Toniolo and Linder, Equation (13)

Percentage Accurate: 50.5% → 63.9%

Time: 26.4s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.5% 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: 63.9% accurate, 0.5× speedup

Math

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

FPCore

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

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $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] + N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(n * N[(U - U$42$), $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[Power[l, 2.0], $MachinePrecision] * -2.0), $MachinePrecision] - N[(N[(N[Power[l, 2.0], $MachinePrecision] * t$95$3), $MachinePrecision] / Om), $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[(N[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(U * N[(N[Power[l, 2.0], $MachinePrecision] * N[(n * N[(2.0 + N[(t$95$3 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $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 16.2%

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

   3. Simplified 36.4%

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

   5. Taylor expanded in Om around -inf 42.2%

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

   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 66.5%

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

   3. Simplified 74.0%

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

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

   1. Initial program 0.0%

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

   3. Simplified 7.8%

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

   5. Taylor expanded in Om around -inf 10.0%

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

   6. Taylor expanded in l around inf 46.1%

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

4. Final simplification 65.1%

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

Alternative 2: 63.5% accurate, 0.5× speedup

Math

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

FPCore

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

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t$95$2 + N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(U * N[(N[Power[l, 2.0], $MachinePrecision] * N[(n * N[(2.0 + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $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 16.2%

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

   3. Simplified 36.4%

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

   5. Taylor expanded in n around 0 39.3%

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

      1. unpow2 39.3%

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

   7. Applied egg-rr 39.3%

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

   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 66.5%

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

   3. Simplified 74.0%

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

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

   1. Initial program 0.0%

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

   3. Simplified 7.8%

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

   5. Taylor expanded in Om around -inf 10.0%

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

   6. Taylor expanded in l around inf 46.1%

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

4. Final simplification 64.7%

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

Alternative 3: 51.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{-171} \lor \neg \left(n \leq 7.8 \cdot 10^{-166}\right):\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= n -1.15e-171) (not (<= n 7.8e-166)))
   (pow (* (* (* 2.0 n) U) (+ t (* -2.0 (/ (pow l 2.0) Om)))) 0.5)
   (sqrt (+ (* -4.0 (/ (* U (* n (pow l 2.0))) Om)) (* 2.0 (* U (* n t)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((n <= -1.15e-171) || !(n <= 7.8e-166)) {
		tmp = pow((((2.0 * n) * U) * (t + (-2.0 * (pow(l, 2.0) / Om)))), 0.5);
	} else {
		tmp = sqrt(((-4.0 * ((U * (n * pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((n <= (-1.15d-171)) .or. (.not. (n <= 7.8d-166))) then
        tmp = (((2.0d0 * n) * u) * (t + ((-2.0d0) * ((l ** 2.0d0) / om)))) ** 0.5d0
    else
        tmp = sqrt((((-4.0d0) * ((u * (n * (l ** 2.0d0))) / om)) + (2.0d0 * (u * (n * t)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((n <= -1.15e-171) || !(n <= 7.8e-166)) {
		tmp = Math.pow((((2.0 * n) * U) * (t + (-2.0 * (Math.pow(l, 2.0) / Om)))), 0.5);
	} else {
		tmp = Math.sqrt(((-4.0 * ((U * (n * Math.pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (n <= -1.15e-171) or not (n <= 7.8e-166):
		tmp = math.pow((((2.0 * n) * U) * (t + (-2.0 * (math.pow(l, 2.0) / Om)))), 0.5)
	else:
		tmp = math.sqrt(((-4.0 * ((U * (n * math.pow(l, 2.0))) / Om)) + (2.0 * (U * (n * t)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((n <= -1.15e-171) || !(n <= 7.8e-166))
		tmp = Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(-2.0 * Float64((l ^ 2.0) / Om)))) ^ 0.5;
	else
		tmp = sqrt(Float64(Float64(-4.0 * Float64(Float64(U * Float64(n * (l ^ 2.0))) / Om)) + Float64(2.0 * Float64(U * Float64(n * t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((n <= -1.15e-171) || ~((n <= 7.8e-166)))
		tmp = (((2.0 * n) * U) * (t + (-2.0 * ((l ^ 2.0) / Om)))) ^ 0.5;
	else
		tmp = sqrt(((-4.0 * ((U * (n * (l ^ 2.0))) / Om)) + (2.0 * (U * (n * t)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[n, -1.15e-171], N[Not[LessEqual[n, 7.8e-166]], $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], 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]]

