Toniolo and Linder, Equation (13)

Percentage Accurate: 50.1% → 62.4%

Time: 32.9s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 62.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - \ell \cdot \left(2 \cdot \frac{\ell}{Om}\right)\\ t_2 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := \left(n \cdot t\_2\right) \cdot \left(U* - U\right)\\ t_4 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t\_3\right)}\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t\_1 + t\_3\right)\right)}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+152}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t\_1 + n \cdot \left(t\_2 \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\left|2 \cdot \frac{n \cdot \left(-2 \cdot \left(U \cdot {\ell}^{2}\right)\right)}{Om}\right|\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (- t (* l (* 2.0 (/ l Om)))))
        (t_2 (pow (/ l Om) 2.0))
        (t_3 (* (* n t_2) (- U* U)))
        (t_4 (sqrt (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l l) Om))) t_3)))))
   (if (<= t_4 0.0)
     (sqrt (* (* 2.0 n) (* U (+ t_1 t_3))))
     (if (<= t_4 2e+152)
       t_4
       (if (<= t_4 INFINITY)
         (sqrt (* (* 2.0 n) (* U (+ t_1 (* n (* t_2 (- U* U)))))))
         (exp
          (*
           (log (fabs (* 2.0 (/ (* n (* -2.0 (* U (pow l 2.0)))) Om))))
           0.5)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t - (l * (2.0 * (l / Om)));
	double t_2 = pow((l / Om), 2.0);
	double t_3 = (n * t_2) * (U_42_ - U);
	double t_4 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_3)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt(((2.0 * n) * (U * (t_1 + t_3))));
	} else if (t_4 <= 2e+152) {
		tmp = t_4;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * n) * (U * (t_1 + (n * (t_2 * (U_42_ - U)))))));
	} else {
		tmp = exp((log(fabs((2.0 * ((n * (-2.0 * (U * pow(l, 2.0)))) / Om)))) * 0.5));
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t - (l * (2.0 * (l / Om)));
	double t_2 = Math.pow((l / Om), 2.0);
	double t_3 = (n * t_2) * (U_42_ - U);
	double t_4 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_3)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t_1 + t_3))));
	} else if (t_4 <= 2e+152) {
		tmp = t_4;
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t_1 + (n * (t_2 * (U_42_ - U)))))));
	} else {
		tmp = Math.exp((Math.log(Math.abs((2.0 * ((n * (-2.0 * (U * Math.pow(l, 2.0)))) / Om)))) * 0.5));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = t - (l * (2.0 * (l / Om)))
	t_2 = math.pow((l / Om), 2.0)
	t_3 = (n * t_2) * (U_42_ - U)
	t_4 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_3)))
	tmp = 0
	if t_4 <= 0.0:
		tmp = math.sqrt(((2.0 * n) * (U * (t_1 + t_3))))
	elif t_4 <= 2e+152:
		tmp = t_4
	elif t_4 <= math.inf:
		tmp = math.sqrt(((2.0 * n) * (U * (t_1 + (n * (t_2 * (U_42_ - U)))))))
	else:
		tmp = math.exp((math.log(math.fabs((2.0 * ((n * (-2.0 * (U * math.pow(l, 2.0)))) / Om)))) * 0.5))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(t - Float64(l * Float64(2.0 * Float64(l / Om))))
	t_2 = Float64(l / Om) ^ 2.0
	t_3 = Float64(Float64(n * t_2) * Float64(U_42_ - U))
	t_4 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_3)))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t_1 + t_3))));
	elseif (t_4 <= 2e+152)
		tmp = t_4;
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t_1 + Float64(n * Float64(t_2 * Float64(U_42_ - U)))))));
	else
		tmp = exp(Float64(log(abs(Float64(2.0 * Float64(Float64(n * Float64(-2.0 * Float64(U * (l ^ 2.0)))) / Om)))) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = t - (l * (2.0 * (l / Om)));
	t_2 = (l / Om) ^ 2.0;
	t_3 = (n * t_2) * (U_42_ - U);
	t_4 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_3)));
	tmp = 0.0;
	if (t_4 <= 0.0)
		tmp = sqrt(((2.0 * n) * (U * (t_1 + t_3))));
	elseif (t_4 <= 2e+152)
		tmp = t_4;
	elseif (t_4 <= Inf)
		tmp = sqrt(((2.0 * n) * (U * (t_1 + (n * (t_2 * (U_42_ - U)))))));
	else
		tmp = exp((log(abs((2.0 * ((n * (-2.0 * (U * (l ^ 2.0)))) / Om)))) * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(t - N[(l * N[(2.0 * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(l / 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[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t$95$1 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, 2e+152], t$95$4, If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t$95$1 + N[(n * N[(t$95$2 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[Log[N[Abs[N[(2.0 * N[(N[(n * N[(-2.0 * N[(U * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 9.7%

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

   3. Simplified 48.6%

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

      1. associate-*r/ 48.6%

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

      2. associate-*l/ 48.6%

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

      3. associate-*r* 48.6%

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

   6. Applied egg-rr 48.6%

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

      1. associate-*r* 51.9%

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

      2. sub-neg 51.9%

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

      3. distribute-lft-in 51.9%

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

   8. Applied egg-rr 51.9%

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

      1. distribute-lft-out 51.9%

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

      2. sub-neg 51.9%

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

      3. *-commutative 51.9%

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

   10. Simplified 51.9%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 2.0000000000000001e152

   1. Initial program 98.6%

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

   if 2.0000000000000001e152 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 33.5%

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

   3. Simplified 35.9%

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

      1. associate-*r/ 35.9%

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

      2. associate-*l/ 39.6%

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

      3. associate-*r* 39.6%

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

   6. Applied egg-rr 39.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in n around 0 8.2%

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

      1. associate-*r* 5.1%

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

      2. *-commutative 5.1%

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

      3. cancel-sign-sub-inv 5.1%

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

      4. metadata-eval 5.1%

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

      5. *-commutative 5.1%

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

      6. associate-*l* 5.1%

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

      7. *-commutative 5.1%

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

   5. Simplified 5.1%

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

      1. pow1/2 35.1%

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

      2. pow-to-exp 4.8%

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

      3. +-commutative 4.8%

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

      4. fma-udef 4.8%

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

      5. *-commutative 4.8%

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

   7. Applied egg-rr 4.8%

      \[\leadsto \color{blue}{e^{\log \left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \mathsf{fma}\left(\frac{{\ell}^{2}}{Om}, -2, t\right)\right) \cdot 0.5}} \]
   8. Step-by-step derivation

