Toniolo and Linder, Equation (13)

Percentage Accurate: 50.7% → 64.1%

Time: 22.0s

Alternatives: 15

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: 50.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 64.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 15.0%

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

   3. Simplified 33.3%

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

      1. sqrt-prod 45.5%

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

      2. fma-undefine 45.5%

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

      3. associate-*r* 45.5%

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

      4. +-commutative 45.5%

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

      5. *-commutative 45.5%

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

      6. fma-define 45.5%

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

      7. associate-*r/ 45.5%

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

      8. pow2 45.5%

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

   6. Applied egg-rr 45.5%

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

      1. *-commutative 45.5%

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

      2. associate-*r/ 45.5%

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

   8. Simplified 45.5%

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

   9. Taylor expanded in n around 0 45.5%

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

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

   1. Initial program 68.1%

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

   3. Simplified 73.7%

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

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

   1. Initial program 0.0%

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

   3. Simplified 14.1%

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

   5. Taylor expanded in n around 0 2.0%

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

      1. associate-*r* 1.8%

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

      2. associate-*r/ 1.8%

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

   7. Simplified 1.8%

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

      1. add-sqr-sqrt 1.8%

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

      2. pow1/2 1.8%

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

      3. pow1/2 26.8%

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

      4. pow-prod-down 26.6%

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

      5. pow2 26.6%

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

      6. associate-*r* 26.6%

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

      7. *-commutative 26.6%

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

      8. associate-/l* 26.6%

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

   9. Applied egg-rr 26.6%

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

       1. unpow1/2 26.6%

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

       2. unpow2 26.6%

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

       3. rem-sqrt-square 27.5%

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

       4. associate-*l* 27.5%

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

       5. associate-*l* 28.6%

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

   11. Simplified 28.6%

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

       1. unpow2 28.6%

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

       2. *-un-lft-identity 28.6%

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

       3. times-frac 41.7%

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

   13. Applied egg-rr 41.7%

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

4. Final simplification 65.8%

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

Alternative 2: 63.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 15.0%

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

   3. Simplified 33.3%

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

   5. Taylor expanded in t around inf 31.0%

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

      1. pow1/2 31.0%

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

      2. associate-*r* 31.0%

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

      3. unpow-prod-down 34.5%

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

      4. pow1/2 34.5%

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

   7. Applied egg-rr 34.5%

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

      1. unpow1/2 34.5%

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

   9. Simplified 34.5%

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

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

   1. Initial program 68.1%

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

   3. Simplified 73.7%

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

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

   1. Initial program 0.0%

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

   3. Simplified 14.1%

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

   5. Taylor expanded in n around 0 2.0%

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

      1. associate-*r* 1.8%

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

      2. associate-*r/ 1.8%

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

   7. Simplified 1.8%

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

      1. add-sqr-sqrt 1.8%

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

      2. pow1/2 1.8%

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

      3. pow1/2 26.8%

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

      4. pow-prod-down 26.6%

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

      5. pow2 26.6%

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

      6. associate-*r* 26.6%

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

      7. *-commutative 26.6%

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

      8. associate-/l* 26.6%

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

   9. Applied egg-rr 26.6%

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

       1. unpow1/2 26.6%

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

       2. unpow2 26.6%

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

       3. rem-sqrt-square 27.5%

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

       4. associate-*l* 27.5%

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

       5. associate-*l* 28.6%

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

   11. Simplified 28.6%

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

       1. unpow2 28.6%

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

       2. *-un-lft-identity 28.6%

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

       3. times-frac 41.7%

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

   13. Applied egg-rr 41.7%

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

4. Final simplification 64.5%

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

Alternative 3: 62.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 12.2%

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

   3. Simplified 32.6%

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

   5. Taylor expanded in n around 0 27.0%

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

      1. associate-*r* 13.1%

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

      2. associate-*r/ 13.1%

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

   7. Simplified 13.1%

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

      1. add-sqr-sqrt 13.1%

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

      2. pow1/2 13.1%

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

      3. pow1/2 13.4%

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

      4. pow-prod-down 13.2%

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

      5. pow2 13.2%

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

      6. associate-*r* 13.2%

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

      7. *-commutative 13.2%

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

      8. associate-/l* 13.2%

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

   9. Applied egg-rr 13.2%

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

       1. unpow1/2 13.2%

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

       2. unpow2 13.2%

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

       3. rem-sqrt-square 14.1%

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

       4. associate-*l* 14.1%

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

       5. associate-*l* 28.8%

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

   11. Simplified 28.8%

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

       1. unpow2 28.8%

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

       2. *-un-lft-identity 28.8%

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

       3. times-frac 34.0%

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

   13. Applied egg-rr 34.0%

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

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

   1. Initial program 68.1%

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

   3. Simplified 73.7%

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

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

   1. Initial program 0.0%

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

   3. Simplified 10.4%

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

   5. Taylor expanded in n around 0 2.3%

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

      1. associate-*r* 1.1%

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

      2. associate-*r/ 1.1%

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

   7. Simplified 1.1%

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

      1. add-sqr-sqrt 1.1%

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

      2. pow1/2 1.1%

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

      3. pow1/2 31.5%

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

      4. pow-prod-down 31.5%

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

      5. pow2 31.5%

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

      6. associate-*r* 31.5%

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

      7. *-commutative 31.5%

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

      8. associate-/l* 31.5%