Derivation

1. Split input into 2 regimes

2. if n < -1.14999999999999989e-171 or 7.79999999999999998e-166 < n

   1. Initial program 49.7%

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

   3. Simplified 55.6%

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

   5. Taylor expanded in n around 0 40.0%

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

      1. pow1/2 48.5%

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

      2. associate-*r* 50.6%

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

      3. cancel-sign-sub-inv 50.6%

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

      4. metadata-eval 50.6%

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

   7. Applied egg-rr 50.6%

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

   if -1.14999999999999989e-171 < n < 7.79999999999999998e-166

   1. Initial program 45.9%

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

   3. Simplified 47.4%

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

   5. Taylor expanded in Om around inf 57.2%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{-171} \lor \neg \left(n \leq 7.8 \cdot 10^{-166}\right):\\ \;\;\;\;{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 42.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{-56}:\\ \;\;\;\;{\left({\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.25}\right)}^{2}\\ \mathbf{elif}\;\ell \leq 2.35 \cdot 10^{+169}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - {\ell}^{2} \cdot \frac{2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(\sqrt{2} \cdot \left(n \cdot \sqrt{U \cdot U*}\right)\right)}{Om}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.15e-56)
   (pow (pow (* 2.0 (* t (* n U))) 0.25) 2.0)
   (if (<= l 2.35e+169)
     (sqrt (* 2.0 (* U (* n (- t (* (pow l 2.0) (/ 2.0 Om)))))))
     (/ (* l (* (sqrt 2.0) (* n (sqrt (* U U*))))) Om))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.15e-56) {
		tmp = pow(pow((2.0 * (t * (n * U))), 0.25), 2.0);
	} else if (l <= 2.35e+169) {
		tmp = sqrt((2.0 * (U * (n * (t - (pow(l, 2.0) * (2.0 / Om)))))));
	} else {
		tmp = (l * (sqrt(2.0) * (n * sqrt((U * U_42_))))) / 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 <= 1.15d-56) then
        tmp = ((2.0d0 * (t * (n * u))) ** 0.25d0) ** 2.0d0
    else if (l <= 2.35d+169) then
        tmp = sqrt((2.0d0 * (u * (n * (t - ((l ** 2.0d0) * (2.0d0 / om)))))))
    else
        tmp = (l * (sqrt(2.0d0) * (n * sqrt((u * u_42))))) / 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 <= 1.15e-56) {
		tmp = Math.pow(Math.pow((2.0 * (t * (n * U))), 0.25), 2.0);
	} else if (l <= 2.35e+169) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (Math.pow(l, 2.0) * (2.0 / Om)))))));
	} else {
		tmp = (l * (Math.sqrt(2.0) * (n * Math.sqrt((U * U_42_))))) / Om;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.15e-56:
		tmp = math.pow(math.pow((2.0 * (t * (n * U))), 0.25), 2.0)
	elif l <= 2.35e+169:
		tmp = math.sqrt((2.0 * (U * (n * (t - (math.pow(l, 2.0) * (2.0 / Om)))))))
	else:
		tmp = (l * (math.sqrt(2.0) * (n * math.sqrt((U * U_42_))))) / Om
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.15e-56)
		tmp = (Float64(2.0 * Float64(t * Float64(n * U))) ^ 0.25) ^ 2.0;
	elseif (l <= 2.35e+169)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64((l ^ 2.0) * Float64(2.0 / Om)))))));
	else
		tmp = Float64(Float64(l * Float64(sqrt(2.0) * Float64(n * sqrt(Float64(U * U_42_))))) / Om);
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.15e-56)
		tmp = ((2.0 * (t * (n * U))) ^ 0.25) ^ 2.0;
	elseif (l <= 2.35e+169)
		tmp = sqrt((2.0 * (U * (n * (t - ((l ^ 2.0) * (2.0 / Om)))))));
	else
		tmp = (l * (sqrt(2.0) * (n * sqrt((U * U_42_))))) / Om;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.15e-56], N[Power[N[Power[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[l, 2.35e+169], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] * N[(n * N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.15000000000000001e-56

