Toniolo and Linder, Equation (13)

Percentage Accurate: 49.3% → 61.4%

Time: 25.1s

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: 49.3% 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: 61.4% accurate, 0.4× speedup

Math

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

FPCore

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

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

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n * Math.pow((l / Om), 2.0)) * (U_42_ - U);
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_1);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t + ((((Math.pow(l, 2.0) * (n * (U_42_ - U))) / Om) - (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 - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = Math.sqrt(((U * -2.0) * ((n * Math.pow(l, 2.0)) * ((2.0 / Om) + (n * ((U - U_42_) / Math.pow(Om, 2.0)))))));
	}
	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 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_1)
	tmp = 0
	if t_2 <= 0.0:
		tmp = math.sqrt(((2.0 * n) * (U * (t + ((((math.pow(l, 2.0) * (n * (U_42_ - U))) / Om) - (2.0 * math.pow(l, 2.0))) / Om)))))
	elif t_2 <= math.inf:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l * (l / Om)))))))
	else:
		tmp = math.sqrt(((U * -2.0) * ((n * math.pow(l, 2.0)) * ((2.0 / Om) + (n * ((U - U_42_) / math.pow(Om, 2.0)))))))
	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 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_1))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t + Float64(Float64(Float64(Float64((l ^ 2.0) * Float64(n * Float64(U_42_ - U))) / Om) - Float64(2.0 * (l ^ 2.0))) / Om)))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(t_1 - Float64(2.0 * Float64(l * Float64(l / Om)))))));
	else
		tmp = sqrt(Float64(Float64(U * -2.0) * Float64(Float64(n * (l ^ 2.0)) * Float64(Float64(2.0 / Om) + Float64(n * Float64(Float64(U - U_42_) / (Om ^ 2.0)))))));
	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 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_1);
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = sqrt(((2.0 * n) * (U * (t + (((((l ^ 2.0) * (n * (U_42_ - U))) / Om) - (2.0 * (l ^ 2.0))) / Om)))));
	elseif (t_2 <= Inf)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l * (l / Om)))))));
	else
		tmp = sqrt(((U * -2.0) * ((n * (l ^ 2.0)) * ((2.0 / Om) + (n * ((U - U_42_) / (Om ^ 2.0)))))));
	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[(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]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t + N[(N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] - N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(t$95$1 - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(U * -2.0), $MachinePrecision] * N[(N[(n * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / Om), $MachinePrecision] + N[(n * N[(N[(U - U$42$), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 7.0%

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

   3. Simplified 27.3%

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

   5. Taylor expanded in Om around inf 36.8%

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

   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 71.0%

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

   3. Simplified 74.3%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.1%

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

   5. Taylor expanded in l around inf 46.7%

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

      1. associate-*r* 46.7%

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

      2. associate-*r* 46.8%

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

      3. *-commutative 46.8%

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

      4. associate-*r/ 46.8%

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

      5. metadata-eval 46.8%

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

      6. associate-/l* 46.8%

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

   7. Simplified 46.8%

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

4. Final simplification 63.9%

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

Alternative 2: 60.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t\_1\right)}\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_1 - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot -2\right) \cdot \left(\left(n \cdot {\ell}^{2}\right) \cdot \left(\frac{2}{Om} + n \cdot \frac{U - U*}{{Om}^{2}}\right)\right)}\\ \end{array} \end{array} \]

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)))))
   (if (<= t_2 2e-157)
     (* (sqrt (* 2.0 U)) (sqrt (* n t)))
     (if (<= t_2 INFINITY)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_1 (* 2.0 (* l (/ l Om)))))))
       (sqrt
        (*
         (* U -2.0)
         (*
          (* n (pow l 2.0))
          (+ (/ 2.0 Om) (* n (/ (- U U*) (pow Om 2.0)))))))))))

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 tmp;
	if (t_2 <= 2e-157) {
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = sqrt(((U * -2.0) * ((n * pow(l, 2.0)) * ((2.0 / Om) + (n * ((U - U_42_) / pow(Om, 2.0)))))));
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (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 tmp;
	if (t_2 <= 2e-157) {
		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 - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = Math.sqrt(((U * -2.0) * ((n * Math.pow(l, 2.0)) * ((2.0 / Om) + (n * ((U - U_42_) / Math.pow(Om, 2.0)))))));
	}
	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)))
	tmp = 0
	if t_2 <= 2e-157:
		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 - (2.0 * (l * (l / Om)))))))
	else:
		tmp = math.sqrt(((U * -2.0) * ((n * math.pow(l, 2.0)) * ((2.0 / Om) + (n * ((U - U_42_) / math.pow(Om, 2.0)))))))
	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)))
	tmp = 0.0
	if (t_2 <= 2e-157)
		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 - Float64(2.0 * Float64(l * Float64(l / Om)))))));
	else
		tmp = sqrt(Float64(Float64(U * -2.0) * Float64(Float64(n * (l ^ 2.0)) * Float64(Float64(2.0 / Om) + Float64(n * Float64(Float64(U - U_42_) / (Om ^ 2.0)))))));
	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)));
	tmp = 0.0;
	if (t_2 <= 2e-157)
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	elseif (t_2 <= Inf)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l * (l / Om)))))));
	else
		tmp = sqrt(((U * -2.0) * ((n * (l ^ 2.0)) * ((2.0 / Om) + (n * ((U - U_42_) / (Om ^ 2.0)))))));
	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]}, If[LessEqual[t$95$2, 2e-157], 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 - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(U * -2.0), $MachinePrecision] * N[(N[(n * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / Om), $MachinePrecision] + N[(n * N[(N[(U - U$42$), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $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*))))) < 1.99999999999999989e-157

   1. Initial program 10.0%

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

   3. Simplified 10.0%

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

   5. Taylor expanded in t around inf 32.7%

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

      1. pow1/2 32.7%

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

      2. associate-*r* 32.8%

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

      3. unpow-prod-down 37.6%

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

      4. pow1/2 37.6%

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

   7. Applied egg-rr 37.6%

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

      1. unpow1/2 37.6%

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

      2. *-commutative 37.6%

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

   9. Simplified 37.6%

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

   if 1.99999999999999989e-157 < (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 71.0%