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

   9. Applied egg-rr 31.5%

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

       1. unpow1/2 31.5%

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

       2. unpow2 31.5%

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

       3. rem-sqrt-square 31.5%

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

       4. associate-*l* 31.5%

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

       5. associate-*l* 33.0%

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

   11. Simplified 33.0%

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

   12. Taylor expanded in t around 0 45.0%

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

       1. associate-*r/ 45.0%

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

       2. associate-*r* 45.0%

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

   14. Simplified 45.0%

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

4. Final simplification 64.6%

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

Alternative 4: 44.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;n \leq 1.06 \cdot 10^{-153} \lor \neg \left(n \leq 5.8 \cdot 10^{+121}\right):\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -1e-310)
   (sqrt (* 2.0 (* (* n U) (- t (/ (* 2.0 (pow l 2.0)) Om)))))
   (if (or (<= n 1.06e-153) (not (<= n 5.8e+121)))
     (* (sqrt (* 2.0 n)) (sqrt (* U t)))
     (sqrt (* (* 2.0 n) (* U (- t (* 2.0 (/ (pow l 2.0) Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -1e-310) {
		tmp = sqrt((2.0 * ((n * U) * (t - ((2.0 * pow(l, 2.0)) / Om)))));
	} else if ((n <= 1.06e-153) || !(n <= 5.8e+121)) {
		tmp = sqrt((2.0 * n)) * sqrt((U * t));
	} else {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (pow(l, 2.0) / Om))))));
	}
	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 (n <= (-1d-310)) then
        tmp = sqrt((2.0d0 * ((n * u) * (t - ((2.0d0 * (l ** 2.0d0)) / om)))))
    else if ((n <= 1.06d-153) .or. (.not. (n <= 5.8d+121))) then
        tmp = sqrt((2.0d0 * n)) * sqrt((u * t))
    else
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * ((l ** 2.0d0) / om))))))
    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 (n <= -1e-310) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t - ((2.0 * Math.pow(l, 2.0)) / Om)))));
	} else if ((n <= 1.06e-153) || !(n <= 5.8e+121)) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((U * t));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * (Math.pow(l, 2.0) / Om))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= -1e-310:
		tmp = math.sqrt((2.0 * ((n * U) * (t - ((2.0 * math.pow(l, 2.0)) / Om)))))
	elif (n <= 1.06e-153) or not (n <= 5.8e+121):
		tmp = math.sqrt((2.0 * n)) * math.sqrt((U * t))
	else:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * (math.pow(l, 2.0) / Om))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -1e-310)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t - Float64(Float64(2.0 * (l ^ 2.0)) / Om)))));
	elseif ((n <= 1.06e-153) || !(n <= 5.8e+121))
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * t)));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64((l ^ 2.0) / Om))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= -1e-310)
		tmp = sqrt((2.0 * ((n * U) * (t - ((2.0 * (l ^ 2.0)) / Om)))));
	elseif ((n <= 1.06e-153) || ~((n <= 5.8e+121)))
		tmp = sqrt((2.0 * n)) * sqrt((U * t));
	else
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * ((l ^ 2.0) / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -1e-310], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t - N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[n, 1.06e-153], N[Not[LessEqual[n, 5.8e+121]], $MachinePrecision]], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -9.999999999999969e-311

   1. Initial program 56.6%

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

   3. Simplified 54.7%

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

   5. Taylor expanded in n around 0 46.0%

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

      1. associate-*r* 48.9%

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

      2. associate-*r/ 48.9%

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

   7. Simplified 48.9%

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

   if -9.999999999999969e-311 < n < 1.06e-153 or 5.7999999999999998e121 < n

   1. Initial program 43.9%

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

   3. Simplified 46.2%

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

      1. sqrt-prod 72.3%

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

      2. fma-undefine 72.3%

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

      3. associate-*r* 72.4%

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

      4. +-commutative 72.4%

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

      5. *-commutative 72.4%

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

      6. fma-define 72.4%

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

      7. associate-*r/ 68.4%

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

      8. pow2 68.4%

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

   6. Applied egg-rr 68.4%

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

      1. *-commutative 68.4%

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

      2. associate-*r/ 68.4%

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

   8. Simplified 68.4%

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

   9. Taylor expanded in t around inf 60.5%

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

   if 1.06e-153 < n < 5.7999999999999998e121

   1. Initial program 52.5%

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

   3. Simplified 62.0%

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

      1. associate-*r* 62.1%

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

      2. sub-neg 62.1%

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

      3. distribute-lft-in 51.8%

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

   6. Applied egg-rr 51.8%

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

      1. *-commutative 51.8%

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

      2. associate-*r* 51.7%

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

      3. unpow2 51.7%

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

      4. times-frac 41.2%

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

      5. unpow2 41.2%

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

      6. unpow2 41.2%

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

      7. *-commutative 41.2%

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

      8. associate-*r* 41.2%

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

      9. unpow2 41.2%

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

      10. times-frac 41.2%

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

      11. unpow2 41.2%

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

      12. unpow2 41.2%

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

      13. distribute-rgt-out 49.8%

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

      14. unpow2 49.8%

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

      15. unpow2 49.8%

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

      16. times-frac 60.3%

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

      17. unpow2 60.3%

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

      18. distribute-rgt-in 62.0%

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

      19. sub-neg 62.0%

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

   8. Simplified 62.0%

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

   9. Taylor expanded in n around 0 49.3%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;n \leq 1.06 \cdot 10^{-153} \lor \neg \left(n \leq 5.8 \cdot 10^{+121}\right):\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 1.6 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-165} \lor \neg \left(n \leq 2.7 \cdot 10^{+107}\right):\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n 1.6e-307)
   (sqrt (* 2.0 (* (* n U) (- t (/ (* 2.0 (pow l 2.0)) Om)))))
   (if (or (<= n 1.05e-165) (not (<= n 2.7e+107)))
     (* (sqrt (* 2.0 n)) (sqrt (* U t)))
     (sqrt (* 2.0 (* U (* n (- t (* 2.0 (/ (pow l 2.0) Om))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 1.6e-307) {
		tmp = sqrt((2.0 * ((n * U) * (t - ((2.0 * pow(l, 2.0)) / Om)))));
	} else if ((n <= 1.05e-165) || !(n <= 2.7e+107)) {
		tmp = sqrt((2.0 * n)) * sqrt((U * t));
	} else {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (pow(l, 2.0) / Om)))))));
	}
	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 (n <= 1.6d-307) then
        tmp = sqrt((2.0d0 * ((n * u) * (t - ((2.0d0 * (l ** 2.0d0)) / om)))))
    else if ((n <= 1.05d-165) .or. (.not. (n <= 2.7d+107))) then
        tmp = sqrt((2.0d0 * n)) * sqrt((u * t))
    else
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * ((l ** 2.0d0) / om)))))))
    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 (n <= 1.6e-307) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t - ((2.0 * Math.pow(l, 2.0)) / Om)))));
	} else if ((n <= 1.05e-165) || !(n <= 2.7e+107)) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((U * t));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (Math.pow(l, 2.0) / Om)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= 1.6e-307:
		tmp = math.sqrt((2.0 * ((n * U) * (t - ((2.0 * math.pow(l, 2.0)) / Om)))))
	elif (n <= 1.05e-165) or not (n <= 2.7e+107):
		tmp = math.sqrt((2.0 * n)) * math.sqrt((U * t))
	else:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (math.pow(l, 2.0) / Om)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= 1.6e-307)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t - Float64(Float64(2.0 * (l ^ 2.0)) / Om)))));
	elseif ((n <= 1.05e-165) || !(n <= 2.7e+107))
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * t)));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64((l ^ 2.0) / Om)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= 1.6e-307)
		tmp = sqrt((2.0 * ((n * U) * (t - ((2.0 * (l ^ 2.0)) / Om)))));
	elseif ((n <= 1.05e-165) || ~((n <= 2.7e+107)))
		tmp = sqrt((2.0 * n)) * sqrt((U * t));
	else
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * ((l ^ 2.0) / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, 1.6e-307], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t - N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[n, 1.05e-165], N[Not[LessEqual[n, 2.7e+107]], $MachinePrecision]], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 1.60000000000000005e-307

