2.0))
        (t_3 (* (* n t_2) (- U* U)))
        (t_4 (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_3))))
   (if (<= t_4 1e-252)
     (sqrt (* (* 2.0 n) (* U (- t (fma 2.0 t_1 (* n (* U t_2)))))))
     (if (<= t_4 INFINITY)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_3 (* 2.0 t_1)))))
       (*
        (sqrt (* U (* n (- (/ (* n (- U* U)) (pow Om 2.0)) (/ 2.0 Om)))))
        (* l_m (sqrt 2.0)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = l_m * (l_m / Om);
	double t_2 = pow((l_m / Om), 2.0);
	double t_3 = (n * t_2) * (U_42_ - U);
	double t_4 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + t_3);
	double tmp;
	if (t_4 <= 1e-252) {
		tmp = sqrt(((2.0 * n) * (U * (t - fma(2.0, t_1, (n * (U * t_2)))))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_3 - (2.0 * t_1)))));
	} else {
		tmp = sqrt((U * (n * (((n * (U_42_ - U)) / pow(Om, 2.0)) - (2.0 / 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(l_m * Float64(l_m / Om))
	t_2 = Float64(l_m / Om) ^ 2.0
	t_3 = Float64(Float64(n * t_2) * Float64(U_42_ - U))
	t_4 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_3))
	tmp = 0.0
	if (t_4 <= 1e-252)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - fma(2.0, t_1, Float64(n * Float64(U * t_2)))))));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(t_3 - Float64(2.0 * t_1)))));
	else
		tmp = Float64(sqrt(Float64(U * Float64(n * Float64(Float64(Float64(n * Float64(U_42_ - U)) / (Om ^ 2.0)) - Float64(2.0 / 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[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(n * t$95$2), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = 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$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 1e-252], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * t$95$1 + N[(n * N[(U * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(t$95$3 - N[(2.0 * t$95$1), $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 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.99999999999999943e-253

   1. Initial program 21.2%

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

   3. Simplified 52.9%

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

   5. Taylor expanded in U around inf 41.7%

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

      1. associate-/l* 41.7%

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

      2. unpow2 41.7%

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

      3. unpow2 41.7%

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

      4. times-frac 52.8%

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

      5. unpow2 52.8%

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

   7. Simplified 52.8%

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

   if 9.99999999999999943e-253 < (*.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 66.0%

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

   3. Simplified 68.9%

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

   if +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 7.2%

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

      1. add-sqr-sqrt 0.4%

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

      2. pow2 0.4%

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

      3. sqrt-prod 0.4%

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

      4. sqrt-pow1 0.6%

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

      5. metadata-eval 0.6%

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

      6. pow1 0.6%

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

   6. Applied egg-rr 0.6%

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

   7. Taylor expanded in l around inf 28.8%

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

      1. associate-*r/ 28.8%

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

      2. metadata-eval 28.8%

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

   9. Simplified 28.8%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{-252}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left(U \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - \frac{2}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 13.3%