      1. pow1 4.8%

         \[\leadsto e^{\log \color{blue}{\left({\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \mathsf{fma}\left(\frac{{\ell}^{2}}{Om}, -2, t\right)\right)}^{1}\right)} \cdot 0.5} \]

      2. metadata-eval 4.8%

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

      3. pow-prod-up 34.8%

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

      4. pow-prod-down 35.1%

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

      5. pow2 35.1%

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

   9. Applied egg-rr 35.1%

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

       1. unpow1/2 35.1%

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

       2. unpow2 35.1%

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

       3. rem-sqrt-square 35.1%

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

       4. associate-*l* 35.1%

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

       5. associate-*l* 39.8%

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

   11. Simplified 39.8%

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

   12. Taylor expanded in l around inf 40.0%

       \[\leadsto e^{\log \left(\left|2 \cdot \color{blue}{\left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}\right|\right) \cdot 0.5} \]
   13. Step-by-step derivation

       1. associate-*r/ 40.0%

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

       2. associate-*r* 40.0%

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

       3. associate-*r* 40.0%

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

   14. Simplified 40.0%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \ell \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \ell \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\left|2 \cdot \frac{n \cdot \left(-2 \cdot \left(U \cdot {\ell}^{2}\right)\right)}{Om}\right|\right) \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - \ell \cdot \left(2 \cdot \frac{\ell}{Om}\right)\\ t_2 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := \left(n \cdot t\_2\right) \cdot \left(U* - U\right)\\ t_4 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t\_3\right)}\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t\_1 + t\_3\right)\right)}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+152}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t\_1 + n \cdot \left(t\_2 \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(\left(n \cdot {\ell}^{2}\right) \cdot \left(\frac{2}{Om} - \frac{n \cdot U*}{{Om}^{2}}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (- t (* l (* 2.0 (/ l Om)))))
        (t_2 (pow (/ l Om) 2.0))
        (t_3 (* (* n t_2) (- U* U)))
        (t_4 (sqrt (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l l) Om))) t_3)))))
   (if (<= t_4 0.0)
     (sqrt (* (* 2.0 n) (* U (+ t_1 t_3))))
     (if (<= t_4 2e+152)
       t_4
       (if (<= t_4 INFINITY)
         (sqrt (* (* 2.0 n) (* U (+ t_1 (* n (* t_2 (- U* U)))))))
         (sqrt
          (*
           -2.0
           (*
            U
            (*
             (* n (pow l 2.0))
             (- (/ 2.0 Om) (/ (* n U*) (pow Om 2.0))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t - (l * (2.0 * (l / Om)));
	double t_2 = pow((l / Om), 2.0);
	double t_3 = (n * t_2) * (U_42_ - U);
	double t_4 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_3)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt(((2.0 * n) * (U * (t_1 + t_3))));
	} else if (t_4 <= 2e+152) {
		tmp = t_4;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * n) * (U * (t_1 + (n * (t_2 * (U_42_ - U)))))));
	} else {
		tmp = sqrt((-2.0 * (U * ((n * pow(l, 2.0)) * ((2.0 / Om) - ((n * U_42_) / pow(Om, 2.0)))))));
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t - (l * (2.0 * (l / Om)));
	double t_2 = Math.pow((l / Om), 2.0);
	double t_3 = (n * t_2) * (U_42_ - U);
	double t_4 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_3)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t_1 + t_3))));
	} else if (t_4 <= 2e+152) {
		tmp = t_4;
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t_1 + (n * (t_2 * (U_42_ - U)))))));
	} else {
		tmp = Math.sqrt((-2.0 * (U * ((n * Math.pow(l, 2.0)) * ((2.0 / Om) - ((n * U_42_) / Math.pow(Om, 2.0)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = t - (l * (2.0 * (l / Om)))
	t_2 = math.pow((l / Om), 2.0)
	t_3 = (n * t_2) * (U_42_ - U)
	t_4 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_3)))
	tmp = 0
	if t_4 <= 0.0:
		tmp = math.sqrt(((2.0 * n) * (U * (t_1 + t_3))))
	elif t_4 <= 2e+152:
		tmp = t_4
	elif t_4 <= math.inf:
		tmp = math.sqrt(((2.0 * n) * (U * (t_1 + (n * (t_2 * (U_42_ - U)))))))
	else:
		tmp = math.sqrt((-2.0 * (U * ((n * math.pow(l, 2.0)) * ((2.0 / Om) - ((n * U_42_) / math.pow(Om, 2.0)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(t - Float64(l * Float64(2.0 * Float64(l / Om))))
	t_2 = Float64(l / Om) ^ 2.0
	t_3 = Float64(Float64(n * t_2) * Float64(U_42_ - U))
	t_4 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_3)))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t_1 + t_3))));
	elseif (t_4 <= 2e+152)
		tmp = t_4;
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t_1 + Float64(n * Float64(t_2 * Float64(U_42_ - U)))))));
	else
		tmp = sqrt(Float64(-2.0 * Float64(U * Float64(Float64(n * (l ^ 2.0)) * Float64(Float64(2.0 / Om) - Float64(Float64(n * U_42_) / (Om ^ 2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = t - (l * (2.0 * (l / Om)));
	t_2 = (l / Om) ^ 2.0;
	t_3 = (n * t_2) * (U_42_ - U);
	t_4 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_3)));
	tmp = 0.0;
	if (t_4 <= 0.0)
		tmp = sqrt(((2.0 * n) * (U * (t_1 + t_3))));
	elseif (t_4 <= 2e+152)
		tmp = t_4;
	elseif (t_4 <= Inf)
		tmp = sqrt(((2.0 * n) * (U * (t_1 + (n * (t_2 * (U_42_ - U)))))));
	else
		tmp = sqrt((-2.0 * (U * ((n * (l ^ 2.0)) * ((2.0 / Om) - ((n * U_42_) / (Om ^ 2.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(t - N[(l * N[(2.0 * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(l / 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[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t$95$1 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, 2e+152], t$95$4, If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t$95$1 + N[(n * N[(t$95$2 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(U * N[(N[(n * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / Om), $MachinePrecision] - N[(N[(n * U$42$), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 9.7%

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

   3. Simplified 48.6%

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

      1. associate-*r/ 48.6%

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

      2. associate-*l/ 48.6%

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

      3. associate-*r* 48.6%

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

   6. Applied egg-rr 48.6%

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

      1. associate-*r* 51.9%

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

      2. sub-neg 51.9%

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

      3. distribute-lft-in 51.9%

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

   8. Applied egg-rr 51.9%

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

      1. distribute-lft-out 51.9%

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

      2. sub-neg 51.9%

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

      3. *-commutative 51.9%

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

   10. Simplified 51.9%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 2.0000000000000001e152

   1. Initial program 98.6%

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

   if 2.0000000000000001e152 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 33.5%