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

   3. Simplified 74.3%

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

   if +inf.0 < (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 0.4%

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

   5. Taylor expanded in l around inf 44.9%

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

      1. associate-*r* 44.9%

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

      2. associate-*r* 45.0%

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

      3. *-commutative 45.0%

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

      4. associate-*r/ 45.0%

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

      5. metadata-eval 45.0%

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

      6. associate-/l* 45.0%

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

   7. Simplified 45.0%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot -2\right) \cdot \left(\left(n \cdot {\ell}^{2}\right) \cdot \left(\frac{2}{Om} + n \cdot \frac{U - U*}{{Om}^{2}}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t\_1\right)}\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_1 - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{2}{Om} - U* \cdot \frac{n}{{Om}^{2}}\right)\right)\right)\right)}\\ \end{array} \end{array} \]

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)))))
   (if (<= t_2 2e-157)
     (* (sqrt (* 2.0 U)) (sqrt (* n t)))
     (if (<= t_2 INFINITY)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_1 (* 2.0 (* l (/ l Om)))))))
       (sqrt
        (*
         -2.0
         (*
          U
          (* (pow l 2.0) (* n (- (/ 2.0 Om) (* U* (/ n (pow Om 2.0)))))))))))))

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 tmp;
	if (t_2 <= 2e-157) {
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = sqrt((-2.0 * (U * (pow(l, 2.0) * (n * ((2.0 / Om) - (U_42_ * (n / pow(Om, 2.0)))))))));
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (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 tmp;
	if (t_2 <= 2e-157) {
		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 - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = Math.sqrt((-2.0 * (U * (Math.pow(l, 2.0) * (n * ((2.0 / Om) - (U_42_ * (n / Math.pow(Om, 2.0)))))))));
	}
	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)))
	tmp = 0
	if t_2 <= 2e-157:
		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 - (2.0 * (l * (l / Om)))))))
	else:
		tmp = math.sqrt((-2.0 * (U * (math.pow(l, 2.0) * (n * ((2.0 / Om) - (U_42_ * (n / math.pow(Om, 2.0)))))))))
	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)))
	tmp = 0.0
	if (t_2 <= 2e-157)
		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 - Float64(2.0 * Float64(l * Float64(l / Om)))))));
	else
		tmp = sqrt(Float64(-2.0 * Float64(U * Float64((l ^ 2.0) * Float64(n * Float64(Float64(2.0 / Om) - Float64(U_42_ * Float64(n / (Om ^ 2.0)))))))));
	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)));
	tmp = 0.0;
	if (t_2 <= 2e-157)
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	elseif (t_2 <= Inf)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l * (l / Om)))))));
	else
		tmp = sqrt((-2.0 * (U * ((l ^ 2.0) * (n * ((2.0 / Om) - (U_42_ * (n / (Om ^ 2.0)))))))));
	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]}, If[LessEqual[t$95$2, 2e-157], 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 - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(U * N[(N[Power[l, 2.0], $MachinePrecision] * N[(n * N[(N[(2.0 / Om), $MachinePrecision] - N[(U$42$ * N[(n / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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*))))) < 1.99999999999999989e-157

   1. Initial program 10.0%

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

   3. Simplified 10.0%

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

   5. Taylor expanded in t around inf 32.7%

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

      1. pow1/2 32.7%

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

      2. associate-*r* 32.8%

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

      3. unpow-prod-down 37.6%

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

      4. pow1/2 37.6%

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

   7. Applied egg-rr 37.6%

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

      1. unpow1/2 37.6%

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

      2. *-commutative 37.6%

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

   9. Simplified 37.6%

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

   if 1.99999999999999989e-157 < (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 71.0%

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

   3. Simplified 74.3%

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

   if +inf.0 < (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 0.4%

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

      1. associate-*r* 0.4%

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

      2. fma-define 13.8%

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

      3. associate-*r* 13.8%

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

   6. Applied egg-rr 13.8%

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

   7. Taylor expanded in U around 0 17.2%

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

      1. associate-/l* 13.9%

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

      2. unpow2 13.9%

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

      3. unpow2 13.9%

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

      4. times-frac 14.1%

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

      5. unpow2 14.1%

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

      6. neg-mul-1 14.1%

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

      7. distribute-lft-neg-out 14.1%

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

      8. *-commutative 14.1%

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

   9. Simplified 14.1%

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

   10. Taylor expanded in l around inf 44.0%

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

       1. +-commutative 44.0%

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

       2. mul-1-neg 44.0%

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

       3. unsub-neg 44.0%

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

       4. associate-*r/ 44.0%

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

       5. metadata-eval 44.0%

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

       6. associate-/l* 44.1%

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

   12. Simplified 44.1%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(\frac{2}{Om} - U* \cdot \frac{n}{{Om}^{2}}\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t\_1\right)}\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_1 - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (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)))))
   (if (<= t_2 2e-157)
     (* (sqrt (* 2.0 U)) (sqrt (* n t)))
     (if (<= t_2 INFINITY)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_1 (* 2.0 (* l (/ l Om)))))))
       (sqrt (fabs (* (+ t (/ (* (pow l 2.0) -2.0) Om)) (* n (* 2.0 U)))))))))

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 tmp;
	if (t_2 <= 2e-157) {
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = sqrt(fabs(((t + ((pow(l, 2.0) * -2.0) / Om)) * (n * (2.0 * U)))));
	}
	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 tmp;
	if (t_2 <= 2e-157) {
		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 - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = Math.sqrt(Math.abs(((t + ((Math.pow(l, 2.0) * -2.0) / Om)) * (n * (2.0 * U)))));
	}
	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)))
	tmp = 0
	if t_2 <= 2e-157:
		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 - (2.0 * (l * (l / Om)))))))
	else:
		tmp = math.sqrt(math.fabs(((t + ((math.pow(l, 2.0) * -2.0) / Om)) * (n * (2.0 * U)))))
	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)))
	tmp = 0.0
	if (t_2 <= 2e-157)
		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 - Float64(2.0 * Float64(l * Float64(l / Om)))))));
	else
		tmp = sqrt(abs(Float64(Float64(t + Float64(Float64((l ^ 2.0) * -2.0) / Om)) * Float64(n * Float64(2.0 * U)))));
	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)));
	tmp = 0.0;
	if (t_2 <= 2e-157)
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	elseif (t_2 <= Inf)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l * (l / Om)))))));
	else
		tmp = sqrt(abs(((t + (((l ^ 2.0) * -2.0) / Om)) * (n * (2.0 * U)))));
	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]}, If[LessEqual[t$95$2, 2e-157], 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 - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(N[(t + N[(N[(N[Power[l, 2.0], $MachinePrecision] * -2.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[(n * N[(2.0 * U), $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*))))) < 1.99999999999999989e-157