   1. Initial program 57.5%

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

   3. Simplified 57.0%

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

   5. Taylor expanded in t around inf 42.5%

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

      1. add-sqr-sqrt 42.3%

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

      2. pow2 42.3%

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

      3. associate-*r* 42.4%

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

   7. Applied egg-rr 42.4%

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

      1. add-cbrt-cube 42.1%

         \[\leadsto {\color{blue}{\left(\sqrt[3]{\left(\sqrt{\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \cdot \sqrt{\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}}\right) \cdot \sqrt{\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}}}\right)}}^{2} \]

      2. pow1/3 39.8%

         \[\leadsto {\color{blue}{\left({\left(\left(\sqrt{\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \cdot \sqrt{\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}}\right) \cdot \sqrt{\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}}\right)}^{0.3333333333333333}\right)}}^{2} \]

      3. add-sqr-sqrt 39.8%

         \[\leadsto {\left({\left(\color{blue}{\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \cdot \sqrt{\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}}\right)}^{0.3333333333333333}\right)}^{2} \]

      4. pow1 39.8%

         \[\leadsto {\left({\left(\color{blue}{{\left(\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\right)}^{1}} \cdot \sqrt{\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}}\right)}^{0.3333333333333333}\right)}^{2} \]

      5. pow1/2 39.8%

         \[\leadsto {\left({\left({\left(\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\right)}^{1} \cdot \color{blue}{{\left(\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\right)}^{0.5}}\right)}^{0.3333333333333333}\right)}^{2} \]

      6. pow-prod-up 39.8%

         \[\leadsto {\left({\color{blue}{\left({\left(\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}\right)}^{2} \]

      7. metadata-eval 39.8%

         \[\leadsto {\left({\left({\left(\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}\right)}^{2} \]

   9. Applied egg-rr 39.8%

      \[\leadsto {\color{blue}{\left({\left({\left(\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}}^{2} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 39.8%

          \[\leadsto {\color{blue}{\left(1 \cdot {\left({\left(\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}}^{2} \]

       2. pow-pow 42.4%

          \[\leadsto {\left(1 \cdot \color{blue}{{\left(\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}\right)}^{2} \]

       3. sqrt-pow2 44.6%

          \[\leadsto {\left(1 \cdot \color{blue}{{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}}\right)}^{2} \]

       4. associate-*l* 44.6%

          \[\leadsto {\left(1 \cdot {\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}\right)}^{2} \]

       5. metadata-eval 44.6%

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

       6. metadata-eval 44.6%

          \[\leadsto {\left(1 \cdot {\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   11. Applied egg-rr 44.6%

       \[\leadsto {\color{blue}{\left(1 \cdot {\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.25}\right)}}^{2} \]
   12. Step-by-step derivation

       1. *-lft-identity 44.6%

          \[\leadsto {\color{blue}{\left({\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.25}\right)}}^{2} \]

       2. associate-*r* 46.8%

          \[\leadsto {\left({\left(2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}\right)}^{0.25}\right)}^{2} \]

   13. Simplified 46.8%

       \[\leadsto {\color{blue}{\left({\left(2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)}^{0.25}\right)}}^{2} \]