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

   3. Simplified 47.4%

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

   5. Taylor expanded in t around inf 47.4%

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

      1. add-sqr-sqrt 47.4%

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

      2. pow1/2 47.4%

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

      3. pow1/2 47.4%

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

      4. pow-prod-down 22.7%

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

      5. pow2 22.7%

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

   7. Applied egg-rr 22.7%

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

      1. unpow1/2 22.7%

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

      2. unpow2 22.7%

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

      3. rem-sqrt-square 47.9%

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

      4. associate-*l* 47.9%

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

   9. Simplified 47.9%

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

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

   1. Initial program 66.6%

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

   3. Simplified 69.3%

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

   if +inf.0 < (*.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 7.2%

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

      1. add-sqr-sqrt 0.4%

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

      2. pow2 0.4%

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

      3. sqrt-prod 0.4%

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

      4. sqrt-pow1 0.6%

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

      5. metadata-eval 0.6%

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

      6. pow1 0.6%

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

   6. Applied egg-rr 0.6%

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

   7. Taylor expanded in l around inf 28.8%

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

      1. associate-*r/ 28.8%

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

      2. metadata-eval 28.8%

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

   9. Simplified 28.8%

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

4. Final simplification 59.1%

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

Alternative 5: 63.7% accurate, 0.4× speedup

Math

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

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 13.3%

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

   3. Simplified 47.4%

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

   5. Taylor expanded in t around inf 47.4%

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

      1. add-sqr-sqrt 47.4%

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

      2. pow1/2 47.4%

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

      3. pow1/2 47.4%

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

      4. pow-prod-down 22.7%

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

      5. pow2 22.7%

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

   7. Applied egg-rr 22.7%

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

      1. unpow1/2 22.7%

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

      2. unpow2 22.7%

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

      3. rem-sqrt-square 47.9%

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

      4. associate-*l* 47.9%

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

   9. Simplified 47.9%

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

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

   1. Initial program 66.6%

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

   3. Simplified 69.3%

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

   if +inf.0 < (*.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 7.2%

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

   5. Taylor expanded in l around inf 28.8%

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

      1. associate-/l* 26.9%

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

      2. associate-*r/ 26.9%

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

      3. metadata-eval 26.9%

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

   7. Simplified 26.9%

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

4. Final simplification 58.7%

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

Alternative 6: 53.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;Om \leq -3.4 \cdot 10^{-52}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot \left({\left(\frac{l\_m}{Om}\right)}^{2} \cdot U*\right) - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 1.65 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{U \cdot \left(U* \cdot \left({l\_m}^{2} \cdot \frac{n}{Om}\right)\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot \left(t - \frac{l\_m}{Om} \cdot \left(2 \cdot l\_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= Om -3.4e-52)
   (sqrt
    (*
     (* 2.0 (* n U))
     (+ t (- (* n (* (pow (/ l_m Om) 2.0) U*)) (* 2.0 (* l_m (/ l_m Om)))))))
   (if (<= Om 1.65e+95)
     (sqrt
      (* (* 2.0 n) (+ (* U t) (/ (* U (* U* (* (pow l_m 2.0) (/ n Om)))) Om))))
     (* (sqrt 2.0) (sqrt (* U (* n (- t (* (/ 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 tmp;
	if (Om <= -3.4e-52) {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * (pow((l_m / Om), 2.0) * U_42_)) - (2.0 * (l_m * (l_m / Om)))))));
	} else if (Om <= 1.65e+95) {
		tmp = sqrt(((2.0 * n) * ((U * t) + ((U * (U_42_ * (pow(l_m, 2.0) * (n / Om)))) / Om))));
	} else {
		tmp = sqrt(2.0) * sqrt((U * (n * (t - ((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) :: tmp
    if (om <= (-3.4d-52)) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + ((n * (((l_m / om) ** 2.0d0) * u_42)) - (2.0d0 * (l_m * (l_m / om)))))))
    else if (om <= 1.65d+95) then
        tmp = sqrt(((2.0d0 * n) * ((u * t) + ((u * (u_42 * ((l_m ** 2.0d0) * (n / om)))) / om))))
    else
        tmp = sqrt(2.0d0) * sqrt((u * (n * (t - ((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 tmp;
	if (Om <= -3.4e-52) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((n * (Math.pow((l_m / Om), 2.0) * U_42_)) - (2.0 * (l_m * (l_m / Om)))))));
	} else if (Om <= 1.65e+95) {
		tmp = Math.sqrt(((2.0 * n) * ((U * t) + ((U * (U_42_ * (Math.pow(l_m, 2.0) * (n / Om)))) / Om))));
	} else {
		tmp = Math.sqrt(2.0) * Math.sqrt((U * (n * (t - ((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_):
	tmp = 0
	if Om <= -3.4e-52:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((n * (math.pow((l_m / Om), 2.0) * U_42_)) - (2.0 * (l_m * (l_m / Om)))))))
	elif Om <= 1.65e+95:
		tmp = math.sqrt(((2.0 * n) * ((U * t) + ((U * (U_42_ * (math.pow(l_m, 2.0) * (n / Om)))) / Om))))
	else:
		tmp = math.sqrt(2.0) * math.sqrt((U * (n * (t - ((l_m / Om) * (2.0 * l_m))))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (Om <= -3.4e-52)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(n * Float64((Float64(l_m / Om) ^ 2.0) * U_42_)) - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))))));
	elseif (Om <= 1.65e+95)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(Float64(U * t) + Float64(Float64(U * Float64(U_42_ * Float64((l_m ^ 2.0) * Float64(n / Om)))) / Om))));
	else
		tmp = Float64(sqrt(2.0) * sqrt(Float64(U * Float64(n * Float64(t - 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_)
	tmp = 0.0;
	if (Om <= -3.4e-52)
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * (((l_m / Om) ^ 2.0) * U_42_)) - (2.0 * (l_m * (l_m / Om)))))));
	elseif (Om <= 1.65e+95)
		tmp = sqrt(((2.0 * n) * ((U * t) + ((U * (U_42_ * ((l_m ^ 2.0) * (n / Om)))) / Om))));
	else
		tmp = sqrt(2.0) * sqrt((U * (n * (t - ((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$42$_] := If[LessEqual[Om, -3.4e-52], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(n * N[(N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision] * U$42$), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 1.65e+95], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(N[(U * t), $MachinePrecision] + N[(N[(U * N[(U$42$ * N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(U * N[(n * N[(t - N[(N[(l$95$m / Om), $MachinePrecision] * N[(2.0 * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if Om < -3.40000000000000017e-52

   1. Initial program 53.8%

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

   3. Simplified 59.9%

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

      1. associate-*r* 59.9%

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

      2. pow1 59.9%

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

   6. Applied egg-rr 59.9%

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

      1. unpow1 59.9%

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

   8. Simplified 59.9%

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

   9. Taylor expanded in U around 0 50.2%

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

       1. mul-1-neg 49.4%

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

       2. associate-/l* 52.5%

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

       3. unpow2 52.5%

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

       4. unpow2 52.5%

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

       5. times-frac 60.3%

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

       6. unpow2 60.3%

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

   11. Simplified 60.1%

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

   if -3.40000000000000017e-52 < Om < 1.6499999999999999e95

   1. Initial program 41.9%

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

   3. Simplified 46.7%

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

   5. Taylor expanded in Om around -inf 50.2%

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

   6. Taylor expanded in Om around inf 50.2%

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

      1. fma-define 50.2%

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

      2. mul-1-neg 50.2%

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

      3. fmm-undef 50.2%

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

      4. associate-*r* 50.2%

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

      5. associate-/l* 51.0%

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

      6. associate-/l* 48.8%

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

   8. Simplified 48.8%

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

   9. Taylor expanded in U* around inf 56.9%

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

       1. associate-/l* 57.8%

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

       2. associate-/l* 57.8%

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

       3. associate-*r/ 59.0%

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

   11. Simplified 59.0%

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

   if 1.6499999999999999e95 < Om

   1. Initial program 46.8%

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

   3. Simplified 55.3%

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

   5. Taylor expanded in n around 0 47.3%

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

      1. metadata-eval 47.3%

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

      2. cancel-sign-sub-inv 47.3%

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

      3. associate-*r/ 47.3%

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

   7. Simplified 47.3%

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

      1. associate-/l* 47.3%

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

      2. unpow2 47.3%

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

      3. associate-*r/ 55.7%

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

      4. associate-*r* 55.7%

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

   9. Applied egg-rr 55.7%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -3.4 \cdot 10^{-52}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot U*\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 1.65 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot \frac{n}{Om}\right)\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U* \cdot \left({l\_m}^{2} \cdot \frac{n}{Om}\right)\\ \mathbf{if}\;n \leq -0.0027:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{t\_1}{Om}\right)\right)}\\ \mathbf{elif}\;n \leq 1.7 \cdot 10^{-81}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot \left(t - \frac{l\_m}{Om} \cdot \left(2 \cdot l\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{U \cdot t\_1}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U* (* (pow l_m 2.0) (/ n Om)))))
   (if (<= n -0.0027)
     (sqrt (* (* 2.0 n) (* U (+ t (/ t_1 Om)))))
     (if (<= n 1.7e-81)
       (* (sqrt 2.0) (sqrt (* U (* n (- t (* (/ l_m Om) (* 2.0 l_m)))))))
       (sqrt (* (* 2.0 n) (+ (* U t) (/ (* U t_1) 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 = U_42_ * (pow(l_m, 2.0) * (n / Om));
	double tmp;
	if (n <= -0.0027) {
		tmp = sqrt(((2.0 * n) * (U * (t + (t_1 / Om)))));
	} else if (n <= 1.7e-81) {
		tmp = sqrt(2.0) * sqrt((U * (n * (t - ((l_m / Om) * (2.0 * l_m))))));
	} else {
		tmp = sqrt(((2.0 * n) * ((U * t) + ((U * t_1) / 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 = u_42 * ((l_m ** 2.0d0) * (n / om))
    if (n <= (-0.0027d0)) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + (t_1 / om)))))
    else if (n <= 1.7d-81) then
        tmp = sqrt(2.0d0) * sqrt((u * (n * (t - ((l_m / om) * (2.0d0 * l_m))))))
    else
        tmp = sqrt(((2.0d0 * n) * ((u * t) + ((u * t_1) / 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 = U_42_ * (Math.pow(l_m, 2.0) * (n / Om));
	double tmp;
	if (n <= -0.0027) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + (t_1 / Om)))));
	} else if (n <= 1.7e-81) {
		tmp = Math.sqrt(2.0) * Math.sqrt((U * (n * (t - ((l_m / Om) * (2.0 * l_m))))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * ((U * t) + ((U * t_1) / Om))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = U_42_ * (math.pow(l_m, 2.0) * (n / Om))
	tmp = 0
	if n <= -0.0027:
		tmp = math.sqrt(((2.0 * n) * (U * (t + (t_1 / Om)))))
	elif n <= 1.7e-81:
		tmp = math.sqrt(2.0) * math.sqrt((U * (n * (t - ((l_m / Om) * (2.0 * l_m))))))
	else:
		tmp = math.sqrt(((2.0 * n) * ((U * t) + ((U * t_1) / Om))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U_42_ * Float64((l_m ^ 2.0) * Float64(n / Om)))
	tmp = 0.0
	if (n <= -0.0027)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(t_1 / Om)))));
	elseif (n <= 1.7e-81)
		tmp = Float64(sqrt(2.0) * sqrt(Float64(U * Float64(n * Float64(t - Float64(Float64(l_m / Om) * Float64(2.0 * l_m)))))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(Float64(U * t) + Float64(Float64(U * t_1) / Om))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = U_42_ * ((l_m ^ 2.0) * (n / Om));
	tmp = 0.0;
	if (n <= -0.0027)
		tmp = sqrt(((2.0 * n) * (U * (t + (t_1 / Om)))));
	elseif (n <= 1.7e-81)
		tmp = sqrt(2.0) * sqrt((U * (n * (t - ((l_m / Om) * (2.0 * l_m))))));
	else
		tmp = sqrt(((2.0 * n) * ((U * t) + ((U * t_1) / 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[(U$42$ * N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -0.0027], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(t$95$1 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 1.7e-81], N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(U * N[(n * N[(t - N[(N[(l$95$m / Om), $MachinePrecision] * N[(2.0 * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(N[(U * t), $MachinePrecision] + N[(N[(U * t$95$1), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -0.0027000000000000001