   1. Initial program 10.0%

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

   3. Simplified 10.0%

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

   5. Taylor expanded in t around inf 32.7%

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

      1. pow1/2 32.7%

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

      2. associate-*r* 32.8%

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

      3. unpow-prod-down 37.6%

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

      4. pow1/2 37.6%

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

   7. Applied egg-rr 37.6%

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

      1. unpow1/2 37.6%

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

      2. *-commutative 37.6%

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

   9. Simplified 37.6%

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

   if 1.99999999999999989e-157 < (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 71.0%

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

   3. Simplified 74.3%

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

   if +inf.0 < (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 3.0%

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

   5. Taylor expanded in Om around inf 11.3%

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

      1. add-sqr-sqrt 11.3%

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

      2. pow1/2 11.3%

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

      3. pow1/2 46.8%

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

      4. pow-prod-down 42.9%

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

      5. pow2 42.9%

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

      6. associate-*r* 41.0%

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

      7. cancel-sign-sub-inv 41.0%

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

      8. metadata-eval 41.0%

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

   7. Applied egg-rr 41.0%

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

      1. unpow1/2 41.0%

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

      2. unpow2 41.0%

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

      3. rem-sqrt-square 41.0%

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

      4. metadata-eval 41.0%

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

      5. cancel-sign-sub-inv 41.0%

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

      6. *-commutative 41.0%

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

      7. cancel-sign-sub-inv 41.0%

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

      8. metadata-eval 41.0%

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

      9. associate-*r/ 41.0%

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

      10. associate-*l* 41.0%

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

      11. *-commutative 41.0%

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

      12. associate-*r* 41.0%

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

      13. *-commutative 41.0%

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

      14. *-commutative 41.0%

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

   9. Simplified 41.0%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\left(t + \frac{{\ell}^{2} \cdot -2}{Om}\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \sqrt{t\_2 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t\_1\right)}\\ \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_1 - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(t\_2 \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l Om) 2.0)) (- U* U)))
        (t_2 (* (* 2.0 n) U))
        (t_3 (sqrt (* t_2 (+ (- t (* 2.0 (/ (* l l) Om))) t_1)))))
   (if (<= t_3 2e-157)
     (* (sqrt (* 2.0 U)) (sqrt (* n t)))
     (if (<= t_3 INFINITY)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_1 (* 2.0 (* l (/ l Om)))))))
       (pow (* t_2 (+ t (* -2.0 (/ (pow l 2.0) Om)))) 0.5)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n * pow((l / Om), 2.0)) * (U_42_ - U);
	double t_2 = (2.0 * n) * U;
	double t_3 = sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + t_1)));
	double tmp;
	if (t_3 <= 2e-157) {
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = pow((t_2 * (t + (-2.0 * (pow(l, 2.0) / Om)))), 0.5);
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n * Math.pow((l / Om), 2.0)) * (U_42_ - U);
	double t_2 = (2.0 * n) * U;
	double t_3 = Math.sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + t_1)));
	double tmp;
	if (t_3 <= 2e-157) {
		tmp = Math.sqrt((2.0 * U)) * Math.sqrt((n * t));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = Math.pow((t_2 * (t + (-2.0 * (Math.pow(l, 2.0) / Om)))), 0.5);
	}
	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 = (2.0 * n) * U
	t_3 = math.sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + t_1)))
	tmp = 0
	if t_3 <= 2e-157:
		tmp = math.sqrt((2.0 * U)) * math.sqrt((n * t))
	elif t_3 <= math.inf:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l * (l / Om)))))))
	else:
		tmp = math.pow((t_2 * (t + (-2.0 * (math.pow(l, 2.0) / Om)))), 0.5)
	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 = Float64(Float64(2.0 * n) * U)
	t_3 = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_1)))
	tmp = 0.0
	if (t_3 <= 2e-157)
		tmp = Float64(sqrt(Float64(2.0 * U)) * sqrt(Float64(n * t)));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(t_1 - Float64(2.0 * Float64(l * Float64(l / Om)))))));
	else
		tmp = Float64(t_2 * Float64(t + Float64(-2.0 * Float64((l ^ 2.0) / Om)))) ^ 0.5;
	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 = (2.0 * n) * U;
	t_3 = sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + t_1)));
	tmp = 0.0;
	if (t_3 <= 2e-157)
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	elseif (t_3 <= Inf)
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_1 - (2.0 * (l * (l / Om)))))));
	else
		tmp = (t_2 * (t + (-2.0 * ((l ^ 2.0) / Om)))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 2e-157], N[(N[Sqrt[N[(2.0 * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * t), $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$1 - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(t$95$2 * N[(t + N[(-2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1.99999999999999989e-157

   1. Initial program 10.0%

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

   3. Simplified 10.0%

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

   5. Taylor expanded in t around inf 32.7%

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

      1. pow1/2 32.7%

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

      2. associate-*r* 32.8%

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

      3. unpow-prod-down 37.6%

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

      4. pow1/2 37.6%

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

   7. Applied egg-rr 37.6%

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

      1. unpow1/2 37.6%

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

      2. *-commutative 37.6%

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

   9. Simplified 37.6%

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

   if 1.99999999999999989e-157 < (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 71.0%