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

   3. Simplified 35.9%

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

      1. associate-*r/ 35.9%

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

      2. associate-*l/ 39.6%

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

      3. associate-*r* 39.6%

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

   6. Applied egg-rr 39.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in l around inf 37.3%

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

      1. associate-*r/ 37.3%

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

      2. metadata-eval 37.3%

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

      3. associate-/l* 36.5%

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

   5. Simplified 36.5%

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

   6. Taylor expanded in U around 0 37.3%

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

      1. associate-*r* 37.5%

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

      2. *-commutative 37.5%

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

      3. +-commutative 37.5%

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

      4. mul-1-neg 37.5%

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

      5. unsub-neg 37.5%

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

      6. associate-*r/ 37.5%

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

      7. metadata-eval 37.5%

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

      8. *-commutative 37.5%

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

   8. Simplified 37.5%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \ell \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \ell \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left(\left(n \cdot {\ell}^{2}\right) \cdot \left(\frac{2}{Om} - \frac{n \cdot U*}{{Om}^{2}}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - \ell \cdot \left(2 \cdot \frac{\ell}{Om}\right)\\ t_2 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := \left(n \cdot t\_2\right) \cdot \left(U* - U\right)\\ t_4 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t\_3\right)}\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t\_1 + t\_3\right)\right)}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+152}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t\_1 + n \cdot \left(t\_2 \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \mathsf{fma}\left(\frac{{\ell}^{2}}{Om}, -2, t\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (- t (* l (* 2.0 (/ l Om)))))
        (t_2 (pow (/ l Om) 2.0))
        (t_3 (* (* n t_2) (- U* U)))
        (t_4 (sqrt (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l l) Om))) t_3)))))
   (if (<= t_4 0.0)
     (sqrt (* (* 2.0 n) (* U (+ t_1 t_3))))
     (if (<= t_4 2e+152)
       t_4
       (if (<= t_4 INFINITY)
         (sqrt (* (* 2.0 n) (* U (+ t_1 (* n (* t_2 (- U* U)))))))
         (pow (* (* 2.0 (* n U)) (fma (/ (pow l 2.0) Om) -2.0 t)) 0.5))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t - (l * (2.0 * (l / Om)));
	double t_2 = pow((l / Om), 2.0);
	double t_3 = (n * t_2) * (U_42_ - U);
	double t_4 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_3)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt(((2.0 * n) * (U * (t_1 + t_3))));
	} else if (t_4 <= 2e+152) {
		tmp = t_4;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * n) * (U * (t_1 + (n * (t_2 * (U_42_ - U)))))));
	} else {
		tmp = pow(((2.0 * (n * U)) * fma((pow(l, 2.0) / Om), -2.0, t)), 0.5);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(t - Float64(l * Float64(2.0 * Float64(l / Om))))
	t_2 = Float64(l / Om) ^ 2.0
	t_3 = Float64(Float64(n * t_2) * Float64(U_42_ - U))
	t_4 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_3)))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t_1 + t_3))));
	elseif (t_4 <= 2e+152)
		tmp = t_4;
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t_1 + Float64(n * Float64(t_2 * Float64(U_42_ - U)))))));
	else
		tmp = Float64(Float64(2.0 * Float64(n * U)) * fma(Float64((l ^ 2.0) / Om), -2.0, t)) ^ 0.5;
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(t - N[(l * N[(2.0 * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(l / 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[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t$95$1 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, 2e+152], t$95$4, If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t$95$1 + N[(n * N[(t$95$2 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 9.7%

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

   3. Simplified 48.6%

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

      1. associate-*r/ 48.6%

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

      2. associate-*l/ 48.6%

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

      3. associate-*r* 48.6%

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

   6. Applied egg-rr 48.6%

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

      1. associate-*r* 51.9%

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

      2. sub-neg 51.9%

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

      3. distribute-lft-in 51.9%

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

   8. Applied egg-rr 51.9%

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

      1. distribute-lft-out 51.9%

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

      2. sub-neg 51.9%

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

      3. *-commutative 51.9%

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

   10. Simplified 51.9%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 2.0000000000000001e152

   1. Initial program 98.6%

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

   if 2.0000000000000001e152 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 33.5%