   if 1.15000000000000001e-56 < l < 2.3499999999999999e169

   1. Initial program 41.6%

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

   3. Simplified 45.7%

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

      1. associate-*r* 45.7%

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

      2. fma-define 45.7%

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

      3. associate-*r* 49.9%

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

   6. Applied egg-rr 49.9%

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

   7. Taylor expanded in n around 0 35.5%

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

      1. associate-*r/ 35.5%

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

      2. associate-*l/ 35.5%

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

   9. Simplified 35.5%

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

   if 2.3499999999999999e169 < l

   1. Initial program 2.2%

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

   3. Simplified 34.2%

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

   5. Taylor expanded in U* around inf 8.6%

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

      1. associate-*l/ 8.7%

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

   7. Simplified 8.7%

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

      1. pow1 8.7%

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

      2. associate-*l* 9.0%

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

      3. *-commutative 9.0%

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

   9. Applied egg-rr 9.0%

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

       1. unpow1 9.0%

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

       2. associate-*l* 9.0%

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

   11. Simplified 9.0%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{-56}:\\ \;\;\;\;{\left({\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.25}\right)}^{2}\\ \mathbf{elif}\;\ell \leq 2.35 \cdot 10^{+169}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - {\ell}^{2} \cdot \frac{2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(\sqrt{2} \cdot \left(n \cdot \sqrt{U \cdot U*}\right)\right)}{Om}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 42.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.3 \cdot 10^{-56}:\\ \;\;\;\;{\left({\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.25}\right)}^{2}\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+169}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(\sqrt{2} \cdot \left(n \cdot \sqrt{U \cdot U*}\right)\right)}{Om}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 3.3e-56)
   (pow (pow (* 2.0 (* t (* n U))) 0.25) 2.0)
   (if (<= l 1.4e+169)
     (sqrt (* 2.0 (* U (* n (- t (* 2.0 (/ (pow l 2.0) Om)))))))
     (/ (* l (* (sqrt 2.0) (* n (sqrt (* U U*))))) Om))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 3.3e-56) {
		tmp = pow(pow((2.0 * (t * (n * U))), 0.25), 2.0);
	} else if (l <= 1.4e+169) {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (pow(l, 2.0) / Om)))))));
	} else {
		tmp = (l * (sqrt(2.0) * (n * sqrt((U * U_42_))))) / 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 <= 3.3d-56) then
        tmp = ((2.0d0 * (t * (n * u))) ** 0.25d0) ** 2.0d0
    else if (l <= 1.4d+169) then
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * ((l ** 2.0d0) / om)))))))
    else
        tmp = (l * (sqrt(2.0d0) * (n * sqrt((u * u_42))))) / 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 <= 3.3e-56) {
		tmp = Math.pow(Math.pow((2.0 * (t * (n * U))), 0.25), 2.0);
	} else if (l <= 1.4e+169) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (Math.pow(l, 2.0) / Om)))))));
	} else {
		tmp = (l * (Math.sqrt(2.0) * (n * Math.sqrt((U * U_42_))))) / Om;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 3.3e-56:
		tmp = math.pow(math.pow((2.0 * (t * (n * U))), 0.25), 2.0)
	elif l <= 1.4e+169:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (math.pow(l, 2.0) / Om)))))))
	else:
		tmp = (l * (math.sqrt(2.0) * (n * math.sqrt((U * U_42_))))) / Om
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 3.3e-56)
		tmp = (Float64(2.0 * Float64(t * Float64(n * U))) ^ 0.25) ^ 2.0;
	elseif (l <= 1.4e+169)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64((l ^ 2.0) / Om)))))));
	else
		tmp = Float64(Float64(l * Float64(sqrt(2.0) * Float64(n * sqrt(Float64(U * U_42_))))) / Om);
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 3.3e-56)
		tmp = ((2.0 * (t * (n * U))) ^ 0.25) ^ 2.0;
	elseif (l <= 1.4e+169)
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * ((l ^ 2.0) / Om)))))));
	else
		tmp = (l * (sqrt(2.0) * (n * sqrt((U * U_42_))))) / Om;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 3.3e-56], N[Power[N[Power[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[l, 1.4e+169], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] * N[(n * N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.29999999999999984e-56

   1. Initial program 57.5%

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

   3. Simplified 57.0%

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

   5. Taylor expanded in t around inf 42.5%

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

      1. add-sqr-sqrt 42.3%

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

      2. pow2 42.3%

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

      3. associate-*r* 42.4%

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

   7. Applied egg-rr 42.4%

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

      1. add-cbrt-cube 42.1%

         \[\leadsto {\color{blue}{\left(\sqrt[3]{\left(\sqrt{\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \cdot \sqrt{\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}}\right) \cdot \sqrt{\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}}}\right)}}^{2} \]

      2. pow1/3 39.8%

         \[\leadsto {\color{blue}{\left({\left(\left(\sqrt{\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \cdot \sqrt{\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}}\right) \cdot \sqrt{\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}}\right)}^{0.3333333333333333}\right)}}^{2} \]

      3. add-sqr-sqrt 39.8%

         \[\leadsto {\left({\left(\color{blue}{\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \cdot \sqrt{\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}}\right)}^{0.3333333333333333}\right)}^{2} \]