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

   3. Simplified 74.3%

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

   if +inf.0 < (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 3.0%

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

   5. Taylor expanded in Om around inf 11.3%

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

      1. pow1/2 46.8%

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

      2. associate-*r* 40.8%

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

      3. cancel-sign-sub-inv 40.8%

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

      4. metadata-eval 40.8%

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

   7. Applied egg-rr 40.8%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.05 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot U*\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right)}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -1.05e-73)
   (sqrt
    (*
     (* 2.0 (* n U))
     (+ t (- (* n (* (pow (/ l Om) 2.0) U*)) (* (/ l Om) (* 2.0 l))))))
   (if (<= n 1.05e-190)
     (sqrt (+ (* -4.0 (/ (* U (* n (pow l 2.0))) Om)) (* 2.0 (* U (* n t)))))
     (pow (* (* (* 2.0 n) U) (+ t (* -2.0 (/ (pow l 2.0) Om)))) 0.5))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -1.05e-73) {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * (pow((l / Om), 2.0) * U_42_)) - ((l / Om) * (2.0 * l))))));
	} else if (n <= 1.05e-190) {
		tmp = sqrt(((-4.0 * ((U * (n * pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	} else {
		tmp = pow((((2.0 * n) * U) * (t + (-2.0 * (pow(l, 2.0) / Om)))), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= (-1.05d-73)) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + ((n * (((l / om) ** 2.0d0) * u_42)) - ((l / om) * (2.0d0 * l))))))
    else if (n <= 1.05d-190) then
        tmp = sqrt((((-4.0d0) * ((u * (n * (l ** 2.0d0))) / om)) + (2.0d0 * (u * (n * t)))))
    else
        tmp = (((2.0d0 * n) * u) * (t + ((-2.0d0) * ((l ** 2.0d0) / om)))) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -1.05e-73) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((n * (Math.pow((l / Om), 2.0) * U_42_)) - ((l / Om) * (2.0 * l))))));
	} else if (n <= 1.05e-190) {
		tmp = Math.sqrt(((-4.0 * ((U * (n * Math.pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	} else {
		tmp = Math.pow((((2.0 * n) * U) * (t + (-2.0 * (Math.pow(l, 2.0) / Om)))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= -1.05e-73:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((n * (math.pow((l / Om), 2.0) * U_42_)) - ((l / Om) * (2.0 * l))))))
	elif n <= 1.05e-190:
		tmp = math.sqrt(((-4.0 * ((U * (n * math.pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))))
	else:
		tmp = math.pow((((2.0 * n) * U) * (t + (-2.0 * (math.pow(l, 2.0) / Om)))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -1.05e-73)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(n * Float64((Float64(l / Om) ^ 2.0) * U_42_)) - Float64(Float64(l / Om) * Float64(2.0 * l))))));
	elseif (n <= 1.05e-190)
		tmp = sqrt(Float64(Float64(-4.0 * Float64(Float64(U * Float64(n * (l ^ 2.0))) / Om)) + Float64(2.0 * Float64(U * Float64(n * t)))));
	else
		tmp = Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(-2.0 * Float64((l ^ 2.0) / Om)))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= -1.05e-73)
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * (((l / Om) ^ 2.0) * U_42_)) - ((l / Om) * (2.0 * l))))));
	elseif (n <= 1.05e-190)
		tmp = sqrt(((-4.0 * ((U * (n * (l ^ 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	else
		tmp = (((2.0 * n) * U) * (t + (-2.0 * ((l ^ 2.0) / Om)))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -1.05e-73], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(n * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * U$42$), $MachinePrecision]), $MachinePrecision] - N[(N[(l / Om), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 1.05e-190], N[Sqrt[N[(N[(-4.0 * N[(N[(U * N[(n * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(-2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.0499999999999999e-73

   1. Initial program 49.6%

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

   3. Simplified 53.9%

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

      1. associate-*r* 53.9%

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

      2. fma-define 56.9%

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

      3. associate-*r* 56.9%

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

   6. Applied egg-rr 56.9%

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

   7. Taylor expanded in U around 0 47.6%

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

      1. associate-/l* 47.0%

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

      2. unpow2 47.0%

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

      3. unpow2 47.0%

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

      4. times-frac 56.9%

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

      5. unpow2 56.9%

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

      6. neg-mul-1 56.9%

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

      7. distribute-lft-neg-out 56.9%

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

      8. *-commutative 56.9%

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

   9. Simplified 56.9%

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

       1. fma-undefine 53.9%

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

   11. Applied egg-rr 53.9%

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

   if -1.0499999999999999e-73 < n < 1.04999999999999996e-190

   1. Initial program 38.9%

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

   3. Simplified 40.8%

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

   5. Taylor expanded in Om around inf 49.7%

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

   if 1.04999999999999996e-190 < n

   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 60.6%

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

   5. Taylor expanded in Om around inf 51.8%

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

      1. pow1/2 67.2%

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

      2. associate-*r* 67.1%

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

      3. cancel-sign-sub-inv 67.1%

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

      4. metadata-eval 67.1%

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

   7. Applied egg-rr 67.1%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.05 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot U*\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right)}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 38.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot \sqrt{2}\\ t_2 := \sqrt{U \cdot U*}\\ \mathbf{if}\;Om \leq -7.2 \cdot 10^{-141}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}\\ \mathbf{elif}\;Om \leq -9.5 \cdot 10^{-206}:\\ \;\;\;\;\frac{\ell \cdot t\_1}{Om} \cdot t\_2\\ \mathbf{elif}\;Om \leq 4.2 \cdot 10^{-73}:\\ \;\;\;\;t\_2 \cdot \left(\ell \cdot \frac{t\_1}{-Om}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left|t \cdot \left(n \cdot U\right)\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (sqrt 2.0))) (t_2 (sqrt (* U U*))))
   (if (<= Om -7.2e-141)
     (pow (* (* 2.0 U) (* n t)) 0.5)
     (if (<= Om -9.5e-206)
       (* (/ (* l t_1) Om) t_2)
       (if (<= Om 4.2e-73)
         (* t_2 (* l (/ t_1 (- Om))))
         (sqrt (* 2.0 (fabs (* t (* n U))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * sqrt(2.0);
	double t_2 = sqrt((U * U_42_));
	double tmp;
	if (Om <= -7.2e-141) {
		tmp = pow(((2.0 * U) * (n * t)), 0.5);
	} else if (Om <= -9.5e-206) {
		tmp = ((l * t_1) / Om) * t_2;
	} else if (Om <= 4.2e-73) {
		tmp = t_2 * (l * (t_1 / -Om));
	} else {
		tmp = sqrt((2.0 * fabs((t * (n * U)))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = n * sqrt(2.0d0)
    t_2 = sqrt((u * u_42))
    if (om <= (-7.2d-141)) then
        tmp = ((2.0d0 * u) * (n * t)) ** 0.5d0
    else if (om <= (-9.5d-206)) then
        tmp = ((l * t_1) / om) * t_2
    else if (om <= 4.2d-73) then
        tmp = t_2 * (l * (t_1 / -om))
    else
        tmp = sqrt((2.0d0 * abs((t * (n * u)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * Math.sqrt(2.0);
	double t_2 = Math.sqrt((U * U_42_));
	double tmp;
	if (Om <= -7.2e-141) {
		tmp = Math.pow(((2.0 * U) * (n * t)), 0.5);
	} else if (Om <= -9.5e-206) {
		tmp = ((l * t_1) / Om) * t_2;
	} else if (Om <= 4.2e-73) {
		tmp = t_2 * (l * (t_1 / -Om));
	} else {
		tmp = Math.sqrt((2.0 * Math.abs((t * (n * U)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = n * math.sqrt(2.0)
	t_2 = math.sqrt((U * U_42_))
	tmp = 0
	if Om <= -7.2e-141:
		tmp = math.pow(((2.0 * U) * (n * t)), 0.5)
	elif Om <= -9.5e-206:
		tmp = ((l * t_1) / Om) * t_2
	elif Om <= 4.2e-73:
		tmp = t_2 * (l * (t_1 / -Om))
	else:
		tmp = math.sqrt((2.0 * math.fabs((t * (n * U)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * sqrt(2.0))
	t_2 = sqrt(Float64(U * U_42_))
	tmp = 0.0
	if (Om <= -7.2e-141)
		tmp = Float64(Float64(2.0 * U) * Float64(n * t)) ^ 0.5;
	elseif (Om <= -9.5e-206)
		tmp = Float64(Float64(Float64(l * t_1) / Om) * t_2);
	elseif (Om <= 4.2e-73)
		tmp = Float64(t_2 * Float64(l * Float64(t_1 / Float64(-Om))));
	else
		tmp = sqrt(Float64(2.0 * abs(Float64(t * Float64(n * U)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = n * sqrt(2.0);
	t_2 = sqrt((U * U_42_));
	tmp = 0.0;
	if (Om <= -7.2e-141)
		tmp = ((2.0 * U) * (n * t)) ^ 0.5;
	elseif (Om <= -9.5e-206)
		tmp = ((l * t_1) / Om) * t_2;
	elseif (Om <= 4.2e-73)
		tmp = t_2 * (l * (t_1 / -Om));
	else
		tmp = sqrt((2.0 * abs((t * (n * U)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -7.2e-141], N[Power[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[Om, -9.5e-206], N[(N[(N[(l * t$95$1), $MachinePrecision] / Om), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[Om, 4.2e-73], N[(t$95$2 * N[(l * N[(t$95$1 / (-Om)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[Abs[N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if Om < -7.2000000000000003e-141