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

   3. Simplified 35.9%

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

      1. associate-*r/ 35.9%

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

      2. associate-*l/ 39.6%

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

      3. associate-*r* 39.6%

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

   6. Applied egg-rr 39.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in n around 0 8.2%

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

      1. associate-*r* 5.1%

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

      2. *-commutative 5.1%

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

      3. cancel-sign-sub-inv 5.1%

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

      4. metadata-eval 5.1%

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

      5. *-commutative 5.1%

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

      6. associate-*l* 5.1%

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

      7. *-commutative 5.1%

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

   5. Simplified 5.1%

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

      1. pow1/2 35.1%

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

      2. +-commutative 35.1%

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

      3. fma-udef 35.1%

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

      4. *-commutative 35.1%

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

   7. Applied egg-rr 35.1%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \ell \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \ell \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \mathsf{fma}\left(\frac{{\ell}^{2}}{Om}, -2, t\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - \ell \cdot \left(2 \cdot \frac{\ell}{Om}\right)\\ t_2 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := \left(n \cdot t\_2\right) \cdot \left(U* - U\right)\\ t_4 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t\_3\right)}\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t\_1 + t\_3\right)\right)}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+152}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t\_1 + n \cdot \left(t\_2 \cdot \left(U* - U\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (- t (* l (* 2.0 (/ l Om)))))
        (t_2 (pow (/ l Om) 2.0))
        (t_3 (* (* n t_2) (- U* U)))
        (t_4 (sqrt (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l l) Om))) t_3)))))
   (if (<= t_4 0.0)
     (sqrt (* (* 2.0 n) (* U (+ t_1 t_3))))
     (if (<= t_4 2e+152)
       t_4
       (sqrt (* (* 2.0 n) (* U (+ t_1 (* n (* t_2 (- U* U)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t - (l * (2.0 * (l / Om)));
	double t_2 = pow((l / Om), 2.0);
	double t_3 = (n * t_2) * (U_42_ - U);
	double t_4 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_3)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt(((2.0 * n) * (U * (t_1 + t_3))));
	} else if (t_4 <= 2e+152) {
		tmp = t_4;
	} else {
		tmp = sqrt(((2.0 * n) * (U * (t_1 + (n * (t_2 * (U_42_ - U)))))));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = t - (l * (2.0d0 * (l / om)))
    t_2 = (l / om) ** 2.0d0
    t_3 = (n * t_2) * (u_42 - u)
    t_4 = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) + t_3)))
    if (t_4 <= 0.0d0) then
        tmp = sqrt(((2.0d0 * n) * (u * (t_1 + t_3))))
    else if (t_4 <= 2d+152) then
        tmp = t_4
    else
        tmp = sqrt(((2.0d0 * n) * (u * (t_1 + (n * (t_2 * (u_42 - u)))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t - (l * (2.0 * (l / Om)));
	double t_2 = Math.pow((l / Om), 2.0);
	double t_3 = (n * t_2) * (U_42_ - U);
	double t_4 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_3)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t_1 + t_3))));
	} else if (t_4 <= 2e+152) {
		tmp = t_4;
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * (t_1 + (n * (t_2 * (U_42_ - U)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = t - (l * (2.0 * (l / Om)))
	t_2 = math.pow((l / Om), 2.0)
	t_3 = (n * t_2) * (U_42_ - U)
	t_4 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_3)))
	tmp = 0
	if t_4 <= 0.0:
		tmp = math.sqrt(((2.0 * n) * (U * (t_1 + t_3))))
	elif t_4 <= 2e+152:
		tmp = t_4
	else:
		tmp = math.sqrt(((2.0 * n) * (U * (t_1 + (n * (t_2 * (U_42_ - U)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(t - Float64(l * Float64(2.0 * Float64(l / Om))))
	t_2 = Float64(l / Om) ^ 2.0
	t_3 = Float64(Float64(n * t_2) * Float64(U_42_ - U))
	t_4 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_3)))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t_1 + t_3))));
	elseif (t_4 <= 2e+152)
		tmp = t_4;
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t_1 + Float64(n * Float64(t_2 * Float64(U_42_ - U)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = t - (l * (2.0 * (l / Om)));
	t_2 = (l / Om) ^ 2.0;
	t_3 = (n * t_2) * (U_42_ - U);
	t_4 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_3)));
	tmp = 0.0;
	if (t_4 <= 0.0)
		tmp = sqrt(((2.0 * n) * (U * (t_1 + t_3))));
	elseif (t_4 <= 2e+152)
		tmp = t_4;
	else
		tmp = sqrt(((2.0 * n) * (U * (t_1 + (n * (t_2 * (U_42_ - U)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(t - N[(l * N[(2.0 * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(l / 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[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t$95$1 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, 2e+152], t$95$4, N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t$95$1 + N[(n * N[(t$95$2 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 9.7%

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

   3. Simplified 48.6%

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

      1. associate-*r/ 48.6%

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

      2. associate-*l/ 48.6%

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

      3. associate-*r* 48.6%

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

   6. Applied egg-rr 48.6%

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

      1. associate-*r* 51.9%

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

      2. sub-neg 51.9%

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

      3. distribute-lft-in 51.9%

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

   8. Applied egg-rr 51.9%

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

      1. distribute-lft-out 51.9%

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

      2. sub-neg 51.9%

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

      3. *-commutative 51.9%

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

   10. Simplified 51.9%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 2.0000000000000001e152

   1. Initial program 98.6%

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

   if 2.0000000000000001e152 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))

   1. Initial program 20.8%

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

   3. Simplified 23.4%

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

      1. associate-*r/ 23.4%

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

      2. associate-*l/ 29.7%

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

      3. associate-*r* 29.7%

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

   6. Applied egg-rr 29.7%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \ell \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \ell \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt
  (*
   (* 2.0 n)
   (*
    U
    (+ (- t (* l (* 2.0 (/ 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 - (l * (2.0 * (l / Om)))) + (n * (pow((l / Om), 2.0) * (U_42_ - U)))))));
}

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 - (l * (2.0d0 * (l / om)))) + (n * (((l / om) ** 2.0d0) * (u_42 - u)))))))
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 - (l * (2.0 * (l / Om)))) + (n * (Math.pow((l / Om), 2.0) * (U_42_ - U)))))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.2%

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

3. Simplified 52.2%

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

   1. associate-*r/ 52.2%

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

   2. associate-*l/ 55.3%

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

   3. associate-*r* 55.3%

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

6. Applied egg-rr 55.3%

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

7. Final simplification 55.3%

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

Alternative 6: 53.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt
  (*
   (* 2.0 n)
   (*
    U
    (+ (- t (* l (* 2.0 (/ 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 - (l * (2.0 * (l / Om)))) + ((n * pow((l / Om), 2.0)) * (U_42_ - U))))));
}

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 - (l * (2.0d0 * (l / om)))) + ((n * ((l / om) ** 2.0d0)) * (u_42 - u))))))
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 - (l * (2.0 * (l / Om)))) + ((n * Math.pow((l / Om), 2.0)) * (U_42_ - U))))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.2%

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

3. Simplified 52.2%

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

   1. associate-*r/ 52.2%

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

   2. associate-*l/ 55.3%

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

   3. associate-*r* 55.3%

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

6. Applied egg-rr 55.3%

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

   1. associate-*r* 56.0%

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

   2. sub-neg 56.0%

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

   3. distribute-lft-in 48.2%

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

8. Applied egg-rr 48.2%

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

   1. distribute-lft-out 56.0%

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

   2. sub-neg 56.0%

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

   3. *-commutative 56.0%

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

10. Simplified 56.0%

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

11. Final simplification 56.0%

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

Alternative 7: 43.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{+218}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - {\ell}^{2} \cdot \frac{2}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 7.5e+218)
   (sqrt (* (* 2.0 n) (* U (- t (* (pow l 2.0) (/ 2.0 Om))))))
   (sqrt (fabs (* 2.0 (* U (* n t)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 7.5e+218) {
		tmp = sqrt(((2.0 * n) * (U * (t - (pow(l, 2.0) * (2.0 / Om))))));
	} else {
		tmp = sqrt(fabs((2.0 * (U * (n * t)))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (t <= 7.5d+218) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - ((l ** 2.0d0) * (2.0d0 / om))))))
    else
        tmp = sqrt(abs((2.0d0 * (u * (n * t)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 7.5e+218) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (Math.pow(l, 2.0) * (2.0 / Om))))));
	} else {
		tmp = Math.sqrt(Math.abs((2.0 * (U * (n * t)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= 7.5e+218:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (math.pow(l, 2.0) * (2.0 / Om))))))
	else:
		tmp = math.sqrt(math.fabs((2.0 * (U * (n * t)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 7.5e+218)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64((l ^ 2.0) * Float64(2.0 / Om))))));
	else
		tmp = sqrt(abs(Float64(2.0 * Float64(U * Float64(n * t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= 7.5e+218)
		tmp = sqrt(((2.0 * n) * (U * (t - ((l ^ 2.0) * (2.0 / Om))))));
	else
		tmp = sqrt(abs((2.0 * (U * (n * t)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 7.5e+218], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7.4999999999999993e218