   1. Initial program 56.6%

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

   3. Simplified 54.7%

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

   5. Taylor expanded in n around 0 46.0%

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

      1. associate-*r* 48.9%

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

      2. associate-*r/ 48.9%

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

   7. Simplified 48.9%

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

   if 1.60000000000000005e-307 < n < 1.04999999999999997e-165 or 2.7000000000000001e107 < n

   1. Initial program 48.1%

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

   3. Simplified 50.2%

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

      1. sqrt-prod 75.0%

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

      2. fma-undefine 75.0%

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

      3. associate-*r* 75.0%

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

      4. +-commutative 75.0%

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

      5. *-commutative 75.0%

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

      6. fma-define 75.0%

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

      7. associate-*r/ 72.5%

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

      8. pow2 72.5%

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

   6. Applied egg-rr 72.5%

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

      1. *-commutative 72.5%

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

      2. associate-*r/ 72.5%

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

   8. Simplified 72.5%

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

   9. Taylor expanded in t around inf 63.7%

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

   if 1.04999999999999997e-165 < n < 2.7000000000000001e107

   1. Initial program 47.3%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in n around 0 43.4%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 1.6 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-165} \lor \neg \left(n \leq 2.7 \cdot 10^{+107}\right):\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(n \cdot U\right)\\ \mathbf{if}\;t \leq -0.026:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right|}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+75}:\\ \;\;\;\;{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot t\_1\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* n U))))
   (if (<= t -0.026)
     (sqrt (fabs (* 2.0 (* n (* U t)))))
     (if (<= t 1.35e+75)
       (pow (* (- t (* 2.0 (/ (pow l 2.0) Om))) t_1) 0.5)
       (* (sqrt t_1) (sqrt t))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double tmp;
	if (t <= -0.026) {
		tmp = sqrt(fabs((2.0 * (n * (U * t)))));
	} else if (t <= 1.35e+75) {
		tmp = pow(((t - (2.0 * (pow(l, 2.0) / Om))) * t_1), 0.5);
	} else {
		tmp = sqrt(t_1) * sqrt(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) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 * (n * u)
    if (t <= (-0.026d0)) then
        tmp = sqrt(abs((2.0d0 * (n * (u * t)))))
    else if (t <= 1.35d+75) then
        tmp = ((t - (2.0d0 * ((l ** 2.0d0) / om))) * t_1) ** 0.5d0
    else
        tmp = sqrt(t_1) * sqrt(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 t_1 = 2.0 * (n * U);
	double tmp;
	if (t <= -0.026) {
		tmp = Math.sqrt(Math.abs((2.0 * (n * (U * t)))));
	} else if (t <= 1.35e+75) {
		tmp = Math.pow(((t - (2.0 * (Math.pow(l, 2.0) / Om))) * t_1), 0.5);
	} else {
		tmp = Math.sqrt(t_1) * Math.sqrt(t);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = 2.0 * (n * U)
	tmp = 0
	if t <= -0.026:
		tmp = math.sqrt(math.fabs((2.0 * (n * (U * t)))))
	elif t <= 1.35e+75:
		tmp = math.pow(((t - (2.0 * (math.pow(l, 2.0) / Om))) * t_1), 0.5)
	else:
		tmp = math.sqrt(t_1) * math.sqrt(t)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(2.0 * Float64(n * U))
	tmp = 0.0
	if (t <= -0.026)
		tmp = sqrt(abs(Float64(2.0 * Float64(n * Float64(U * t)))));
	elseif (t <= 1.35e+75)
		tmp = Float64(Float64(t - Float64(2.0 * Float64((l ^ 2.0) / Om))) * t_1) ^ 0.5;
	else
		tmp = Float64(sqrt(t_1) * sqrt(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = 2.0 * (n * U);
	tmp = 0.0;
	if (t <= -0.026)
		tmp = sqrt(abs((2.0 * (n * (U * t)))));
	elseif (t <= 1.35e+75)
		tmp = ((t - (2.0 * ((l ^ 2.0) / Om))) * t_1) ^ 0.5;
	else
		tmp = sqrt(t_1) * sqrt(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.026], N[Sqrt[N[Abs[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 1.35e+75], N[Power[N[(N[(t - N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], 0.5], $MachinePrecision], N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.0259999999999999988