   1. Initial program 46.4%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in t around inf 38.7%

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

      1. pow1/2 44.6%

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

      2. associate-*r* 44.6%

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

   7. Applied egg-rr 44.6%

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

   if -7.2000000000000003e-141 < Om < -9.49999999999999979e-206

   1. Initial program 31.4%

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

   3. Simplified 31.4%

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

   5. Taylor expanded in U* around inf 40.4%

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

   if -9.49999999999999979e-206 < Om < 4.1999999999999997e-73

   1. Initial program 39.5%

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

   3. Simplified 38.1%

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

   5. Taylor expanded in Om around 0 40.0%

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

   6. Taylor expanded in U around 0 40.0%

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

      1. mul-1-neg 40.0%

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

      2. distribute-rgt-neg-in 40.0%

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

      3. *-commutative 40.0%

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

      4. distribute-rgt-neg-in 40.0%

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

   8. Simplified 40.0%

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

   9. Taylor expanded in l around -inf 29.9%

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

       1. mul-1-neg 29.9%

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

       2. distribute-rgt-neg-in 29.9%

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

       3. associate-/l* 31.4%

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

   11. Simplified 31.4%

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

   if 4.1999999999999997e-73 < Om

   1. Initial program 63.0%

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

   3. Simplified 64.3%

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

   5. Taylor expanded in t around inf 49.5%

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

      1. associate-*r* 50.1%

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

      2. add-sqr-sqrt 50.0%

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

      3. pow1/2 50.0%

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

      4. pow1/2 53.7%

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

      5. pow-prod-down 35.3%

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

      6. pow2 35.3%

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

      7. associate-*r* 34.1%

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

   7. Applied egg-rr 34.1%

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

      1. unpow1/2 34.1%

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

      2. unpow2 34.1%

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

      3. rem-sqrt-square 50.3%

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

      4. associate-*r* 54.3%

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

   9. Simplified 54.3%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -7.2 \cdot 10^{-141}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}\\ \mathbf{elif}\;Om \leq -9.5 \cdot 10^{-206}:\\ \;\;\;\;\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{U \cdot U*}\\ \mathbf{elif}\;Om \leq 4.2 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{U \cdot U*} \cdot \left(\ell \cdot \frac{n \cdot \sqrt{2}}{-Om}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left|t \cdot \left(n \cdot U\right)\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 39.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \left(n \cdot \sqrt{2}\right)\\ t_2 := \sqrt{U \cdot U*}\\ \mathbf{if}\;Om \leq -1 \cdot 10^{-141}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}\\ \mathbf{elif}\;Om \leq -1.5 \cdot 10^{-205}:\\ \;\;\;\;\frac{t\_1}{Om} \cdot t\_2\\ \mathbf{elif}\;Om \leq 6 \cdot 10^{-91}:\\ \;\;\;\;\frac{t\_1 \cdot t\_2}{-Om}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left|t \cdot \left(n \cdot U\right)\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (* n (sqrt 2.0)))) (t_2 (sqrt (* U U*))))
   (if (<= Om -1e-141)
     (pow (* (* 2.0 U) (* n t)) 0.5)
     (if (<= Om -1.5e-205)
       (* (/ t_1 Om) t_2)
       (if (<= Om 6e-91)
         (/ (* t_1 t_2) (- Om))
         (sqrt (* 2.0 (fabs (* t (* n U))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (n * sqrt(2.0));
	double t_2 = sqrt((U * U_42_));
	double tmp;
	if (Om <= -1e-141) {
		tmp = pow(((2.0 * U) * (n * t)), 0.5);
	} else if (Om <= -1.5e-205) {
		tmp = (t_1 / Om) * t_2;
	} else if (Om <= 6e-91) {
		tmp = (t_1 * t_2) / -Om;
	} else {
		tmp = sqrt((2.0 * fabs((t * (n * U)))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = l * (n * sqrt(2.0d0))
    t_2 = sqrt((u * u_42))
    if (om <= (-1d-141)) then
        tmp = ((2.0d0 * u) * (n * t)) ** 0.5d0
    else if (om <= (-1.5d-205)) then
        tmp = (t_1 / om) * t_2
    else if (om <= 6d-91) then
        tmp = (t_1 * t_2) / -om
    else
        tmp = sqrt((2.0d0 * abs((t * (n * u)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (n * Math.sqrt(2.0));
	double t_2 = Math.sqrt((U * U_42_));
	double tmp;
	if (Om <= -1e-141) {
		tmp = Math.pow(((2.0 * U) * (n * t)), 0.5);
	} else if (Om <= -1.5e-205) {
		tmp = (t_1 / Om) * t_2;
	} else if (Om <= 6e-91) {
		tmp = (t_1 * t_2) / -Om;
	} else {
		tmp = Math.sqrt((2.0 * Math.abs((t * (n * U)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = l * (n * math.sqrt(2.0))
	t_2 = math.sqrt((U * U_42_))
	tmp = 0
	if Om <= -1e-141:
		tmp = math.pow(((2.0 * U) * (n * t)), 0.5)
	elif Om <= -1.5e-205:
		tmp = (t_1 / Om) * t_2
	elif Om <= 6e-91:
		tmp = (t_1 * t_2) / -Om
	else:
		tmp = math.sqrt((2.0 * math.fabs((t * (n * U)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(n * sqrt(2.0)))
	t_2 = sqrt(Float64(U * U_42_))