   1. Initial program 51.7%

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

   3. Simplified 56.4%

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

   5. Taylor expanded in Om around inf 44.7%

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

      1. *-commutative 44.7%

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

      2. associate-*l/ 44.7%

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

      3. associate-*r/ 44.7%

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

   7. Simplified 44.7%

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

   if 7.4999999999999993e218 < t

   1. Initial program 36.1%

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

   3. Taylor expanded in t around inf 51.5%

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

      1. add-sqr-sqrt 51.5%

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

      2. pow1/2 51.5%

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

      3. pow1/2 56.1%

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

      4. pow-prod-down 45.2%

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

      5. pow2 45.2%

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

      6. associate-*r* 45.2%

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

   5. Applied egg-rr 45.2%

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

      1. unpow1/2 45.2%

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

      2. unpow2 45.2%

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

      3. rem-sqrt-square 57.1%

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

      4. associate-*r* 57.1%

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

   7. Simplified 57.1%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{+218}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - {\ell}^{2} \cdot \frac{2}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 40.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq -5000000000000:\\ \;\;\;\;\sqrt{\left|t \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right|}\\ \mathbf{elif}\;U \leq 1.9 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{\left|n \cdot \left(t \cdot \left(2 \cdot U\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U -5000000000000.0)
   (sqrt (fabs (* t (* n (* 2.0 U)))))
   (if (<= U 1.9e+124)
     (sqrt (fabs (* n (* t (* 2.0 U)))))
     (* (sqrt (* 2.0 U)) (sqrt (* n t))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -5000000000000.0) {
		tmp = sqrt(fabs((t * (n * (2.0 * U)))));
	} else if (U <= 1.9e+124) {
		tmp = sqrt(fabs((n * (t * (2.0 * U)))));
	} else {
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (u <= (-5000000000000.0d0)) then
        tmp = sqrt(abs((t * (n * (2.0d0 * u)))))
    else if (u <= 1.9d+124) then
        tmp = sqrt(abs((n * (t * (2.0d0 * u)))))
    else
        tmp = sqrt((2.0d0 * u)) * sqrt((n * t))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -5000000000000.0) {
		tmp = Math.sqrt(Math.abs((t * (n * (2.0 * U)))));
	} else if (U <= 1.9e+124) {
		tmp = Math.sqrt(Math.abs((n * (t * (2.0 * U)))));
	} else {
		tmp = Math.sqrt((2.0 * U)) * Math.sqrt((n * t));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= -5000000000000.0:
		tmp = math.sqrt(math.fabs((t * (n * (2.0 * U)))))
	elif U <= 1.9e+124:
		tmp = math.sqrt(math.fabs((n * (t * (2.0 * U)))))
	else:
		tmp = math.sqrt((2.0 * U)) * math.sqrt((n * t))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= -5000000000000.0)
		tmp = sqrt(abs(Float64(t * Float64(n * Float64(2.0 * U)))));
	elseif (U <= 1.9e+124)
		tmp = sqrt(abs(Float64(n * Float64(t * Float64(2.0 * U)))));
	else
		tmp = Float64(sqrt(Float64(2.0 * U)) * sqrt(Float64(n * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= -5000000000000.0)
		tmp = sqrt(abs((t * (n * (2.0 * U)))));
	elseif (U <= 1.9e+124)
		tmp = sqrt(abs((n * (t * (2.0 * U)))));
	else
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, -5000000000000.0], N[Sqrt[N[Abs[N[(t * N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[U, 1.9e+124], N[Sqrt[N[Abs[N[(n * N[(t * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if U < -5e12