   1. Initial program 46.3%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in n around 0 46.0%

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

      1. associate-*r* 43.4%

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

      2. associate-*r/ 43.4%

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

   7. Simplified 43.4%

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

      1. add-sqr-sqrt 43.4%

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

      2. pow1/2 43.4%

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

      3. pow1/2 43.6%

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

      4. pow-prod-down 29.1%

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

      5. pow2 29.1%

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

      6. associate-*r* 29.1%

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

      7. *-commutative 29.1%

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

      8. associate-/l* 29.1%

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

   9. Applied egg-rr 29.1%

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

       1. unpow1/2 29.1%

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

       2. unpow2 29.1%

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

       3. rem-sqrt-square 43.9%

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

       4. associate-*l* 43.9%

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

       5. associate-*l* 46.6%

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

   11. Simplified 46.6%

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

   12. Taylor expanded in t around inf 49.6%

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

       1. associate-*r* 47.2%

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

       2. *-commutative 47.2%

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

       3. associate-*r* 51.2%

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

   14. Simplified 51.2%

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

   if -0.0259999999999999988 < t < 1.34999999999999999e75

   1. Initial program 52.1%

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

   3. Simplified 54.2%

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

   5. Taylor expanded in n around 0 39.4%

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

      1. associate-*r* 43.6%

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

      2. associate-*r/ 43.6%

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

   7. Simplified 43.6%

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

      1. pow1/2 51.2%

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

      2. associate-*r* 51.2%

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

      3. *-commutative 51.2%

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

      4. associate-/l* 51.2%

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

   9. Applied egg-rr 51.2%

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

   if 1.34999999999999999e75 < t

   1. Initial program 60.0%

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

   3. Simplified 52.1%

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

      1. associate-*r* 54.1%

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

      2. sub-neg 54.1%

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

      3. distribute-lft-in 47.4%

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

   6. Applied egg-rr 47.4%

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

      1. *-commutative 47.4%

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

      2. associate-*r* 47.2%

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

      3. unpow2 47.2%

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

      4. times-frac 38.4%

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

      5. unpow2 38.4%

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

      6. unpow2 38.4%

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

      7. *-commutative 38.4%

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

      8. associate-*r* 35.8%

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

      9. unpow2 35.8%

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

      10. times-frac 35.8%

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

      11. unpow2 35.8%

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

      12. unpow2 35.8%

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

      13. distribute-rgt-out 40.5%

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

      14. unpow2 40.5%

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

      15. unpow2 40.5%

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

      16. times-frac 49.2%

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

      17. unpow2 49.2%

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

      18. distribute-rgt-in 51.4%

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

      19. sub-neg 51.4%

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

   8. Simplified 51.4%

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

   9. Taylor expanded in t around inf 46.6%

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

       1. associate-*r* 52.6%

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

       2. associate-*r* 52.6%

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

       3. sqrt-prod 70.9%

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

   11. Applied egg-rr 70.9%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.026:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right|}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+75}:\\ \;\;\;\;{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{-265}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-195}:\\ \;\;\;\;\sqrt{U \cdot U*} \cdot \left(\left(n \cdot \ell\right) \cdot \frac{\sqrt{2}}{-Om}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t -8.5e-265)
   (sqrt (* 2.0 (* U (* n (- t (* 2.0 (/ (pow l 2.0) Om)))))))
   (if (<= t 1.55e-195)
     (* (sqrt (* U U*)) (* (* n l) (/ (sqrt 2.0) (- Om))))
     (* (sqrt (* 2.0 (* n U))) (sqrt t)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -8.5e-265) {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (pow(l, 2.0) / Om)))))));
	} else if (t <= 1.55e-195) {
		tmp = sqrt((U * U_42_)) * ((n * l) * (sqrt(2.0) / -Om));
	} else {
		tmp = sqrt((2.0 * (n * U))) * sqrt(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 <= (-8.5d-265)) then
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * ((l ** 2.0d0) / om)))))))
    else if (t <= 1.55d-195) then
        tmp = sqrt((u * u_42)) * ((n * l) * (sqrt(2.0d0) / -om))
    else
        tmp = sqrt((2.0d0 * (n * u))) * sqrt(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 <= -8.5e-265) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (Math.pow(l, 2.0) / Om)))))));
	} else if (t <= 1.55e-195) {
		tmp = Math.sqrt((U * U_42_)) * ((n * l) * (Math.sqrt(2.0) / -Om));
	} else {
		tmp = Math.sqrt((2.0 * (n * U))) * Math.sqrt(t);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= -8.5e-265:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (math.pow(l, 2.0) / Om)))))))
	elif t <= 1.55e-195:
		tmp = math.sqrt((U * U_42_)) * ((n * l) * (math.sqrt(2.0) / -Om))
	else:
		tmp = math.sqrt((2.0 * (n * U))) * math.sqrt(t)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= -8.5e-265)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64((l ^ 2.0) / Om)))))));
	elseif (t <= 1.55e-195)
		tmp = Float64(sqrt(Float64(U * U_42_)) * Float64(Float64(n * l) * Float64(sqrt(2.0) / Float64(-Om))));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(n * U))) * sqrt(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= -8.5e-265)
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * ((l ^ 2.0) / Om)))))));
	elseif (t <= 1.55e-195)
		tmp = sqrt((U * U_42_)) * ((n * l) * (sqrt(2.0) / -Om));
	else
		tmp = sqrt((2.0 * (n * U))) * sqrt(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, -8.5e-265], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 1.55e-195], N[(N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision] * N[(N[(n * l), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-Om)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.4999999999999997e-265