   1. Initial program 48.3%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in Om around -inf 36.2%

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

   6. Taylor expanded in U around 0 52.2%

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

      1. mul-1-neg 52.2%

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

      2. fma-define 52.2%

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

      3. associate-/l* 52.2%

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

      4. associate-/l* 52.2%

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

   8. Simplified 52.2%

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

   9. Taylor expanded in U* around inf 58.7%

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

       1. mul-1-neg 58.7%

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

       2. associate-/l* 58.7%

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

       3. associate-*r/ 58.7%

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

   11. Simplified 58.7%

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

   if -0.0027000000000000001 < n < 1.6999999999999999e-81

   1. Initial program 50.1%

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

   3. Simplified 57.0%

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

   5. Taylor expanded in n around 0 52.2%

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

      1. metadata-eval 52.2%

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

      2. cancel-sign-sub-inv 52.2%

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

      3. associate-*r/ 52.2%

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

   7. Simplified 52.2%

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

      1. associate-/l* 52.2%

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

      2. unpow2 52.2%

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

      3. associate-*r/ 58.7%

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

      4. associate-*r* 58.7%

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

   9. Applied egg-rr 58.7%

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

   if 1.6999999999999999e-81 < n

   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 47.4%

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

   5. Taylor expanded in Om around -inf 39.0%

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

   6. Taylor expanded in Om around inf 39.0%

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

      1. fma-define 39.0%

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

      2. mul-1-neg 39.0%

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

      3. fmm-undef 39.0%

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

      4. associate-*r* 39.0%

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

      5. associate-/l* 40.4%

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

      6. associate-/l* 37.7%

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

   8. Simplified 37.7%

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

   9. Taylor expanded in U* around inf 52.9%

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

       1. associate-/l* 54.4%

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

       2. associate-/l* 54.4%

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

       3. associate-*r/ 57.8%

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

   11. Simplified 57.8%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -0.0027:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{U* \cdot \left({\ell}^{2} \cdot \frac{n}{Om}\right)}{Om}\right)\right)}\\ \mathbf{elif}\;n \leq 1.7 \cdot 10^{-81}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{U \cdot \left(U* \cdot \left({\ell}^{2} \cdot \frac{n}{Om}\right)\right)}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq -2.65 \cdot 10^{-5} \lor \neg \left(n \leq 1.3 \cdot 10^{-78}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{U* \cdot \left({l\_m}^{2} \cdot \frac{n}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot \left(t - \frac{l\_m}{Om} \cdot \left(2 \cdot l\_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (or (<= n -2.65e-5) (not (<= n 1.3e-78)))
   (sqrt (* (* 2.0 n) (* U (+ t (/ (* U* (* (pow l_m 2.0) (/ n Om))) Om)))))
   (* (sqrt 2.0) (sqrt (* U (* n (- t (* (/ 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 tmp;
	if ((n <= -2.65e-5) || !(n <= 1.3e-78)) {
		tmp = sqrt(((2.0 * n) * (U * (t + ((U_42_ * (pow(l_m, 2.0) * (n / Om))) / Om)))));
	} else {
		tmp = sqrt(2.0) * sqrt((U * (n * (t - ((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) :: tmp
    if ((n <= (-2.65d-5)) .or. (.not. (n <= 1.3d-78))) then
        tmp = sqrt(((2.0d0 * n) * (u * (t + ((u_42 * ((l_m ** 2.0d0) * (n / om))) / om)))))
    else
        tmp = sqrt(2.0d0) * sqrt((u * (n * (t - ((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 tmp;
	if ((n <= -2.65e-5) || !(n <= 1.3e-78)) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((U_42_ * (Math.pow(l_m, 2.0) * (n / Om))) / Om)))));
	} else {
		tmp = Math.sqrt(2.0) * Math.sqrt((U * (n * (t - ((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_):
	tmp = 0
	if (n <= -2.65e-5) or not (n <= 1.3e-78):
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((U_42_ * (math.pow(l_m, 2.0) * (n / Om))) / Om)))))
	else:
		tmp = math.sqrt(2.0) * math.sqrt((U * (n * (t - ((l_m / Om) * (2.0 * l_m))))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if ((n <= -2.65e-5) || !(n <= 1.3e-78))
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(U_42_ * Float64((l_m ^ 2.0) * Float64(n / Om))) / Om)))));
	else
		tmp = Float64(sqrt(2.0) * sqrt(Float64(U * Float64(n * Float64(t - 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_)
	tmp = 0.0;
	if ((n <= -2.65e-5) || ~((n <= 1.3e-78)))
		tmp = sqrt(((2.0 * n) * (U * (t + ((U_42_ * ((l_m ^ 2.0) * (n / Om))) / Om)))));
	else
		tmp = sqrt(2.0) * sqrt((U * (n * (t - ((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$42$_] := If[Or[LessEqual[n, -2.65e-5], N[Not[LessEqual[n, 1.3e-78]], $MachinePrecision]], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(U$42$ * N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(n / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(U * N[(n * N[(t - N[(N[(l$95$m / Om), $MachinePrecision] * N[(2.0 * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2.65e-5 or 1.3000000000000001e-78 < n