   1. Initial program 65.5%

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

   3. Taylor expanded in t around inf 38.6%

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

      1. associate-*r* 43.0%

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

      2. *-commutative 43.0%

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

      3. associate-*l* 43.0%

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

      4. *-commutative 43.0%

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

      5. associate-*r* 43.0%

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

      6. *-commutative 43.0%

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

   5. Simplified 43.0%

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

      1. add-sqr-sqrt 43.0%

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

      2. pow1/2 43.0%

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

      3. pow1/2 49.8%

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

      4. pow-prod-down 32.9%

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

      5. pow2 32.9%

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

      6. *-commutative 32.9%

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

      7. associate-*r* 32.9%

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

   7. Applied egg-rr 32.9%

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

      1. unpow1/2 32.9%

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

      2. *-commutative 32.9%

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

      3. *-commutative 32.9%

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

      4. associate-*r* 32.9%

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

      5. associate-*r* 28.5%

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

      6. unpow2 28.5%

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

      7. rem-sqrt-square 41.6%

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

      8. associate-*r* 50.5%

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

      9. associate-*r* 50.5%

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

      10. *-commutative 50.5%

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

      11. *-commutative 50.5%

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

      12. *-commutative 50.5%

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

      13. associate-*r* 50.5%

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

      14. *-commutative 50.5%

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

   9. Simplified 50.5%

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

   if -5e12 < U < 1.8999999999999999e124

   1. Initial program 45.7%

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

   3. Simplified 51.9%

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

      1. associate-*r/ 51.9%

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

      2. associate-*l/ 56.1%

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

      3. associate-*r* 56.1%

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

   6. Applied egg-rr 56.1%

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

   7. Taylor expanded in t around inf 34.5%

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

      1. add-sqr-sqrt 34.5%

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

      2. pow1/2 34.5%

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

      3. pow1/2 35.6%

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

      4. pow-prod-down 25.6%

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

      5. pow2 25.6%

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

      6. associate-*r* 24.1%

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

   9. Applied egg-rr 24.1%

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

       1. unpow1/2 24.1%

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

       2. unpow2 24.1%

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

       3. rem-sqrt-square 31.2%

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

       4. *-commutative 31.2%

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

       5. associate-*l* 31.2%

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

       6. *-commutative 31.2%

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

       7. associate-*r* 31.2%

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

       8. associate-*r* 36.7%

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

   11. Simplified 36.7%

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

   if 1.8999999999999999e124 < U

   1. Initial program 58.5%

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

   3. Taylor expanded in t around inf 49.6%

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

      1. pow1/2 49.6%

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

      2. associate-*r* 49.6%

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

      3. unpow-prod-down 75.0%

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

      4. pow1/2 75.0%

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

   5. Applied egg-rr 75.0%

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

      1. unpow1/2 75.0%

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

   7. Simplified 75.0%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -5000000000000:\\ \;\;\;\;\sqrt{\left|t \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right|}\\ \mathbf{elif}\;U \leq 1.9 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{\left|n \cdot \left(t \cdot \left(2 \cdot U\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 40.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq -50000000:\\ \;\;\;\;\sqrt{\left|t \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right|}\\ \mathbf{elif}\;U \leq 5 \cdot 10^{+122}:\\ \;\;\;\;\sqrt{\left|n \cdot \left(t \cdot \left(2 \cdot U\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U -50000000.0)
   (sqrt (fabs (* t (* n (* 2.0 U)))))
   (if (<= U 5e+122)
     (sqrt (fabs (* n (* t (* 2.0 U)))))
     (* (sqrt (* (* 2.0 n) t)) (sqrt U)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -50000000.0) {
		tmp = sqrt(fabs((t * (n * (2.0 * U)))));
	} else if (U <= 5e+122) {
		tmp = sqrt(fabs((n * (t * (2.0 * U)))));
	} else {
		tmp = sqrt(((2.0 * n) * t)) * sqrt(U);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (u <= (-50000000.0d0)) then
        tmp = sqrt(abs((t * (n * (2.0d0 * u)))))
    else if (u <= 5d+122) then
        tmp = sqrt(abs((n * (t * (2.0d0 * u)))))
    else
        tmp = sqrt(((2.0d0 * n) * t)) * sqrt(u)
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -50000000.0) {
		tmp = Math.sqrt(Math.abs((t * (n * (2.0 * U)))));
	} else if (U <= 5e+122) {
		tmp = Math.sqrt(Math.abs((n * (t * (2.0 * U)))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * t)) * Math.sqrt(U);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= -50000000.0:
		tmp = math.sqrt(math.fabs((t * (n * (2.0 * U)))))
	elif U <= 5e+122:
		tmp = math.sqrt(math.fabs((n * (t * (2.0 * U)))))
	else:
		tmp = math.sqrt(((2.0 * n) * t)) * math.sqrt(U)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= -50000000.0)
		tmp = sqrt(abs(Float64(t * Float64(n * Float64(2.0 * U)))));
	elseif (U <= 5e+122)
		tmp = sqrt(abs(Float64(n * Float64(t * Float64(2.0 * U)))));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * t)) * sqrt(U));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= -50000000.0)
		tmp = sqrt(abs((t * (n * (2.0 * U)))));
	elseif (U <= 5e+122)
		tmp = sqrt(abs((n * (t * (2.0 * U)))));
	else
		tmp = sqrt(((2.0 * n) * t)) * sqrt(U);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, -50000000.0], N[Sqrt[N[Abs[N[(t * N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[U, 5e+122], N[Sqrt[N[Abs[N[(n * N[(t * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if U < -5e7