   1. Initial program 49.2%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in n around 0 42.5%

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

   if -8.4999999999999997e-265 < t < 1.55000000000000001e-195

   1. Initial program 33.0%

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

   3. Simplified 36.9%

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

      1. associate-*r* 36.9%

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

      2. sub-neg 36.9%

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

      3. distribute-lft-in 25.2%

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

   6. Applied egg-rr 25.2%

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

      1. *-commutative 25.2%

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

      2. associate-*r* 21.4%

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

      3. unpow2 21.4%

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

      4. times-frac 17.1%

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

      5. unpow2 17.1%

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

      6. unpow2 17.1%

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

      7. *-commutative 17.1%

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

      8. associate-*r* 17.1%

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

      9. unpow2 17.1%

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

      10. times-frac 17.1%

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

      11. unpow2 17.1%

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

      12. unpow2 17.1%

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

      13. distribute-rgt-out 25.0%

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

      14. unpow2 25.0%

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

      15. unpow2 25.0%

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

      16. times-frac 29.2%

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

      17. unpow2 29.2%

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

      18. distribute-rgt-in 36.9%

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

      19. sub-neg 36.9%

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

   8. Simplified 36.9%

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

   9. Taylor expanded in U* around inf 31.2%

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

       1. associate-/l* 31.3%

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

       2. associate-/l* 33.1%

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

       3. associate-/l* 33.1%

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

   11. Simplified 33.1%

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

   12. Taylor expanded in n around -inf 20.5%

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

       1. mul-1-neg 20.5%

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

       2. associate-/l* 16.9%

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

       3. associate-*r/ 16.9%

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

       4. *-commutative 16.9%

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

       5. distribute-rgt-neg-in 16.9%

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

       6. associate-*r/ 16.9%

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

       7. associate-/l* 20.5%

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

       8. distribute-frac-neg2 20.5%

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

       9. associate-*r* 20.6%

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

       10. associate-/l* 20.6%

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

       11. *-commutative 20.6%

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

   14. Simplified 20.6%

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

   if 1.55000000000000001e-195 < t

   1. Initial program 59.5%

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

   3. Simplified 59.0%

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

      1. associate-*r* 59.8%

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

      2. sub-neg 59.8%

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

      3. distribute-lft-in 52.4%

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

   6. Applied egg-rr 52.4%

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

      1. *-commutative 52.4%

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

      2. associate-*r* 52.2%

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

      3. unpow2 52.2%

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

      4. times-frac 43.0%

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

      5. unpow2 43.0%

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

      6. unpow2 43.0%

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

      7. *-commutative 43.0%

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

      8. associate-*r* 41.0%

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

      9. unpow2 41.0%

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

      10. times-frac 41.0%

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

      11. unpow2 41.0%

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

      12. unpow2 41.0%

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

      13. distribute-rgt-out 45.8%

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

      14. unpow2 45.8%

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

      15. unpow2 45.8%

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

      16. times-frac 55.0%

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

      17. unpow2 55.0%

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

      18. distribute-rgt-in 57.7%

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

      19. sub-neg 57.7%

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

   8. Simplified 57.7%

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

   9. Taylor expanded in t around inf 42.1%

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

       1. associate-*r* 46.9%

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

       2. associate-*r* 46.9%

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

       3. sqrt-prod 57.3%

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

   11. Applied egg-rr 57.3%

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

Alternative 8: 39.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{-250}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right|}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-195}:\\ \;\;\;\;\sqrt{U \cdot U*} \cdot \left(\left(n \cdot \ell\right) \cdot \frac{\sqrt{2}}{-Om}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t -8.8e-250)
   (sqrt (fabs (* 2.0 (* n (* U t)))))
   (if (<= t 1.45e-195)
     (* (sqrt (* U U*)) (* (* n l) (/ (sqrt 2.0) (- Om))))
     (* (sqrt (* 2.0 (* n U))) (sqrt t)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -8.8e-250) {
		tmp = sqrt(fabs((2.0 * (n * (U * t)))));
	} else if (t <= 1.45e-195) {
		tmp = sqrt((U * U_42_)) * ((n * l) * (sqrt(2.0) / -Om));
	} else {
		tmp = sqrt((2.0 * (n * U))) * sqrt(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 <= (-8.8d-250)) then
        tmp = sqrt(abs((2.0d0 * (n * (u * t)))))
    else if (t <= 1.45d-195) then
        tmp = sqrt((u * u_42)) * ((n * l) * (sqrt(2.0d0) / -om))
    else
        tmp = sqrt((2.0d0 * (n * u))) * sqrt(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 <= -8.8e-250) {
		tmp = Math.sqrt(Math.abs((2.0 * (n * (U * t)))));
	} else if (t <= 1.45e-195) {
		tmp = Math.sqrt((U * U_42_)) * ((n * l) * (Math.sqrt(2.0) / -Om));
	} else {
		tmp = Math.sqrt((2.0 * (n * U))) * Math.sqrt(t);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= -8.8e-250:
		tmp = math.sqrt(math.fabs((2.0 * (n * (U * t)))))
	elif t <= 1.45e-195:
		tmp = math.sqrt((U * U_42_)) * ((n * l) * (math.sqrt(2.0) / -Om))
	else:
		tmp = math.sqrt((2.0 * (n * U))) * math.sqrt(t)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= -8.8e-250)
		tmp = sqrt(abs(Float64(2.0 * Float64(n * Float64(U * t)))));
	elseif (t <= 1.45e-195)
		tmp = Float64(sqrt(Float64(U * U_42_)) * Float64(Float64(n * l) * Float64(sqrt(2.0) / Float64(-Om))));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(n * U))) * sqrt(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= -8.8e-250)
		tmp = sqrt(abs((2.0 * (n * (U * t)))));
	elseif (t <= 1.45e-195)
		tmp = sqrt((U * U_42_)) * ((n * l) * (sqrt(2.0) / -Om));
	else
		tmp = sqrt((2.0 * (n * U))) * sqrt(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, -8.8e-250], N[Sqrt[N[Abs[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 1.45e-195], N[(N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision] * N[(N[(n * l), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-Om)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.8e-250