   1. Initial program 44.1%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in Om around -inf 38.0%

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

   6. Taylor expanded in U around 0 49.2%

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

      1. mul-1-neg 49.2%

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

      2. fma-define 49.2%

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

      3. associate-/l* 49.2%

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

      4. associate-/l* 51.5%

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

   8. Simplified 51.5%

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

   9. Taylor expanded in U* around inf 55.3%

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

       1. mul-1-neg 55.3%

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

       2. associate-/l* 55.3%

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

       3. associate-*r/ 57.5%

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

   11. Simplified 57.5%

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

   if -2.65e-5 < n < 1.3000000000000001e-78

   1. Initial program 50.1%

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

   3. Simplified 57.0%

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

   5. Taylor expanded in n around 0 52.2%

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

      1. metadata-eval 52.2%

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

      2. cancel-sign-sub-inv 52.2%

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

      3. associate-*r/ 52.2%

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

   7. Simplified 52.2%

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

      1. associate-/l* 52.2%

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

      2. unpow2 52.2%

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

      3. associate-*r/ 58.7%

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

      4. associate-*r* 58.7%

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

   9. Applied egg-rr 58.7%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.65 \cdot 10^{-5} \lor \neg \left(n \leq 1.3 \cdot 10^{-78}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \frac{U* \cdot \left({\ell}^{2} \cdot \frac{n}{Om}\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;Om \leq -1.7 \cdot 10^{+62} \lor \neg \left(Om \leq 4 \cdot 10^{+84}\right):\\ \;\;\;\;\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot \left(t - \frac{l\_m}{Om} \cdot \left(2 \cdot l\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\frac{U \cdot \left(\left(l\_m \cdot l\_m\right) \cdot \left(n \cdot \left(U* - U\right)\right)\right)}{Om} - 2 \cdot \left(U \cdot \left(l\_m \cdot l\_m\right)\right)}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (or (<= Om -1.7e+62) (not (<= Om 4e+84)))
   (* (sqrt 2.0) (sqrt (* U (* n (- t (* (/ l_m Om) (* 2.0 l_m)))))))
   (sqrt
    (*
     (* 2.0 n)
     (+
      (* U t)
      (/
       (-
        (/ (* U (* (* l_m l_m) (* n (- U* U)))) Om)
        (* 2.0 (* U (* l_m l_m))))
       Om))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((Om <= -1.7e+62) || !(Om <= 4e+84)) {
		tmp = sqrt(2.0) * sqrt((U * (n * (t - ((l_m / Om) * (2.0 * l_m))))));
	} else {
		tmp = sqrt(((2.0 * n) * ((U * t) + ((((U * ((l_m * l_m) * (n * (U_42_ - U)))) / Om) - (2.0 * (U * (l_m * l_m)))) / Om))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((om <= (-1.7d+62)) .or. (.not. (om <= 4d+84))) then
        tmp = sqrt(2.0d0) * sqrt((u * (n * (t - ((l_m / om) * (2.0d0 * l_m))))))
    else
        tmp = sqrt(((2.0d0 * n) * ((u * t) + ((((u * ((l_m * l_m) * (n * (u_42 - u)))) / om) - (2.0d0 * (u * (l_m * l_m)))) / om))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((Om <= -1.7e+62) || !(Om <= 4e+84)) {
		tmp = Math.sqrt(2.0) * Math.sqrt((U * (n * (t - ((l_m / Om) * (2.0 * l_m))))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * ((U * t) + ((((U * ((l_m * l_m) * (n * (U_42_ - U)))) / Om) - (2.0 * (U * (l_m * l_m)))) / Om))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if (Om <= -1.7e+62) or not (Om <= 4e+84):
		tmp = math.sqrt(2.0) * math.sqrt((U * (n * (t - ((l_m / Om) * (2.0 * l_m))))))
	else:
		tmp = math.sqrt(((2.0 * n) * ((U * t) + ((((U * ((l_m * l_m) * (n * (U_42_ - U)))) / Om) - (2.0 * (U * (l_m * l_m)))) / Om))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if ((Om <= -1.7e+62) || !(Om <= 4e+84))
		tmp = Float64(sqrt(2.0) * sqrt(Float64(U * Float64(n * Float64(t - Float64(Float64(l_m / Om) * Float64(2.0 * l_m)))))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(Float64(U * t) + Float64(Float64(Float64(Float64(U * Float64(Float64(l_m * l_m) * Float64(n * Float64(U_42_ - U)))) / Om) - Float64(2.0 * Float64(U * Float64(l_m * l_m)))) / Om))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if ((Om <= -1.7e+62) || ~((Om <= 4e+84)))
		tmp = sqrt(2.0) * sqrt((U * (n * (t - ((l_m / Om) * (2.0 * l_m))))));
	else
		tmp = sqrt(((2.0 * n) * ((U * t) + ((((U * ((l_m * l_m) * (n * (U_42_ - U)))) / Om) - (2.0 * (U * (l_m * l_m)))) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[Or[LessEqual[Om, -1.7e+62], N[Not[LessEqual[Om, 4e+84]], $MachinePrecision]], N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(U * N[(n * N[(t - N[(N[(l$95$m / Om), $MachinePrecision] * N[(2.0 * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(N[(U * t), $MachinePrecision] + N[(N[(N[(N[(U * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - N[(2.0 * N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -1.70000000000000007e62 or 4.00000000000000023e84 < Om