   1. Initial program 65.5%

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

   3. Taylor expanded in t around inf 38.6%

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

      1. associate-*r* 43.0%

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

      2. *-commutative 43.0%

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

      3. associate-*l* 43.0%

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

      4. *-commutative 43.0%

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

      5. associate-*r* 43.0%

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

      6. *-commutative 43.0%

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

   5. Simplified 43.0%

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

      1. add-sqr-sqrt 43.0%

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

      2. pow1/2 43.0%

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

      3. pow1/2 49.8%

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

      4. pow-prod-down 32.9%

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

      5. pow2 32.9%

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

      6. *-commutative 32.9%

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

      7. associate-*r* 32.9%

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

   7. Applied egg-rr 32.9%

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

      1. unpow1/2 32.9%

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

      2. *-commutative 32.9%

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

      3. *-commutative 32.9%

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

      4. associate-*r* 32.9%

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

      5. associate-*r* 28.5%

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

      6. unpow2 28.5%

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

      7. rem-sqrt-square 41.6%

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

      8. associate-*r* 50.5%

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

      9. associate-*r* 50.5%

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

      10. *-commutative 50.5%

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

      11. *-commutative 50.5%

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

      12. *-commutative 50.5%

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

      13. associate-*r* 50.5%

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

      14. *-commutative 50.5%

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

   9. Simplified 50.5%

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

   if -5e7 < U < 4.99999999999999989e122

   1. Initial program 45.7%

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

   3. Simplified 51.9%

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

      1. associate-*r/ 51.9%

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

      2. associate-*l/ 56.1%

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

      3. associate-*r* 56.1%

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

   6. Applied egg-rr 56.1%

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

   7. Taylor expanded in t around inf 34.5%

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

      1. add-sqr-sqrt 34.5%

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

      2. pow1/2 34.5%

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

      3. pow1/2 35.6%

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

      4. pow-prod-down 25.6%

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

      5. pow2 25.6%

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

      6. associate-*r* 24.1%

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

   9. Applied egg-rr 24.1%

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

       1. unpow1/2 24.1%

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

       2. unpow2 24.1%

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

       3. rem-sqrt-square 31.2%

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

       4. *-commutative 31.2%

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

       5. associate-*l* 31.2%

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

       6. *-commutative 31.2%

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

       7. associate-*r* 31.2%

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

       8. associate-*r* 36.7%

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

   11. Simplified 36.7%

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

   if 4.99999999999999989e122 < U

   1. Initial program 58.5%

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

   3. Taylor expanded in t around inf 49.6%

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

      1. associate-*r* 49.7%

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

      2. *-commutative 49.7%

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

      3. associate-*l* 49.7%

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

      4. *-commutative 49.7%

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

      5. associate-*r* 49.7%

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

      6. *-commutative 49.7%

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

   5. Simplified 49.7%

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

      1. pow1/2 49.7%

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

      2. associate-*r* 49.6%

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

      3. unpow-prod-down 75.1%

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

      4. *-commutative 75.1%

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

      5. pow1/2 75.1%

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

   7. Applied egg-rr 75.1%

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

      1. unpow1/2 75.1%

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

      2. *-commutative 75.1%

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

   9. Simplified 75.1%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -50000000:\\ \;\;\;\;\sqrt{\left|t \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right|}\\ \mathbf{elif}\;U \leq 5 \cdot 10^{+122}:\\ \;\;\;\;\sqrt{\left|n \cdot \left(t \cdot \left(2 \cdot U\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot t} \cdot \sqrt{U}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 38.5% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t -2e+79)
   (sqrt (fabs (* 2.0 (* U (* n t)))))
   (sqrt (fabs (* t (* n (* 2.0 U)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -2e+79) {
		tmp = sqrt(fabs((2.0 * (U * (n * t)))));
	} else {
		tmp = sqrt(fabs((t * (n * (2.0 * U)))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (t <= (-2d+79)) then
        tmp = sqrt(abs((2.0d0 * (u * (n * t)))))
    else
        tmp = sqrt(abs((t * (n * (2.0d0 * u)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -2e+79) {
		tmp = Math.sqrt(Math.abs((2.0 * (U * (n * t)))));
	} else {
		tmp = Math.sqrt(Math.abs((t * (n * (2.0 * U)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= -2e+79:
		tmp = math.sqrt(math.fabs((2.0 * (U * (n * t)))))
	else:
		tmp = math.sqrt(math.fabs((t * (n * (2.0 * U)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= -2e+79)
		tmp = sqrt(abs(Float64(2.0 * Float64(U * Float64(n * t)))));
	else
		tmp = sqrt(abs(Float64(t * Float64(n * Float64(2.0 * U)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= -2e+79)
		tmp = sqrt(abs((2.0 * (U * (n * t)))));
	else
		tmp = sqrt(abs((t * (n * (2.0 * U)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, -2e+79], N[Sqrt[N[Abs[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(t * N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.99999999999999993e79

   1. Initial program 49.4%

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

   3. Taylor expanded in t around inf 56.5%

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

      1. add-sqr-sqrt 56.5%

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

      2. pow1/2 56.5%

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

      3. pow1/2 56.5%

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

      4. pow-prod-down 38.0%

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

      5. pow2 38.0%

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

      6. associate-*r* 38.0%

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

   5. Applied egg-rr 38.0%

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

      1. unpow1/2 38.0%

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

      2. unpow2 38.0%

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

      3. rem-sqrt-square 57.6%

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

      4. associate-*r* 57.5%

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

   7. Simplified 57.5%

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

   if -1.99999999999999993e79 < t

   1. Initial program 50.4%

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

   3. Taylor expanded in t around inf 31.3%

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

      1. associate-*r* 32.6%

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

      2. *-commutative 32.6%

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

      3. associate-*l* 32.6%

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

      4. *-commutative 32.6%

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

      5. associate-*r* 32.6%

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

      6. *-commutative 32.6%

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

   5. Simplified 32.6%

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

      1. add-sqr-sqrt 32.6%

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

      2. pow1/2 32.6%

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

      3. pow1/2 34.5%

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

      4. pow-prod-down 24.2%

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

      5. pow2 24.2%

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

      6. *-commutative 24.2%

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

      7. associate-*r* 24.2%

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

   7. Applied egg-rr 24.2%

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

      1. unpow1/2 24.2%

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

      2. *-commutative 24.2%

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

      3. *-commutative 24.2%

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

      4. associate-*r* 24.2%

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

      5. associate-*r* 23.8%

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

      6. unpow2 23.8%

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

      7. rem-sqrt-square 33.4%

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

      8. associate-*r* 35.5%

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

      9. associate-*r* 35.5%

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

      10. *-commutative 35.5%

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

      11. *-commutative 35.5%

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

      12. *-commutative 35.5%

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

      13. associate-*r* 35.5%

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

      14. *-commutative 35.5%

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

   9. Simplified 35.5%

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

4. Final simplification 39.1%

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

Alternative 11: 38.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq -10000000000:\\ \;\;\;\;\sqrt{\left|t \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|n \cdot \left(t \cdot \left(2 \cdot U\right)\right)\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U -10000000000.0)
   (sqrt (fabs (* t (* n (* 2.0 U)))))
   (sqrt (fabs (* n (* t (* 2.0 U)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -10000000000.0) {
		tmp = sqrt(fabs((t * (n * (2.0 * U)))));
	} else {
		tmp = sqrt(fabs((n * (t * (2.0 * U)))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (u <= (-10000000000.0d0)) then
        tmp = sqrt(abs((t * (n * (2.0d0 * u)))))
    else
        tmp = sqrt(abs((n * (t * (2.0d0 * u)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -10000000000.0) {
		tmp = Math.sqrt(Math.abs((t * (n * (2.0 * U)))));
	} else {
		tmp = Math.sqrt(Math.abs((n * (t * (2.0 * U)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= -10000000000.0:
		tmp = math.sqrt(math.fabs((t * (n * (2.0 * U)))))
	else:
		tmp = math.sqrt(math.fabs((n * (t * (2.0 * U)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= -10000000000.0)
		tmp = sqrt(abs(Float64(t * Float64(n * Float64(2.0 * U)))));
	else
		tmp = sqrt(abs(Float64(n * Float64(t * Float64(2.0 * U)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= -10000000000.0)
		tmp = sqrt(abs((t * (n * (2.0 * U)))));
	else
		tmp = sqrt(abs((n * (t * (2.0 * U)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, -10000000000.0], N[Sqrt[N[Abs[N[(t * N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(n * N[(t * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -1e10

   1. Initial program 65.5%

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

   3. Taylor expanded in t around inf 38.6%

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

      1. associate-*r* 43.0%

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

      2. *-commutative 43.0%

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

      3. associate-*l* 43.0%

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

      4. *-commutative 43.0%

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

      5. associate-*r* 43.0%

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

      6. *-commutative 43.0%

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

   5. Simplified 43.0%

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

      1. add-sqr-sqrt 43.0%

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

      2. pow1/2 43.0%

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

      3. pow1/2 49.8%

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

      4. pow-prod-down 32.9%

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

      5. pow2 32.9%

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

      6. *-commutative 32.9%

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

      7. associate-*r* 32.9%

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

   7. Applied egg-rr 32.9%

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

      1. unpow1/2 32.9%

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

      2. *-commutative 32.9%

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

      3. *-commutative 32.9%

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

      4. associate-*r* 32.9%

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

      5. associate-*r* 28.5%

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

      6. unpow2 28.5%

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

      7. rem-sqrt-square 41.6%

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

      8. associate-*r* 50.5%

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

      9. associate-*r* 50.5%

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

      10. *-commutative 50.5%

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

      11. *-commutative 50.5%

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

      12. *-commutative 50.5%

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

      13. associate-*r* 50.5%

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

      14. *-commutative 50.5%

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

   9. Simplified 50.5%

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

   if -1e10 < U

   1. Initial program 47.1%

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

   3. Simplified 52.1%

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

      1. associate-*r/ 52.1%

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

      2. associate-*l/ 55.9%

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

      3. associate-*r* 55.9%

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

   6. Applied egg-rr 55.9%

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

   7. Taylor expanded in t around inf 35.7%

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

      1. add-sqr-sqrt 35.7%

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

      2. pow1/2 35.7%

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

      3. pow1/2 36.8%

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

      4. pow-prod-down 26.5%

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

      5. pow2 26.5%

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

      6. associate-*r* 25.1%

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

   9. Applied egg-rr 25.1%

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

       1. unpow1/2 25.1%

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

       2. unpow2 25.1%

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

       3. rem-sqrt-square 33.2%

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

       4. *-commutative 33.2%

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

       5. associate-*l* 33.2%

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

       6. *-commutative 33.2%

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

       7. associate-*r* 33.2%

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

       8. associate-*r* 37.7%

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

   11. Simplified 37.7%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -10000000000:\\ \;\;\;\;\sqrt{\left|t \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|n \cdot \left(t \cdot \left(2 \cdot U\right)\right)\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 37.8% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt (fabs (* 2.0 (* U (* n t))))))