   1. Initial program 48.6%

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

   3. Simplified 52.7%

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

   5. Taylor expanded in n around 0 42.4%

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

      1. associate-*r* 42.6%

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

      2. associate-*r/ 42.6%

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

   7. Simplified 42.6%

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

      1. add-sqr-sqrt 42.6%

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

      2. pow1/2 42.6%

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

      3. pow1/2 46.4%

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

      4. pow-prod-down 32.4%

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

      5. pow2 32.4%

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

      6. associate-*r* 32.4%

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

      7. *-commutative 32.4%

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

      8. associate-/l* 32.4%

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

   9. Applied egg-rr 32.4%

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

       1. unpow1/2 32.4%

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

       2. unpow2 32.4%

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

       3. rem-sqrt-square 46.8%

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

       4. associate-*l* 46.8%

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

       5. associate-*l* 46.8%

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

   11. Simplified 46.8%

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

   12. Taylor expanded in t around inf 41.4%

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

       1. associate-*r* 42.6%

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

       2. *-commutative 42.6%

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

       3. associate-*r* 43.2%

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

   14. Simplified 43.2%

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

   if -8.8e-250 < t < 1.4500000000000001e-195

   1. Initial program 38.5%

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

   3. Simplified 41.5%

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

      1. associate-*r* 41.5%

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

      2. sub-neg 41.5%

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

      3. distribute-lft-in 26.2%

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

   6. Applied egg-rr 26.2%

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

      1. *-commutative 26.2%

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

      2. associate-*r* 23.2%

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

      3. unpow2 23.2%

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

      4. times-frac 19.8%

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

      5. unpow2 19.8%

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

      6. unpow2 19.8%

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

      7. *-commutative 19.8%

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

      8. associate-*r* 19.8%

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

      9. unpow2 19.8%

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

      10. times-frac 19.8%

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

      11. unpow2 19.8%

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

      12. unpow2 19.8%

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

      13. distribute-rgt-out 32.0%

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

      14. unpow2 32.0%

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

      15. unpow2 32.0%

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

      16. times-frac 35.3%

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

      17. unpow2 35.3%

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

      18. distribute-rgt-in 41.4%

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

      19. sub-neg 41.4%

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

   8. Simplified 41.4%

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

   9. Taylor expanded in U* around inf 34.1%

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

       1. associate-/l* 34.2%

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

       2. associate-/l* 35.6%

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

       3. associate-/l* 35.6%

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

   11. Simplified 35.6%

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

   12. Taylor expanded in n around -inf 25.7%

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

       1. mul-1-neg 25.7%

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

       2. associate-/l* 22.9%

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

       3. associate-*r/ 22.9%

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

       4. *-commutative 22.9%

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

       5. distribute-rgt-neg-in 22.9%

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

       6. associate-*r/ 22.9%

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

       7. associate-/l* 25.7%

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

       8. distribute-frac-neg2 25.7%

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

       9. associate-*r* 25.8%

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

       10. associate-/l* 25.7%

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

       11. *-commutative 25.7%

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

   14. Simplified 25.7%

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

   if 1.4500000000000001e-195 < t

   1. Initial program 59.5%

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

   3. Simplified 59.0%

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

      1. associate-*r* 59.8%

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

      2. sub-neg 59.8%

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

      3. distribute-lft-in 52.4%

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

   6. Applied egg-rr 52.4%

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

      1. *-commutative 52.4%

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

      2. associate-*r* 52.2%

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

      3. unpow2 52.2%

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

      4. times-frac 43.0%

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

      5. unpow2 43.0%

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

      6. unpow2 43.0%

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

      7. *-commutative 43.0%

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

      8. associate-*r* 41.0%

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

      9. unpow2 41.0%

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

      10. times-frac 41.0%

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

      11. unpow2 41.0%

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

      12. unpow2 41.0%

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

      13. distribute-rgt-out 45.8%

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

      14. unpow2 45.8%

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

      15. unpow2 45.8%

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

      16. times-frac 55.0%

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

      17. unpow2 55.0%

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

      18. distribute-rgt-in 57.7%

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

      19. sub-neg 57.7%

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

   8. Simplified 57.7%

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

   9. Taylor expanded in t around inf 42.1%

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

       1. associate-*r* 46.9%

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

       2. associate-*r* 46.9%

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

       3. sqrt-prod 57.3%

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

   11. Applied egg-rr 57.3%

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

Alternative 9: 39.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-250}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right|}\\ \mathbf{elif}\;t \leq 1.56 \cdot 10^{-195}:\\ \;\;\;\;\sqrt{U \cdot U*} \cdot \left(\ell \cdot \left(n \cdot \frac{\sqrt{2}}{-Om}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t -1.15e-250)
   (sqrt (fabs (* 2.0 (* n (* U t)))))
   (if (<= t 1.56e-195)
     (* (sqrt (* U U*)) (* l (* n (/ (sqrt 2.0) (- Om)))))
     (* (sqrt (* 2.0 (* n U))) (sqrt t)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -1.15e-250) {
		tmp = sqrt(fabs((2.0 * (n * (U * t)))));
	} else if (t <= 1.56e-195) {
		tmp = sqrt((U * U_42_)) * (l * (n * (sqrt(2.0) / -Om)));
	} else {
		tmp = sqrt((2.0 * (n * U))) * sqrt(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 <= (-1.15d-250)) then
        tmp = sqrt(abs((2.0d0 * (n * (u * t)))))
    else if (t <= 1.56d-195) then
        tmp = sqrt((u * u_42)) * (l * (n * (sqrt(2.0d0) / -om)))
    else
        tmp = sqrt((2.0d0 * (n * u))) * sqrt(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 <= -1.15e-250) {
		tmp = Math.sqrt(Math.abs((2.0 * (n * (U * t)))));
	} else if (t <= 1.56e-195) {
		tmp = Math.sqrt((U * U_42_)) * (l * (n * (Math.sqrt(2.0) / -Om)));
	} else {
		tmp = Math.sqrt((2.0 * (n * U))) * Math.sqrt(t);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= -1.15e-250:
		tmp = math.sqrt(math.fabs((2.0 * (n * (U * t)))))
	elif t <= 1.56e-195:
		tmp = math.sqrt((U * U_42_)) * (l * (n * (math.sqrt(2.0) / -Om)))
	else:
		tmp = math.sqrt((2.0 * (n * U))) * math.sqrt(t)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= -1.15e-250)
		tmp = sqrt(abs(Float64(2.0 * Float64(n * Float64(U * t)))));
	elseif (t <= 1.56e-195)
		tmp = Float64(sqrt(Float64(U * U_42_)) * Float64(l * Float64(n * Float64(sqrt(2.0) / Float64(-Om)))));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(n * U))) * sqrt(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= -1.15e-250)
		tmp = sqrt(abs((2.0 * (n * (U * t)))));
	elseif (t <= 1.56e-195)
		tmp = sqrt((U * U_42_)) * (l * (n * (sqrt(2.0) / -Om)));
	else
		tmp = sqrt((2.0 * (n * U))) * sqrt(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, -1.15e-250], N[Sqrt[N[Abs[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 1.56e-195], N[(N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision] * N[(l * N[(n * N[(N[Sqrt[2.0], $MachinePrecision] / (-Om)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.15e-250