   1. Initial program 47.6%

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

   3. Simplified 55.0%

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

   5. Taylor expanded in n around 0 47.4%

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

      1. metadata-eval 47.4%

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

      2. cancel-sign-sub-inv 47.4%

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

      3. associate-*r/ 47.4%

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

   7. Simplified 47.4%

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

      1. associate-/l* 47.4%

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

      2. unpow2 47.4%

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

      3. associate-*r/ 55.4%

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

      4. associate-*r* 55.4%

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

   9. Applied egg-rr 55.4%

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

   if -1.70000000000000007e62 < Om < 4.00000000000000023e84

   1. Initial program 46.9%

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

   3. Simplified 51.7%

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

   5. Taylor expanded in Om around -inf 52.6%

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

      1. unpow2 42.9%

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

   7. Applied egg-rr 52.6%

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

      1. unpow2 42.9%

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

   9. Applied egg-rr 52.6%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1.7 \cdot 10^{+62} \lor \neg \left(Om \leq 4 \cdot 10^{+84}\right):\\ \;\;\;\;\sqrt{2} \cdot \sqrt{U \cdot \left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U* - U\right)\right)\right)}{Om} - 2 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(l\_m \cdot l\_m\right)\\ \mathbf{if}\;l\_m \leq 3.9 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t + -2 \cdot \frac{t\_1}{Om}\right)\right)}\\ \mathbf{elif}\;l\_m \leq 8.4 \cdot 10^{+136}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\frac{U \cdot \left(\left(l\_m \cdot l\_m\right) \cdot \left(n \cdot \left(U* - U\right)\right)\right)}{Om} - 2 \cdot t\_1}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \frac{n \cdot \left(U \cdot {l\_m}^{2}\right)}{Om}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* l_m l_m))))
   (if (<= l_m 3.9e-17)
     (sqrt (* 2.0 (* n (+ (* U t) (* -2.0 (/ t_1 Om))))))
     (if (<= l_m 8.4e+136)
       (sqrt
        (*
         (* 2.0 n)
         (+
          (* U t)
          (/ (- (/ (* U (* (* l_m l_m) (* n (- U* U)))) Om) (* 2.0 t_1)) Om))))
       (pow (* -4.0 (/ (* n (* U (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 = U * (l_m * l_m);
	double tmp;
	if (l_m <= 3.9e-17) {
		tmp = sqrt((2.0 * (n * ((U * t) + (-2.0 * (t_1 / Om))))));
	} else if (l_m <= 8.4e+136) {
		tmp = sqrt(((2.0 * n) * ((U * t) + ((((U * ((l_m * l_m) * (n * (U_42_ - U)))) / Om) - (2.0 * t_1)) / Om))));
	} else {
		tmp = pow((-4.0 * ((n * (U * pow(l_m, 2.0))) / Om)), 0.5);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = u * (l_m * l_m)
    if (l_m <= 3.9d-17) then
        tmp = sqrt((2.0d0 * (n * ((u * t) + ((-2.0d0) * (t_1 / om))))))
    else if (l_m <= 8.4d+136) then
        tmp = sqrt(((2.0d0 * n) * ((u * t) + ((((u * ((l_m * l_m) * (n * (u_42 - u)))) / om) - (2.0d0 * t_1)) / om))))
    else
        tmp = ((-4.0d0) * ((n * (u * (l_m ** 2.0d0))) / om)) ** 0.5d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (l_m * l_m);
	double tmp;
	if (l_m <= 3.9e-17) {
		tmp = Math.sqrt((2.0 * (n * ((U * t) + (-2.0 * (t_1 / Om))))));
	} else if (l_m <= 8.4e+136) {
		tmp = Math.sqrt(((2.0 * n) * ((U * t) + ((((U * ((l_m * l_m) * (n * (U_42_ - U)))) / Om) - (2.0 * t_1)) / Om))));
	} else {
		tmp = Math.pow((-4.0 * ((n * (U * 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 = U * (l_m * l_m)
	tmp = 0
	if l_m <= 3.9e-17:
		tmp = math.sqrt((2.0 * (n * ((U * t) + (-2.0 * (t_1 / Om))))))
	elif l_m <= 8.4e+136:
		tmp = math.sqrt(((2.0 * n) * ((U * t) + ((((U * ((l_m * l_m) * (n * (U_42_ - U)))) / Om) - (2.0 * t_1)) / Om))))
	else:
		tmp = math.pow((-4.0 * ((n * (U * 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(U * Float64(l_m * l_m))
	tmp = 0.0
	if (l_m <= 3.9e-17)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(Float64(U * t) + Float64(-2.0 * Float64(t_1 / Om))))));
	elseif (l_m <= 8.4e+136)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(Float64(U * t) + Float64(Float64(Float64(Float64(U * Float64(Float64(l_m * l_m) * Float64(n * Float64(U_42_ - U)))) / Om) - Float64(2.0 * t_1)) / Om))));
	else
		tmp = Float64(-4.0 * Float64(Float64(n * Float64(U * (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 = U * (l_m * l_m);
	tmp = 0.0;
	if (l_m <= 3.9e-17)
		tmp = sqrt((2.0 * (n * ((U * t) + (-2.0 * (t_1 / Om))))));
	elseif (l_m <= 8.4e+136)
		tmp = sqrt(((2.0 * n) * ((U * t) + ((((U * ((l_m * l_m) * (n * (U_42_ - U)))) / Om) - (2.0 * t_1)) / Om))));
	else
		tmp = (-4.0 * ((n * (U * (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[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l$95$m, 3.9e-17], N[Sqrt[N[(2.0 * N[(n * N[(N[(U * t), $MachinePrecision] + N[(-2.0 * N[(t$95$1 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 8.4e+136], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(N[(U * t), $MachinePrecision] + N[(N[(N[(N[(U * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(-4.0 * N[(N[(n * N[(U * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.89999999999999989e-17