	tmp = 0.0
	if (Om <= -1e-141)
		tmp = Float64(Float64(2.0 * U) * Float64(n * t)) ^ 0.5;
	elseif (Om <= -1.5e-205)
		tmp = Float64(Float64(t_1 / Om) * t_2);
	elseif (Om <= 6e-91)
		tmp = Float64(Float64(t_1 * t_2) / Float64(-Om));
	else
		tmp = sqrt(Float64(2.0 * abs(Float64(t * Float64(n * U)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l * (n * sqrt(2.0));
	t_2 = sqrt((U * U_42_));
	tmp = 0.0;
	if (Om <= -1e-141)
		tmp = ((2.0 * U) * (n * t)) ^ 0.5;
	elseif (Om <= -1.5e-205)
		tmp = (t_1 / Om) * t_2;
	elseif (Om <= 6e-91)
		tmp = (t_1 * t_2) / -Om;
	else
		tmp = sqrt((2.0 * abs((t * (n * U)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -1e-141], N[Power[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[Om, -1.5e-205], N[(N[(t$95$1 / Om), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[Om, 6e-91], N[(N[(t$95$1 * t$95$2), $MachinePrecision] / (-Om)), $MachinePrecision], N[Sqrt[N[(2.0 * N[Abs[N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if Om < -1e-141

   1. Initial program 46.4%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in t around inf 38.7%

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

      1. pow1/2 44.6%

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

      2. associate-*r* 44.6%

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

   7. Applied egg-rr 44.6%

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

   if -1e-141 < Om < -1.5e-205

   1. Initial program 31.4%

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

   3. Simplified 31.4%

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

   5. Taylor expanded in U* around inf 40.4%

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

   if -1.5e-205 < Om < 6.0000000000000004e-91

   1. Initial program 40.6%

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

   3. Simplified 37.6%

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

   5. Taylor expanded in Om around 0 39.6%

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

   6. Taylor expanded in U around 0 39.5%

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

      1. mul-1-neg 39.5%

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

      2. distribute-rgt-neg-in 39.5%

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

      3. *-commutative 39.5%

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

      4. distribute-rgt-neg-in 39.5%

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

   8. Simplified 39.5%

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

   9. Taylor expanded in l around -inf 30.8%

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

       1. mul-1-neg 30.8%

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

       2. associate-*l/ 30.7%

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

   11. Simplified 30.7%

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

   if 6.0000000000000004e-91 < Om

   1. Initial program 61.6%

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

   3. Simplified 62.8%

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

   5. Taylor expanded in t around inf 49.6%

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

      1. associate-*r* 49.0%

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

      2. add-sqr-sqrt 48.9%

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

      3. pow1/2 48.9%

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

      4. pow1/2 52.5%

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

      5. pow-prod-down 34.6%

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

      6. pow2 34.6%

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

      7. associate-*r* 34.5%

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

   7. Applied egg-rr 34.5%

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

      1. unpow1/2 34.5%

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

      2. unpow2 34.5%

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

      3. rem-sqrt-square 50.3%

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

      4. associate-*r* 53.1%

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

   9. Simplified 53.1%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1 \cdot 10^{-141}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}\\ \mathbf{elif}\;Om \leq -1.5 \cdot 10^{-205}:\\ \;\;\;\;\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{U \cdot U*}\\ \mathbf{elif}\;Om \leq 6 \cdot 10^{-91}:\\ \;\;\;\;\frac{\left(\ell \cdot \left(n \cdot \sqrt{2}\right)\right) \cdot \sqrt{U \cdot U*}}{-Om}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left|t \cdot \left(n \cdot U\right)\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq -1.3 \cdot 10^{+29} \lor \neg \left(U \leq 1.5 \cdot 10^{-38}\right):\\ \;\;\;\;\sqrt{2 \cdot \left|t \cdot \left(n \cdot U\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= U -1.3e+29) (not (<= U 1.5e-38)))
   (sqrt (* 2.0 (fabs (* t (* n U)))))
   (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 ((U <= -1.3e+29) || !(U <= 1.5e-38)) {
		tmp = sqrt((2.0 * fabs((t * (n * U)))));
	} 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 ((u <= (-1.3d+29)) .or. (.not. (u <= 1.5d-38))) then
        tmp = sqrt((2.0d0 * abs((t * (n * u)))))
    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 ((U <= -1.3e+29) || !(U <= 1.5e-38)) {
		tmp = Math.sqrt((2.0 * Math.abs((t * (n * U)))));
	} 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 (U <= -1.3e+29) or not (U <= 1.5e-38):
		tmp = math.sqrt((2.0 * math.fabs((t * (n * U)))))
	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 ((U <= -1.3e+29) || !(U <= 1.5e-38))
		tmp = sqrt(Float64(2.0 * abs(Float64(t * Float64(n * U)))));
	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 ((U <= -1.3e+29) || ~((U <= 1.5e-38)))
		tmp = sqrt((2.0 * abs((t * (n * U)))));
	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[Or[LessEqual[U, -1.3e+29], N[Not[LessEqual[U, 1.5e-38]], $MachinePrecision]], N[Sqrt[N[(2.0 * N[Abs[N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]], $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 2 regimes