C

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

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(abs((2.0d0 * (u * (n * t)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.2%

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

3. Taylor expanded in t around inf 35.4%

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

   1. add-sqr-sqrt 35.4%

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

   2. pow1/2 35.4%

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

   3. pow1/2 36.3%

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

   4. pow-prod-down 26.1%

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

   5. pow2 26.1%

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

   6. associate-*r* 26.1%

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

5. Applied egg-rr 26.1%

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

   1. unpow1/2 26.1%

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

   2. unpow2 26.1%

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

   3. rem-sqrt-square 37.4%

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

   4. associate-*r* 37.3%

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

7. Simplified 37.3%

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

8. Final simplification 37.3%

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

Alternative 13: 37.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(t \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t -4.8e+84)
   (sqrt (* 2.0 (* U (* n t))))
   (pow (* t (* 2.0 (* n U))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -4.8e+84) {
		tmp = sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = pow((t * (2.0 * (n * U))), 0.5);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (t <= (-4.8d+84)) then
        tmp = sqrt((2.0d0 * (u * (n * t))))
    else
        tmp = (t * (2.0d0 * (n * u))) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -4.8e+84) {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = Math.pow((t * (2.0 * (n * U))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= -4.8e+84:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	else:
		tmp = math.pow((t * (2.0 * (n * U))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= -4.8e+84)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	else
		tmp = Float64(t * Float64(2.0 * Float64(n * U))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= -4.8e+84)
		tmp = sqrt((2.0 * (U * (n * t))));
	else
		tmp = (t * (2.0 * (n * U))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, -4.8e+84], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(t * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.7999999999999999e84

   1. Initial program 49.4%

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

   3. Taylor expanded in t around inf 56.5%

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

   if -4.7999999999999999e84 < t

   1. Initial program 50.4%

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

   3. Taylor expanded in t around inf 31.3%

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

      1. associate-*r* 32.6%

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

      2. *-commutative 32.6%

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

      3. associate-*l* 32.6%

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

      4. *-commutative 32.6%

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

      5. associate-*r* 32.6%

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

      6. *-commutative 32.6%

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

   5. Simplified 32.6%

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

      1. pow1/2 34.5%

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

      2. *-commutative 34.5%

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

      3. associate-*r* 34.5%

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

   7. Applied egg-rr 34.5%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(t \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 37.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq -1.35 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om -1.35e+79)
   (sqrt (* (* 2.0 n) (* U t)))
   (pow (* (* n t) (* 2.0 U)) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= -1.35e+79) {
		tmp = sqrt(((2.0 * n) * (U * t)));
	} else {
		tmp = pow(((n * t) * (2.0 * U)), 0.5);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (om <= (-1.35d+79)) then
        tmp = sqrt(((2.0d0 * n) * (u * t)))
    else
        tmp = ((n * t) * (2.0d0 * u)) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= -1.35e+79) {
		tmp = Math.sqrt(((2.0 * n) * (U * t)));
	} else {
		tmp = Math.pow(((n * t) * (2.0 * U)), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if Om <= -1.35e+79:
		tmp = math.sqrt(((2.0 * n) * (U * t)))
	else:
		tmp = math.pow(((n * t) * (2.0 * U)), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (Om <= -1.35e+79)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t)));
	else
		tmp = Float64(Float64(n * t) * Float64(2.0 * U)) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (Om <= -1.35e+79)
		tmp = sqrt(((2.0 * n) * (U * t)));
	else
		tmp = ((n * t) * (2.0 * U)) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[Om, -1.35e+79], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -1.35e79

   1. Initial program 58.6%

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

   3. Simplified 60.8%

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

      1. associate-*r/ 60.8%

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

      2. associate-*l/ 65.5%

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

      3. associate-*r* 65.5%

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

   6. Applied egg-rr 65.5%

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

   7. Taylor expanded in t around inf 53.3%

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

   if -1.35e79 < Om

   1. Initial program 47.4%

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

   3. Taylor expanded in t around inf 31.8%

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

      1. pow1/2 32.9%

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

      2. associate-*r* 33.0%

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

   5. Applied egg-rr 33.0%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1.35 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 35.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 10^{-105}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1e-105) (sqrt (* (* 2.0 n) (* U t))) (sqrt (* 2.0 (* U (* n t))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1e-105) {
		tmp = sqrt(((2.0 * n) * (U * t)));
	} else {
		tmp = sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (l <= 1d-105) then
        tmp = sqrt(((2.0d0 * n) * (u * t)))
    else
        tmp = sqrt((2.0d0 * (u * (n * t))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1e-105) {
		tmp = Math.sqrt(((2.0 * n) * (U * t)));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1e-105:
		tmp = math.sqrt(((2.0 * n) * (U * t)))
	else:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1e-105)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t)));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1e-105)
		tmp = sqrt(((2.0 * n) * (U * t)));
	else
		tmp = sqrt((2.0 * (U * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1e-105], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 9.99999999999999965e-106

   1. Initial program 56.3%

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

   3. Simplified 57.4%

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

      1. associate-*r/ 57.4%

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

      2. associate-*l/ 59.6%

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

      3. associate-*r* 59.6%

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

   6. Applied egg-rr 59.6%

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

   7. Taylor expanded in t around inf 40.9%

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

   if 9.99999999999999965e-106 < l

   1. Initial program 36.8%

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

   3. Taylor expanded in t around inf 27.1%

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

4. Final simplification 36.6%

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

Alternative 16: 35.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.2%

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

3. Taylor expanded in t around inf 35.4%

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

4. Final simplification 35.4%

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

Reproduce

herbie shell --seed 2024039 
(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*))))))