   1. Initial program 48.6%

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

   3. Simplified 52.7%

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

   5. Taylor expanded in n around 0 42.4%

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

      1. associate-*r* 42.6%

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

      2. associate-*r/ 42.6%

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

   7. Simplified 42.6%

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

      1. add-sqr-sqrt 42.6%

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

      2. pow1/2 42.6%

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

      3. pow1/2 46.4%

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

      4. pow-prod-down 32.4%

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

      5. pow2 32.4%

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

      6. associate-*r* 32.4%

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

      7. *-commutative 32.4%

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

      8. associate-/l* 32.4%

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

   9. Applied egg-rr 32.4%

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

       1. unpow1/2 32.4%

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

       2. unpow2 32.4%

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

       3. rem-sqrt-square 46.8%

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

       4. associate-*l* 46.8%

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

       5. associate-*l* 46.8%

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

   11. Simplified 46.8%

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

   12. Taylor expanded in t around inf 41.4%

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

       1. associate-*r* 42.6%

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

       2. *-commutative 42.6%

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

       3. associate-*r* 43.2%

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

   14. Simplified 43.2%

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

   if -1.15e-250 < t < 1.56000000000000006e-195

   1. Initial program 38.5%

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

   3. Simplified 41.5%

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

      1. associate-*r* 41.5%

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

      2. sub-neg 41.5%

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

      3. distribute-lft-in 26.2%

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

   6. Applied egg-rr 26.2%

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

      1. *-commutative 26.2%

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

      2. associate-*r* 23.2%

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

      3. unpow2 23.2%

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

      4. times-frac 19.8%

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

      5. unpow2 19.8%

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

      6. unpow2 19.8%

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

      7. *-commutative 19.8%

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

      8. associate-*r* 19.8%

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

      9. unpow2 19.8%

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

      10. times-frac 19.8%

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

      11. unpow2 19.8%

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

      12. unpow2 19.8%

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

      13. distribute-rgt-out 32.0%

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

      14. unpow2 32.0%

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

      15. unpow2 32.0%

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

      16. times-frac 35.3%

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

      17. unpow2 35.3%

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

      18. distribute-rgt-in 41.4%

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

      19. sub-neg 41.4%

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

   8. Simplified 41.4%

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

   9. Taylor expanded in U* around inf 34.1%

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

       1. associate-/l* 34.2%

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

       2. associate-/l* 35.6%

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

       3. associate-/l* 35.6%

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

   11. Simplified 35.6%

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

   12. Taylor expanded in n around -inf 25.7%

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

       1. mul-1-neg 25.7%

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

       2. associate-/l* 22.9%

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

       3. associate-*r/ 22.9%

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

       4. *-commutative 22.9%

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

   14. Simplified 22.9%

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

   if 1.56000000000000006e-195 < t

   1. Initial program 59.5%

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

   3. Simplified 59.0%

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

      1. associate-*r* 59.8%

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

      2. sub-neg 59.8%

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

      3. distribute-lft-in 52.4%

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

   6. Applied egg-rr 52.4%

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

      1. *-commutative 52.4%

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

      2. associate-*r* 52.2%

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

      3. unpow2 52.2%

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

      4. times-frac 43.0%

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

      5. unpow2 43.0%

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

      6. unpow2 43.0%

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

      7. *-commutative 43.0%

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

      8. associate-*r* 41.0%

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

      9. unpow2 41.0%

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

      10. times-frac 41.0%

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

      11. unpow2 41.0%

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

      12. unpow2 41.0%

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

      13. distribute-rgt-out 45.8%

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

      14. unpow2 45.8%

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

      15. unpow2 45.8%

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

      16. times-frac 55.0%

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

      17. unpow2 55.0%

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

      18. distribute-rgt-in 57.7%

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

      19. sub-neg 57.7%

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

   8. Simplified 57.7%

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

   9. Taylor expanded in t around inf 42.1%

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

       1. associate-*r* 46.9%

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

       2. associate-*r* 46.9%

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

       3. sqrt-prod 57.3%

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

   11. Applied egg-rr 57.3%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-250}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right|}\\ \mathbf{elif}\;t \leq 1.56 \cdot 10^{-195}:\\ \;\;\;\;\sqrt{U \cdot U*} \cdot \left(\ell \cdot \left(n \cdot \frac{\sqrt{2}}{-Om}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n 1e+122)
   (sqrt (fabs (* 2.0 (* n (* U (- t (* 2.0 (* l (/ l Om)))))))))
   (* (sqrt (* 2.0 n)) (sqrt (* U t)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 1e+122) {
		tmp = sqrt(fabs((2.0 * (n * (U * (t - (2.0 * (l * (l / Om)))))))));
	} else {
		tmp = sqrt((2.0 * n)) * sqrt((U * 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 (n <= 1d+122) then
        tmp = sqrt(abs((2.0d0 * (n * (u * (t - (2.0d0 * (l * (l / om)))))))))
    else
        tmp = sqrt((2.0d0 * n)) * sqrt((u * 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 (n <= 1e+122) {
		tmp = Math.sqrt(Math.abs((2.0 * (n * (U * (t - (2.0 * (l * (l / Om)))))))));
	} else {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((U * t));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= 1e+122:
		tmp = math.sqrt(math.fabs((2.0 * (n * (U * (t - (2.0 * (l * (l / Om)))))))))
	else:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((U * t))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= 1e+122)
		tmp = sqrt(abs(Float64(2.0 * Float64(n * Float64(U * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))))))));
	else
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= 1e+122)
		tmp = sqrt(abs((2.0 * (n * (U * (t - (2.0 * (l * (l / Om)))))))));
	else
		tmp = sqrt((2.0 * n)) * sqrt((U * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, 1e+122], N[Sqrt[N[Abs[N[(2.0 * N[(n * N[(U * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 1.00000000000000001e122

   1. Initial program 53.0%

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

   3. Simplified 55.7%

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

   5. Taylor expanded in n around 0 44.7%

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

      1. associate-*r* 47.1%

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

      2. associate-*r/ 47.1%

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

   7. Simplified 47.1%

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

      1. add-sqr-sqrt 47.1%

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

      2. pow1/2 47.1%

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

      3. pow1/2 50.9%

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

      4. pow-prod-down 35.2%

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

      5. pow2 35.2%

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

      6. associate-*r* 35.2%

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

      7. *-commutative 35.2%

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

      8. associate-/l* 35.2%

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

   9. Applied egg-rr 35.2%

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

       1. unpow1/2 35.2%

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

       2. unpow2 35.2%

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

       3. rem-sqrt-square 51.4%

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

       4. associate-*l* 51.4%

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

       5. associate-*l* 50.2%

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

   11. Simplified 50.2%

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

       1. unpow2 50.2%

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

       2. *-un-lft-identity 50.2%

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

       3. times-frac 55.4%

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

   13. Applied egg-rr 55.4%

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

   if 1.00000000000000001e122 < n

   1. Initial program 45.8%

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

   3. Simplified 43.0%

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

      1. sqrt-prod 79.1%

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

      2. fma-undefine 79.1%

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

      3. associate-*r* 79.1%

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

      4. +-commutative 79.1%

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

      5. *-commutative 79.1%

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

      6. fma-define 79.1%

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

      7. associate-*r/ 79.1%

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

      8. pow2 79.1%

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

   6. Applied egg-rr 79.1%

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

      1. *-commutative 79.1%

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

      2. associate-*r/ 79.1%

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

   8. Simplified 79.1%

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

   9. Taylor expanded in t around inf 67.9%

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

4. Final simplification 57.1%

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

Alternative 11: 40.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.5 \cdot 10^{-169}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -5.5e-169)
   (pow (* 2.0 (* t (* n U))) 0.5)
   (if (<= n -1e-310)
     (sqrt (fabs (* 2.0 (* n (* U t)))))
     (* (sqrt (* 2.0 n)) (sqrt (* U t))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -5.5e-169) {
		tmp = pow((2.0 * (t * (n * U))), 0.5);
	} else if (n <= -1e-310) {
		tmp = sqrt(fabs((2.0 * (n * (U * t)))));
	} else {
		tmp = sqrt((2.0 * n)) * sqrt((U * 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 (n <= (-5.5d-169)) then
        tmp = (2.0d0 * (t * (n * u))) ** 0.5d0
    else if (n <= (-1d-310)) then
        tmp = sqrt(abs((2.0d0 * (n * (u * t)))))
    else
        tmp = sqrt((2.0d0 * n)) * sqrt((u * 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 (n <= -5.5e-169) {
		tmp = Math.pow((2.0 * (t * (n * U))), 0.5);
	} else if (n <= -1e-310) {
		tmp = Math.sqrt(Math.abs((2.0 * (n * (U * t)))));
	} else {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((U * t));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= -5.5e-169:
		tmp = math.pow((2.0 * (t * (n * U))), 0.5)
	elif n <= -1e-310:
		tmp = math.sqrt(math.fabs((2.0 * (n * (U * t)))))
	else:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((U * t))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -5.5e-169)
		tmp = Float64(2.0 * Float64(t * Float64(n * U))) ^ 0.5;
	elseif (n <= -1e-310)
		tmp = sqrt(abs(Float64(2.0 * Float64(n * Float64(U * t)))));
	else
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= -5.5e-169)
		tmp = (2.0 * (t * (n * U))) ^ 0.5;
	elseif (n <= -1e-310)
		tmp = sqrt(abs((2.0 * (n * (U * t)))));
	else
		tmp = sqrt((2.0 * n)) * sqrt((U * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -5.5e-169], N[Power[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[n, -1e-310], N[Sqrt[N[Abs[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.4999999999999994e-169