   1. Initial program 52.3%

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

   3. Simplified 58.1%

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

   5. Taylor expanded in Om around inf 50.6%

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

      1. unpow2 50.6%

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

   7. Applied egg-rr 50.6%

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

   if 3.89999999999999989e-17 < l < 8.3999999999999996e136

   1. Initial program 44.6%

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

   3. Simplified 48.7%

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

   5. Taylor expanded in Om around -inf 49.0%

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

      1. unpow2 29.1%

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

   7. Applied egg-rr 49.0%

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

      1. unpow2 29.1%

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

   9. Applied egg-rr 49.0%

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

   if 8.3999999999999996e136 < l

   1. Initial program 21.0%

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

   3. Simplified 29.9%

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

   5. Taylor expanded in Om around inf 17.1%

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

   6. Taylor expanded in l around inf 19.8%

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

      1. pow1/2 29.1%

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

      2. associate-*r* 29.1%

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

      3. metadata-eval 29.1%

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

      4. associate-*r* 29.4%

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

   8. Applied egg-rr 29.4%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.9 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t + -2 \cdot \frac{U \cdot \left(\ell \cdot \ell\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 8.4 \cdot 10^{+136}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U* - U\right)\right)\right)}{Om} - 2 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \frac{n \cdot \left(U \cdot {\ell}^{2}\right)}{Om}\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 45.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(l\_m \cdot l\_m\right)\\ \mathbf{if}\;l\_m \leq 6 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t + -2 \cdot \frac{t\_1}{Om}\right)\right)}\\ \mathbf{elif}\;l\_m \leq 6.1 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\frac{U \cdot \left(\left(l\_m \cdot l\_m\right) \cdot \left(n \cdot \left(U* - U\right)\right)\right)}{Om} - 2 \cdot t\_1}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \left(\left(n \cdot \frac{\sqrt{2}}{Om}\right) \cdot \sqrt{U \cdot U*}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* l_m l_m))))
   (if (<= l_m 6e-18)
     (sqrt (* 2.0 (* n (+ (* U t) (* -2.0 (/ t_1 Om))))))
     (if (<= l_m 6.1e+153)
       (sqrt
        (*
         (* 2.0 n)
         (+
          (* U t)
          (/ (- (/ (* U (* (* l_m l_m) (* n (- U* U)))) Om) (* 2.0 t_1)) Om))))
       (* l_m (* (* n (/ (sqrt 2.0) Om)) (sqrt (* U U*))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (l_m * l_m);
	double tmp;
	if (l_m <= 6e-18) {
		tmp = sqrt((2.0 * (n * ((U * t) + (-2.0 * (t_1 / Om))))));
	} else if (l_m <= 6.1e+153) {
		tmp = sqrt(((2.0 * n) * ((U * t) + ((((U * ((l_m * l_m) * (n * (U_42_ - U)))) / Om) - (2.0 * t_1)) / Om))));
	} else {
		tmp = l_m * ((n * (sqrt(2.0) / Om)) * sqrt((U * U_42_)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = u * (l_m * l_m)
    if (l_m <= 6d-18) then
        tmp = sqrt((2.0d0 * (n * ((u * t) + ((-2.0d0) * (t_1 / om))))))
    else if (l_m <= 6.1d+153) then
        tmp = sqrt(((2.0d0 * n) * ((u * t) + ((((u * ((l_m * l_m) * (n * (u_42 - u)))) / om) - (2.0d0 * t_1)) / om))))
    else
        tmp = l_m * ((n * (sqrt(2.0d0) / om)) * sqrt((u * u_42)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (l_m * l_m);
	double tmp;
	if (l_m <= 6e-18) {
		tmp = Math.sqrt((2.0 * (n * ((U * t) + (-2.0 * (t_1 / Om))))));
	} else if (l_m <= 6.1e+153) {
		tmp = Math.sqrt(((2.0 * n) * ((U * t) + ((((U * ((l_m * l_m) * (n * (U_42_ - U)))) / Om) - (2.0 * t_1)) / Om))));
	} else {
		tmp = l_m * ((n * (Math.sqrt(2.0) / Om)) * Math.sqrt((U * U_42_)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = U * (l_m * l_m)
	tmp = 0
	if l_m <= 6e-18:
		tmp = math.sqrt((2.0 * (n * ((U * t) + (-2.0 * (t_1 / Om))))))
	elif l_m <= 6.1e+153:
		tmp = math.sqrt(((2.0 * n) * ((U * t) + ((((U * ((l_m * l_m) * (n * (U_42_ - U)))) / Om) - (2.0 * t_1)) / Om))))
	else:
		tmp = l_m * ((n * (math.sqrt(2.0) / Om)) * math.sqrt((U * U_42_)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(l_m * l_m))
	tmp = 0.0
	if (l_m <= 6e-18)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(Float64(U * t) + Float64(-2.0 * Float64(t_1 / Om))))));
	elseif (l_m <= 6.1e+153)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(Float64(U * t) + Float64(Float64(Float64(Float64(U * Float64(Float64(l_m * l_m) * Float64(n * Float64(U_42_ - U)))) / Om) - Float64(2.0 * t_1)) / Om))));
	else
		tmp = Float64(l_m * Float64(Float64(n * Float64(sqrt(2.0) / Om)) * sqrt(Float64(U * U_42_))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = U * (l_m * l_m);
	tmp = 0.0;
	if (l_m <= 6e-18)
		tmp = sqrt((2.0 * (n * ((U * t) + (-2.0 * (t_1 / Om))))));
	elseif (l_m <= 6.1e+153)
		tmp = sqrt(((2.0 * n) * ((U * t) + ((((U * ((l_m * l_m) * (n * (U_42_ - U)))) / Om) - (2.0 * t_1)) / Om))));
	else
		tmp = l_m * ((n * (sqrt(2.0) / Om)) * sqrt((U * U_42_)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l$95$m, 6e-18], N[Sqrt[N[(2.0 * N[(n * N[(N[(U * t), $MachinePrecision] + N[(-2.0 * N[(t$95$1 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 6.1e+153], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(N[(U * t), $MachinePrecision] + N[(N[(N[(N[(U * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[(N[(n * N[(N[Sqrt[2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 5.99999999999999966e-18