2. if U < -1.3e29 or 1.49999999999999994e-38 < U

   1. Initial program 55.4%

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

   3. Simplified 56.5%

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

   5. Taylor expanded in t around inf 40.8%

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

      1. associate-*r* 42.7%

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

      2. add-sqr-sqrt 42.5%

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

      3. pow1/2 42.5%

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

      4. pow1/2 48.6%

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

      5. pow-prod-down 32.2%

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

      6. pow2 32.2%

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

      7. associate-*r* 31.3%

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

   7. Applied egg-rr 31.3%

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

      1. unpow1/2 31.3%

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

      2. unpow2 31.3%

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

      3. rem-sqrt-square 43.7%

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

      4. associate-*r* 49.4%

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

   9. Simplified 49.4%

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

   if -1.3e29 < U < 1.49999999999999994e-38

   1. Initial program 45.5%

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

   3. Simplified 54.3%

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

   5. Taylor expanded in Om around inf 47.2%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -1.3 \cdot 10^{+29} \lor \neg \left(U \leq 1.5 \cdot 10^{-38}\right):\\ \;\;\;\;\sqrt{2 \cdot \left|t \cdot \left(n \cdot U\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 38.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.85 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{2 \cdot \left|t \cdot \left(n \cdot U\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{U \cdot U*} \cdot \frac{n \cdot \sqrt{2}}{Om}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.85e+153)
   (sqrt (* 2.0 (fabs (* t (* n U)))))
   (* l (* (sqrt (* U U*)) (/ (* n (sqrt 2.0)) Om)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.85e+153) {
		tmp = sqrt((2.0 * fabs((t * (n * U)))));
	} else {
		tmp = l * (sqrt((U * U_42_)) * ((n * sqrt(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 (l <= 1.85d+153) then
        tmp = sqrt((2.0d0 * abs((t * (n * u)))))
    else
        tmp = l * (sqrt((u * u_42)) * ((n * sqrt(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 (l <= 1.85e+153) {
		tmp = Math.sqrt((2.0 * Math.abs((t * (n * U)))));
	} else {
		tmp = l * (Math.sqrt((U * U_42_)) * ((n * Math.sqrt(2.0)) / Om));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.85e+153:
		tmp = math.sqrt((2.0 * math.fabs((t * (n * U)))))
	else:
		tmp = l * (math.sqrt((U * U_42_)) * ((n * math.sqrt(2.0)) / Om))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.85e+153)
		tmp = sqrt(Float64(2.0 * abs(Float64(t * Float64(n * U)))));
	else
		tmp = Float64(l * Float64(sqrt(Float64(U * U_42_)) * Float64(Float64(n * sqrt(2.0)) / Om)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.85e+153)
		tmp = sqrt((2.0 * abs((t * (n * U)))));
	else
		tmp = l * (sqrt((U * U_42_)) * ((n * sqrt(2.0)) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.85e+153], N[Sqrt[N[(2.0 * N[Abs[N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l * N[(N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision] * N[(N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.8500000000000001e153

   1. Initial program 53.3%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in t around inf 40.3%

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

      1. associate-*r* 39.2%

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

      2. add-sqr-sqrt 39.1%

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

      3. pow1/2 39.1%

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

      4. pow1/2 42.7%

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

      5. pow-prod-down 31.5%

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

      6. pow2 31.5%

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

      7. associate-*r* 32.3%

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

   7. Applied egg-rr 32.3%

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

      1. unpow1/2 32.3%

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

      2. unpow2 32.3%

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

      3. rem-sqrt-square 43.7%

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

      4. associate-*r* 43.5%

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

   9. Simplified 43.5%

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

   if 1.8500000000000001e153 < l

   1. Initial program 20.8%

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

   3. Simplified 35.9%

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

      1. associate-*r* 35.9%

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

      2. fma-define 39.2%

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

      3. associate-*r* 36.0%

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

   6. Applied egg-rr 36.0%

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

   7. Taylor expanded in U around 0 23.1%

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

      1. associate-/l* 23.1%

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

      2. unpow2 23.1%

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

      3. unpow2 23.1%

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

      4. times-frac 36.0%

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

      5. unpow2 36.0%

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

      6. neg-mul-1 36.0%

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

      7. distribute-lft-neg-out 36.0%

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

      8. *-commutative 36.0%

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

   9. Simplified 36.0%

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

   10. Taylor expanded in n around inf 26.8%

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

       1. associate-/l* 29.8%

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

       2. associate-*r/ 29.8%

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

       3. associate-*r* 29.8%

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

       4. *-commutative 29.8%

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

       5. associate-*r/ 29.8%

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

   12. Simplified 29.8%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.85 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{2 \cdot \left|t \cdot \left(n \cdot U\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\sqrt{U \cdot U*} \cdot \frac{n \cdot \sqrt{2}}{Om}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 38.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.9 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{2 \cdot \left|t \cdot \left(n \cdot U\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(n \cdot {\ell}^{2}\right) \cdot \left(U \cdot -4\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 4.9e+143)
   (sqrt (* 2.0 (fabs (* t (* n U)))))
   (sqrt (/ (* (* n (pow l 2.0)) (* U -4.0)) Om))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 4.9e+143) {
		tmp = sqrt((2.0 * fabs((t * (n * U)))));
	} else {
		tmp = sqrt((((n * pow(l, 2.0)) * (U * -4.0)) / 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 (l <= 4.9d+143) then
        tmp = sqrt((2.0d0 * abs((t * (n * u)))))
    else
        tmp = sqrt((((n * (l ** 2.0d0)) * (u * (-4.0d0))) / 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 (l <= 4.9e+143) {
		tmp = Math.sqrt((2.0 * Math.abs((t * (n * U)))));
	} else {
		tmp = Math.sqrt((((n * Math.pow(l, 2.0)) * (U * -4.0)) / Om));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 4.9e+143:
		tmp = math.sqrt((2.0 * math.fabs((t * (n * U)))))
	else:
		tmp = math.sqrt((((n * math.pow(l, 2.0)) * (U * -4.0)) / Om))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 4.9e+143)
		tmp = sqrt(Float64(2.0 * abs(Float64(t * Float64(n * U)))));
	else
		tmp = sqrt(Float64(Float64(Float64(n * (l ^ 2.0)) * Float64(U * -4.0)) / Om));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 4.9e+143)
		tmp = sqrt((2.0 * abs((t * (n * U)))));
	else
		tmp = sqrt((((n * (l ^ 2.0)) * (U * -4.0)) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 4.9e+143], N[Sqrt[N[(2.0 * N[Abs[N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(n * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(U * -4.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.89999999999999986e143