   1. Initial program 59.1%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in t around inf 40.6%

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

      1. pow1/2 42.6%

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

      2. associate-*r* 44.6%

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

      3. *-commutative 44.6%

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

   7. Applied egg-rr 44.6%

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

   if -5.4999999999999994e-169 < n < -9.999999999999969e-311

   1. Initial program 47.0%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in n around 0 33.1%

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

      1. associate-*r* 39.7%

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

      2. associate-*r/ 39.7%

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

   7. Simplified 39.7%

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

      1. add-sqr-sqrt 39.7%

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

      2. pow1/2 39.7%

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

      3. pow1/2 39.7%

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

      4. pow-prod-down 22.5%

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

      5. pow2 22.5%

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

      6. associate-*r* 22.5%

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

      7. *-commutative 22.5%

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

      8. associate-/l* 22.5%

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

   9. Applied egg-rr 22.5%

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

       1. unpow1/2 22.5%

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

       2. unpow2 22.5%

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

       3. rem-sqrt-square 40.9%

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

       4. associate-*l* 40.9%

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

       5. associate-*l* 49.0%

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

   11. Simplified 49.0%

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

   12. Taylor expanded in t around inf 23.2%

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

       1. associate-*r* 30.2%

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

       2. *-commutative 30.2%

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

       3. associate-*r* 45.6%

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

   14. Simplified 45.6%

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

   if -9.999999999999969e-311 < n

   1. Initial program 47.8%

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

   3. Simplified 53.3%

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

      1. sqrt-prod 69.5%

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

      2. fma-undefine 69.5%

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

      3. associate-*r* 69.6%

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

      4. +-commutative 69.6%

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

      5. *-commutative 69.6%

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

      6. fma-define 69.6%

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

      7. associate-*r/ 63.0%

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

      8. pow2 63.0%

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

   6. Applied egg-rr 63.0%

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

      1. *-commutative 63.0%

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

      2. associate-*r/ 63.0%

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

   8. Simplified 63.0%

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

   9. Taylor expanded in t around inf 50.1%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.5 \cdot 10^{-169}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 40.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 1.6e-307)
   (sqrt (fabs (* 2.0 (* n (* U t)))))
   (* (sqrt (* 2.0 (* n U))) (sqrt t))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 1.6e-307) {
		tmp = sqrt(fabs((2.0 * (n * (U * t)))));
	} else {
		tmp = sqrt((2.0 * (n * U))) * sqrt(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 <= 1.6d-307) then
        tmp = sqrt(abs((2.0d0 * (n * (u * t)))))
    else
        tmp = sqrt((2.0d0 * (n * u))) * sqrt(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 <= 1.6e-307) {
		tmp = Math.sqrt(Math.abs((2.0 * (n * (U * t)))));
	} else {
		tmp = Math.sqrt((2.0 * (n * U))) * Math.sqrt(t);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= 1.6e-307:
		tmp = math.sqrt(math.fabs((2.0 * (n * (U * t)))))
	else:
		tmp = math.sqrt((2.0 * (n * U))) * math.sqrt(t)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 1.6e-307)
		tmp = sqrt(abs(Float64(2.0 * Float64(n * Float64(U * t)))));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(n * U))) * sqrt(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= 1.6e-307)
		tmp = sqrt(abs((2.0 * (n * (U * t)))));
	else
		tmp = sqrt((2.0 * (n * U))) * sqrt(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 1.6e-307], N[Sqrt[N[Abs[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.60000000000000005e-307

   1. Initial program 49.3%

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

   3. Simplified 52.9%

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

   5. Taylor expanded in n around 0 41.5%

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

      1. associate-*r* 42.0%

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

      2. associate-*r/ 42.0%

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

   7. Simplified 42.0%

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

      1. add-sqr-sqrt 42.0%

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

      2. pow1/2 42.0%

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

      3. pow1/2 45.4%

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

      4. pow-prod-down 32.2%

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

      5. pow2 32.2%

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

      6. associate-*r* 32.2%

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

      7. *-commutative 32.2%

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

      8. associate-/l* 32.2%

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

   9. Applied egg-rr 32.2%

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

       1. unpow1/2 32.2%

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

       2. unpow2 32.2%

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

       3. rem-sqrt-square 45.9%

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

       4. associate-*l* 45.9%

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

       5. associate-*l* 45.8%

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

   11. Simplified 45.8%

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

   12. Taylor expanded in t around inf 37.0%

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

       1. associate-*r* 38.1%

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

       2. *-commutative 38.1%

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

       3. associate-*r* 38.6%

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

   14. Simplified 38.6%

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

   if 1.60000000000000005e-307 < t

   1. Initial program 54.7%

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

   3. Simplified 55.0%

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

      1. associate-*r* 55.8%

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

      2. sub-neg 55.8%

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

      3. distribute-lft-in 47.8%

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

   6. Applied egg-rr 47.8%

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

      1. *-commutative 47.8%

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

      2. associate-*r* 46.9%

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

      3. unpow2 46.9%

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

      4. times-frac 38.1%

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

      5. unpow2 38.1%

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

      6. unpow2 38.1%

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

      7. *-commutative 38.1%

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

      8. associate-*r* 36.4%

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

      9. unpow2 36.4%

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

      10. times-frac 36.4%

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

      11. unpow2 36.4%

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

      12. unpow2 36.4%

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

      13. distribute-rgt-out 41.3%

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

      14. unpow2 41.3%

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

      15. unpow2 41.3%

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

      16. times-frac 50.1%

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

      17. unpow2 50.1%

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

      18. distribute-rgt-in 54.0%

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

      19. sub-neg 54.0%

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

   8. Simplified 54.0%

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

   9. Taylor expanded in t around inf 37.5%

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

       1. associate-*r* 41.6%

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

       2. associate-*r* 41.6%

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

       3. sqrt-prod 51.2%

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

   11. Applied egg-rr 51.2%

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

Alternative 13: 37.6% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (n U t l Om U*) :precision binary64 (pow (* 2.0 (* t (* n U))) 0.5))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return pow((2.0 * (t * (n * U))), 0.5);
}

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 = (2.0d0 * (t * (n * u))) ** 0.5d0
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.pow((2.0 * (t * (n * U))), 0.5);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.0%

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

3. Simplified 54.0%

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

5. Taylor expanded in t around inf 35.6%

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

   1. pow1/2 36.7%

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

   2. associate-*r* 41.4%

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

   3. *-commutative 41.4%

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

7. Applied egg-rr 41.4%

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

8. Final simplification 41.4%

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

Alternative 14: 36.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.0%

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

3. Simplified 54.0%

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

5. Taylor expanded in t around inf 35.6%

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

   1. associate-*r* 39.1%

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

7. Simplified 39.1%

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

8. Final simplification 39.1%

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

Alternative 15: 35.7% 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 52.0%

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

3. Simplified 54.0%

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

5. Taylor expanded in t around inf 35.6%

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

Reproduce

herbie shell --seed 2024108 
(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*))))))