   1. Initial program 52.3%

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

   3. Simplified 58.1%

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

   5. Taylor expanded in Om around inf 50.6%

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

      1. unpow2 50.6%

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

   7. Applied egg-rr 50.6%

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

   if 5.99999999999999966e-18 < l < 6.0999999999999998e153

   1. Initial program 47.2%

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

   3. Simplified 44.4%

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

   5. Taylor expanded in Om around -inf 44.6%

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

      1. unpow2 28.0%

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

   7. Applied egg-rr 44.6%

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

      1. unpow2 28.0%

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

   9. Applied egg-rr 44.6%

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

   if 6.0999999999999998e153 < l

   1. Initial program 14.4%

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

   3. Simplified 31.3%

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

   5. Taylor expanded in Om around -inf 14.6%

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

   6. Taylor expanded in U around 0 15.2%

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

      1. mul-1-neg 15.2%

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

      2. fma-define 15.2%

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

      3. associate-/l* 15.2%

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

      4. associate-/l* 15.1%

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

   8. Simplified 15.1%

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

   9. Taylor expanded in n around inf 24.8%

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

       1. associate-*r/ 31.0%

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

       2. associate-*l* 34.2%

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

       3. associate-/l* 34.2%

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

   11. Simplified 34.2%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t + -2 \cdot \frac{U \cdot \left(\ell \cdot \ell\right)}{Om}\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.1 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U* - U\right)\right)\right)}{Om} - 2 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\left(n \cdot \frac{\sqrt{2}}{Om}\right) \cdot \sqrt{U \cdot U*}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq -2.85 \cdot 10^{+76}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{+203}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\frac{U \cdot \left(\left(l\_m \cdot l\_m\right) \cdot \left(n \cdot \left(U* - U\right)\right)\right)}{Om} - 2 \cdot \left(U \cdot \left(l\_m \cdot l\_m\right)\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t -2.85e+76)
   (pow (* 2.0 (* U (* n t))) 0.5)
   (if (<= t 6.1e+203)
     (sqrt
      (*
       (* 2.0 n)
       (+
        (* U t)
        (/
         (-
          (/ (* U (* (* l_m l_m) (* n (- U* U)))) Om)
          (* 2.0 (* U (* l_m l_m))))
         Om))))
     (* (sqrt (* n (* 2.0 U))) (sqrt t)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= -2.85e+76) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else if (t <= 6.1e+203) {
		tmp = sqrt(((2.0 * n) * ((U * t) + ((((U * ((l_m * l_m) * (n * (U_42_ - U)))) / Om) - (2.0 * (U * (l_m * l_m)))) / Om))));
	} else {
		tmp = sqrt((n * (2.0 * U))) * sqrt(t);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= (-2.85d+76)) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else if (t <= 6.1d+203) then
        tmp = sqrt(((2.0d0 * n) * ((u * t) + ((((u * ((l_m * l_m) * (n * (u_42 - u)))) / om) - (2.0d0 * (u * (l_m * l_m)))) / om))))
    else
        tmp = sqrt((n * (2.0d0 * u))) * sqrt(t)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= -2.85e+76) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else if (t <= 6.1e+203) {
		tmp = Math.sqrt(((2.0 * n) * ((U * t) + ((((U * ((l_m * l_m) * (n * (U_42_ - U)))) / Om) - (2.0 * (U * (l_m * l_m)))) / Om))));
	} else {
		tmp = Math.sqrt((n * (2.0 * U))) * Math.sqrt(t);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if t <= -2.85e+76:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	elif t <= 6.1e+203:
		tmp = math.sqrt(((2.0 * n) * ((U * t) + ((((U * ((l_m * l_m) * (n * (U_42_ - U)))) / Om) - (2.0 * (U * (l_m * l_m)))) / Om))))
	else:
		tmp = math.sqrt((n * (2.0 * U))) * math.sqrt(t)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= -2.85e+76)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	elseif (t <= 6.1e+203)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(Float64(U * t) + Float64(Float64(Float64(Float64(U * Float64(Float64(l_m * l_m) * Float64(n * Float64(U_42_ - U)))) / Om) - Float64(2.0 * Float64(U * Float64(l_m * l_m)))) / Om))));
	else
		tmp = Float64(sqrt(Float64(n * Float64(2.0 * U))) * sqrt(t));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (t <= -2.85e+76)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	elseif (t <= 6.1e+203)
		tmp = sqrt(((2.0 * n) * ((U * t) + ((((U * ((l_m * l_m) * (n * (U_42_ - U)))) / Om) - (2.0 * (U * (l_m * l_m)))) / Om))));
	else
		tmp = sqrt((n * (2.0 * U))) * sqrt(t);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[t, -2.85e+76], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[t, 6.1e+203], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(N[(U * t), $MachinePrecision] + N[(N[(N[(N[(U * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - N[(2.0 * N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.85000000000000002e76

   1. Initial program 56.2%

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

   3. Simplified 49.3%

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

   5. Taylor expanded in l around 0 56.7%

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

      1. associate-*r* 56.7%

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

   7. Simplified 56.7%

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

      1. pow1/2 59.2%

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

      2. associate-*l* 59.2%

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

   9. Applied egg-rr 59.2%

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

   if -2.85000000000000002e76 < t < 6.10000000000000014e203

   1. Initial program 46.8%

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

   3. Simplified 55.3%

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

   5. Taylor expanded in Om around -inf 48.4%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(-1 \cdot \frac{2 \cdot \left(U \cdot {\ell}^{2}\right) + \frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{Om}}{Om} + U \cdot t\right)}} \]
   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 48.4%

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

      1. unpow2 44.8%

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

   9. Applied egg-rr 48.4%

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

   if 6.10000000000000014e203 < t

   1. Initial program 34.2%

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

   3. Simplified 43.2%

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

   5. Taylor expanded in l around 0 47.3%

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

      1. associate-*r* 47.3%

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

   7. Simplified 47.3%

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

      1. pow1/2 47.6%

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

      2. associate-*r* 43.0%

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

      3. unpow-prod-down 51.4%

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

      4. pow1/2 51.4%

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

   9. Applied egg-rr 51.4%

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

       1. unpow1/2 51.4%

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

   11. Simplified 51.4%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.85 \cdot 10^{+76}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{+203}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U* - U\right)\right)\right)}{Om} - 2 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 45.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+76} \lor \neg \left(t \leq 4.5 \cdot 10^{+184}\right):\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\frac{U \cdot \left(\left(l\_m \cdot l\_m\right) \cdot \left(n \cdot \left(U* - U\right)\right)\right)}{Om} - 2 \cdot \left(U \cdot \left(l\_m \cdot l\_m\right)\right)}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (or (<= t -5e+76) (not (<= t 4.5e+184)))
   (pow (* 2.0 (* U (* n t))) 0.5)
   (sqrt
    (*
     (* 2.0 n)
     (+
      (* U t)
      (/
       (-
        (/ (* U (* (* l_m l_m) (* n (- U* U)))) Om)
        (* 2.0 (* U (* l_m l_m))))
       Om))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((t <= -5e+76) || !(t <= 4.5e+184)) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = sqrt(((2.0 * n) * ((U * t) + ((((U * ((l_m * l_m) * (n * (U_42_ - U)))) / Om) - (2.0 * (U * (l_m * l_m)))) / Om))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((t <= (-5d+76)) .or. (.not. (t <= 4.5d+184))) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else
        tmp = sqrt(((2.0d0 * n) * ((u * t) + ((((u * ((l_m * l_m) * (n * (u_42 - u)))) / om) - (2.0d0 * (u * (l_m * l_m)))) / om))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((t <= -5e+76) || !(t <= 4.5e+184)) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = Math.sqrt(((2.0 * n) * ((U * t) + ((((U * ((l_m * l_m) * (n * (U_42_ - U)))) / Om) - (2.0 * (U * (l_m * l_m)))) / Om))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if (t <= -5e+76) or not (t <= 4.5e+184):
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	else:
		tmp = math.sqrt(((2.0 * n) * ((U * t) + ((((U * ((l_m * l_m) * (n * (U_42_ - U)))) / Om) - (2.0 * (U * (l_m * l_m)))) / Om))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if ((t <= -5e+76) || !(t <= 4.5e+184))
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(Float64(U * t) + Float64(Float64(Float64(Float64(U * Float64(Float64(l_m * l_m) * Float64(n * Float64(U_42_ - U)))) / Om) - Float64(2.0 * Float64(U * Float64(l_m * l_m)))) / Om))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if ((t <= -5e+76) || ~((t <= 4.5e+184)))
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	else
		tmp = sqrt(((2.0 * n) * ((U * t) + ((((U * ((l_m * l_m) * (n * (U_42_ - U)))) / Om) - (2.0 * (U * (l_m * l_m)))) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[Or[LessEqual[t, -5e+76], N[Not[LessEqual[t, 4.5e+184]], $MachinePrecision]], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(N[(U * t), $MachinePrecision] + N[(N[(N[(N[(U * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - N[(2.0 * N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.99999999999999991e76 or 4.50000000000000036e184 < t