   1. Initial program 53.2%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in t around inf 40.4%

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

      1. associate-*r* 39.7%

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

      2. add-sqr-sqrt 39.6%

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

      3. pow1/2 39.6%

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

      4. pow1/2 43.2%

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

      5. pow-prod-down 31.9%

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

      6. pow2 31.9%

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

      7. associate-*r* 32.3%

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

   7. Applied egg-rr 32.3%

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

      1. unpow1/2 32.3%

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

      2. unpow2 32.3%

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

      3. rem-sqrt-square 43.8%

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

      4. associate-*r* 44.0%

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

   9. Simplified 44.0%

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

   if 4.89999999999999986e143 < l

   1. Initial program 24.9%

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

   3. Simplified 39.5%

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

   5. Taylor expanded in Om around inf 28.6%

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

   6. Taylor expanded in t around 0 28.6%

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

      1. associate-*r/ 28.6%

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

      2. associate-*r* 28.6%

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

      3. *-commutative 28.6%

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

   8. Simplified 28.6%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.9 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{2 \cdot \left|t \cdot \left(n \cdot U\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(n \cdot {\ell}^{2}\right) \cdot \left(U \cdot -4\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 48.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ {\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}^{0.5} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (pow (* (* (* 2.0 n) U) (+ t (* -2.0 (/ (pow l 2.0) Om)))) 0.5))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return pow((((2.0 * n) * U) * (t + (-2.0 * (pow(l, 2.0) / Om)))), 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 * n) * u) * (t + ((-2.0d0) * ((l ** 2.0d0) / om)))) ** 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 * n) * U) * (t + (-2.0 * (Math.pow(l, 2.0) / Om)))), 0.5);
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.pow((((2.0 * n) * U) * (t + (-2.0 * (math.pow(l, 2.0) / Om)))), 0.5)

Julia

function code(n, U, t, l, Om, U_42_)
	return Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(-2.0 * Float64((l ^ 2.0) / Om)))) ^ 0.5
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = (((2.0 * n) * U) * (t + (-2.0 * ((l ^ 2.0) / Om)))) ^ 0.5;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Power[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(-2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]

Derivation

1. Initial program 49.4%

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

3. Simplified 49.9%

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

5. Taylor expanded in Om around inf 43.3%

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

   1. pow1/2 52.2%

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

   2. associate-*r* 51.8%

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

   3. cancel-sign-sub-inv 51.8%

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

   4. metadata-eval 51.8%

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

7. Applied egg-rr 51.8%

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

Alternative 13: 37.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 1.06 \cdot 10^{-243}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left|t \cdot \left(n \cdot U\right)\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n 1.06e-243)
   (pow (* (* 2.0 U) (* n t)) 0.5)
   (sqrt (* 2.0 (fabs (* t (* n U)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 1.06e-243) {
		tmp = pow(((2.0 * U) * (n * t)), 0.5);
	} else {
		tmp = sqrt((2.0 * fabs((t * (n * U)))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= 1.06d-243) then
        tmp = ((2.0d0 * u) * (n * t)) ** 0.5d0
    else
        tmp = sqrt((2.0d0 * abs((t * (n * u)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 1.06e-243) {
		tmp = Math.pow(((2.0 * U) * (n * t)), 0.5);
	} else {
		tmp = Math.sqrt((2.0 * Math.abs((t * (n * U)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= 1.06e-243:
		tmp = math.pow(((2.0 * U) * (n * t)), 0.5)
	else:
		tmp = math.sqrt((2.0 * math.fabs((t * (n * U)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= 1.06e-243)
		tmp = Float64(Float64(2.0 * U) * Float64(n * t)) ^ 0.5;
	else
		tmp = sqrt(Float64(2.0 * abs(Float64(t * Float64(n * U)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= 1.06e-243)
		tmp = ((2.0 * U) * (n * t)) ^ 0.5;
	else
		tmp = sqrt((2.0 * abs((t * (n * U)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, 1.06e-243], N[Power[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(2.0 * N[Abs[N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 1.06e-243

   1. Initial program 43.3%

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

   3. Simplified 45.5%

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

   5. Taylor expanded in t around inf 32.2%

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

      1. pow1/2 33.6%

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

      2. associate-*r* 33.6%

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

   7. Applied egg-rr 33.6%

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

   if 1.06e-243 < n

   1. Initial program 56.9%

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

   3. Simplified 59.2%

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

   5. Taylor expanded in t around inf 40.7%

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

      1. associate-*r* 44.1%

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

      2. add-sqr-sqrt 43.9%

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

      3. pow1/2 43.9%

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

      4. pow1/2 51.0%

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

      5. pow-prod-down 39.5%

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

      6. pow2 39.5%

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

      7. associate-*r* 37.8%

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

   7. Applied egg-rr 37.8%

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

      1. unpow1/2 37.8%

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

      2. unpow2 37.8%

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

      3. rem-sqrt-square 46.8%

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

      4. associate-*r* 51.9%

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

   9. Simplified 51.9%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 1.06 \cdot 10^{-243}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left|t \cdot \left(n \cdot U\right)\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 37.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ {\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (pow (* (* 2.0 U) (* n t)) 0.5))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return pow(((2.0 * U) * (n * t)), 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 * u) * (n * t)) ** 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 * U) * (n * t)), 0.5);
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.pow(((2.0 * U) * (n * t)), 0.5)

Julia

function code(n, U, t, l, Om, U_42_)
	return Float64(Float64(2.0 * U) * Float64(n * t)) ^ 0.5
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = ((2.0 * U) * (n * t)) ^ 0.5;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Power[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]

Derivation

1. Initial program 49.4%

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

3. Simplified 51.7%

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

5. Taylor expanded in t around inf 36.0%

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

   1. pow1/2 39.2%

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

   2. associate-*r* 39.2%

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

7. Applied egg-rr 39.2%

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

Alternative 15: 35.6% 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 49.4%

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

3. Simplified 51.7%

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

5. Taylor expanded in t around inf 36.0%

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

Reproduce

herbie shell --seed 2024089 
(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*))))))