   1. Initial program 44.1%

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

   3. Simplified 46.6%

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

   5. Taylor expanded in l around 0 54.5%

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

      1. associate-*r* 54.5%

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

   7. Simplified 54.5%

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

      1. pow1/2 55.9%

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

      2. associate-*l* 55.9%

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

   9. Applied egg-rr 55.9%

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

   if -4.99999999999999991e76 < t < 4.50000000000000036e184

   1. Initial program 48.6%

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

   3. Simplified 56.1%

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

   5. Taylor expanded in Om around -inf 47.6%

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

      1. unpow2 44.3%

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

   7. Applied egg-rr 47.6%

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

      1. unpow2 44.3%

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

   9. Applied egg-rr 47.6%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+76} \lor \neg \left(t \leq 4.5 \cdot 10^{+184}\right):\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t + \frac{\frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U* - U\right)\right)\right)}{Om} - 2 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{+227}:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+188}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t + -2 \cdot \frac{U \cdot \left(l\_m \cdot l\_m\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t -1.75e+227)
   (sqrt (* (* n t) (* 2.0 U)))
   (if (<= t 2.1e+188)
     (sqrt (* 2.0 (* n (+ (* U t) (* -2.0 (/ (* U (* l_m l_m)) Om))))))
     (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 (t <= -1.75e+227) {
		tmp = sqrt(((n * t) * (2.0 * U)));
	} else if (t <= 2.1e+188) {
		tmp = sqrt((2.0 * (n * ((U * t) + (-2.0 * ((U * (l_m * l_m)) / Om))))));
	} 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 (t <= (-1.75d+227)) then
        tmp = sqrt(((n * t) * (2.0d0 * u)))
    else if (t <= 2.1d+188) then
        tmp = sqrt((2.0d0 * (n * ((u * t) + ((-2.0d0) * ((u * (l_m * l_m)) / om))))))
    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 (t <= -1.75e+227) {
		tmp = Math.sqrt(((n * t) * (2.0 * U)));
	} else if (t <= 2.1e+188) {
		tmp = Math.sqrt((2.0 * (n * ((U * t) + (-2.0 * ((U * (l_m * l_m)) / Om))))));
	} 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 t <= -1.75e+227:
		tmp = math.sqrt(((n * t) * (2.0 * U)))
	elif t <= 2.1e+188:
		tmp = math.sqrt((2.0 * (n * ((U * t) + (-2.0 * ((U * (l_m * l_m)) / Om))))))
	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 (t <= -1.75e+227)
		tmp = sqrt(Float64(Float64(n * t) * Float64(2.0 * U)));
	elseif (t <= 2.1e+188)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(Float64(U * t) + Float64(-2.0 * Float64(Float64(U * Float64(l_m * l_m)) / Om))))));
	else
		tmp = Float64(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 (t <= -1.75e+227)
		tmp = sqrt(((n * t) * (2.0 * U)));
	elseif (t <= 2.1e+188)
		tmp = sqrt((2.0 * (n * ((U * t) + (-2.0 * ((U * (l_m * l_m)) / Om))))));
	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[t, -1.75e+227], N[Sqrt[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 2.1e+188], N[Sqrt[N[(2.0 * N[(n * N[(N[(U * t), $MachinePrecision] + N[(-2.0 * N[(N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.75e227

   1. Initial program 54.4%

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

   3. Simplified 35.1%

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

   5. Taylor expanded in l around 0 74.9%

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

      1. associate-*r* 74.9%

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

   7. Simplified 74.9%

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

   if -1.75e227 < t < 2.09999999999999986e188

   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 55.9%

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

   5. Taylor expanded in Om around inf 45.4%

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

      1. unpow2 45.4%

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

   7. Applied egg-rr 45.4%

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

   if 2.09999999999999986e188 < t

   1. Initial program 31.7%

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

   3. Simplified 44.2%

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

   5. Taylor expanded in l around 0 53.4%

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

      1. associate-*r* 53.4%

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

   7. Simplified 53.4%

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

      1. pow1/2 53.5%

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

      2. associate-*l* 53.5%

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

   9. Applied egg-rr 53.5%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{+227}:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+188}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t + -2 \cdot \frac{U \cdot \left(\ell \cdot \ell\right)}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 37.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 5.8 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 5.8e-156)
   (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 (l_m <= 5.8e-156) {
		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 (l_m <= 5.8d-156) 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 (l_m <= 5.8e-156) {
		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 l_m <= 5.8e-156:
		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 (l_m <= 5.8e-156)
		tmp = sqrt(Float64(Float64(2.0 * 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 (l_m <= 5.8e-156)
		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[l$95$m, 5.8e-156], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $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 l < 5.80000000000000041e-156

   1. Initial program 50.3%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in t around inf 46.7%

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

   if 5.80000000000000041e-156 < l

   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 45.5%

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

   5. Taylor expanded in l around 0 25.1%

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

      1. associate-*r* 25.1%

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

   7. Simplified 25.1%

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

      1. pow1/2 29.7%

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

      2. associate-*l* 29.7%

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

   9. Applied egg-rr 29.7%

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

Alternative 16: 35.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 8.6 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 8.6e-156)
   (sqrt (* (* 2.0 n) (* U t)))
   (sqrt (* (* n t) (* 2.0 U)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 8.6e-156) {
		tmp = sqrt(((2.0 * n) * (U * t)));
	} else {
		tmp = sqrt(((n * t) * (2.0 * U)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l_m <= 8.6d-156) then
        tmp = sqrt(((2.0d0 * n) * (u * t)))
    else
        tmp = sqrt(((n * t) * (2.0d0 * u)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 8.6e-156) {
		tmp = Math.sqrt(((2.0 * n) * (U * t)));
	} else {
		tmp = Math.sqrt(((n * t) * (2.0 * U)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 8.6e-156:
		tmp = math.sqrt(((2.0 * n) * (U * t)))
	else:
		tmp = math.sqrt(((n * t) * (2.0 * U)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 8.6e-156)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t)));
	else
		tmp = sqrt(Float64(Float64(n * t) * Float64(2.0 * U)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (l_m <= 8.6e-156)
		tmp = sqrt(((2.0 * n) * (U * t)));
	else
		tmp = sqrt(((n * t) * (2.0 * U)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 8.6e-156], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 8.59999999999999954e-156

   1. Initial program 50.3%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in t around inf 46.7%

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

   if 8.59999999999999954e-156 < l

   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 45.5%

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

   5. Taylor expanded in l around 0 25.1%

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

      1. associate-*r* 25.1%

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

   7. Simplified 25.1%

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

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8.6 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 35.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.3%

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

3. Simplified 53.3%

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

5. Taylor expanded in l around 0 39.7%

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

   1. associate-*r* 39.7%

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

7. Simplified 39.7%

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

8. Final simplification 39.7%

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

Alternative 18: 35.8% accurate, 2.1× speedup

Math

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

FPCore

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

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 * (t * (n * U))));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.3%

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

3. Simplified 53.3%

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

5. Taylor expanded in Om around -inf 44.5%

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

6. Taylor expanded in U around 0 50.0%

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

   1. mul-1-neg 50.0%

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

   2. fma-define 50.0%

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

   3. associate-/l* 50.0%

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

   4. associate-/l* 51.2%

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

8. Simplified 51.2%

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

9. Taylor expanded in t around inf 39.7%

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

    1. associate-*r* 35.5%

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

11. Simplified 35.5%

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

12. Final simplification 35.5%

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

Reproduce

herbie shell --seed 2024149 
(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*))))))