      4. pow1 39.8%

         \[\leadsto {\left({\left(\color{blue}{{\left(\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\right)}^{1}} \cdot \sqrt{\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}}\right)}^{0.3333333333333333}\right)}^{2} \]

      5. pow1/2 39.8%

         \[\leadsto {\left({\left({\left(\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\right)}^{1} \cdot \color{blue}{{\left(\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\right)}^{0.5}}\right)}^{0.3333333333333333}\right)}^{2} \]

      6. pow-prod-up 39.8%

         \[\leadsto {\left({\color{blue}{\left({\left(\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}\right)}^{2} \]

      7. metadata-eval 39.8%

         \[\leadsto {\left({\left({\left(\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}\right)}^{2} \]

   9. Applied egg-rr 39.8%

      \[\leadsto {\color{blue}{\left({\left({\left(\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}}^{2} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 39.8%

          \[\leadsto {\color{blue}{\left(1 \cdot {\left({\left(\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\right)}^{1.5}\right)}^{0.3333333333333333}\right)}}^{2} \]

       2. pow-pow 42.4%

          \[\leadsto {\left(1 \cdot \color{blue}{{\left(\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}\right)}^{2} \]

       3. sqrt-pow2 44.6%

          \[\leadsto {\left(1 \cdot \color{blue}{{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}}\right)}^{2} \]

       4. associate-*l* 44.6%

          \[\leadsto {\left(1 \cdot {\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}\right)}^{2} \]

       5. metadata-eval 44.6%

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

       6. metadata-eval 44.6%

          \[\leadsto {\left(1 \cdot {\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   11. Applied egg-rr 44.6%

       \[\leadsto {\color{blue}{\left(1 \cdot {\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.25}\right)}}^{2} \]
   12. Step-by-step derivation

       1. *-lft-identity 44.6%

          \[\leadsto {\color{blue}{\left({\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.25}\right)}}^{2} \]

       2. associate-*r* 46.8%

          \[\leadsto {\left({\left(2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}\right)}^{0.25}\right)}^{2} \]

   13. Simplified 46.8%

       \[\leadsto {\color{blue}{\left({\left(2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)}^{0.25}\right)}}^{2} \]

   if 3.29999999999999984e-56 < l < 1.4000000000000001e169

   1. Initial program 41.6%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in n around 0 35.5%

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

   if 1.4000000000000001e169 < l

   1. Initial program 2.2%

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

   3. Simplified 34.2%

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

   5. Taylor expanded in U* around inf 8.6%

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

      1. associate-*l/ 8.7%

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

   7. Simplified 8.7%

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

      1. pow1 8.7%

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

      2. associate-*l* 9.0%

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

      3. *-commutative 9.0%

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

   9. Applied egg-rr 9.0%

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

       1. unpow1 9.0%

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

       2. associate-*l* 9.0%

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

   11. Simplified 9.0%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.3 \cdot 10^{-56}:\\ \;\;\;\;{\left({\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.25}\right)}^{2}\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+169}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(\sqrt{2} \cdot \left(n \cdot \sqrt{U \cdot U*}\right)\right)}{Om}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 46.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 6.6 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - {\ell}^{2} \cdot \frac{2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n 6.6e-164)
   (sqrt (* 2.0 (* U (* n (- t (* (pow l 2.0) (/ 2.0 Om)))))))
   (sqrt (* (* 2.0 (* n U)) (- t (/ (* 2.0 (pow l 2.0)) Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 6.6e-164) {
		tmp = sqrt((2.0 * (U * (n * (t - (pow(l, 2.0) * (2.0 / Om)))))));
	} else {
		tmp = sqrt(((2.0 * (n * U)) * (t - ((2.0 * pow(l, 2.0)) / Om))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= 6.6d-164) then
        tmp = sqrt((2.0d0 * (u * (n * (t - ((l ** 2.0d0) * (2.0d0 / om)))))))
    else
        tmp = sqrt(((2.0d0 * (n * u)) * (t - ((2.0d0 * (l ** 2.0d0)) / om))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 6.6e-164) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (Math.pow(l, 2.0) * (2.0 / Om)))))));
	} else {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t - ((2.0 * Math.pow(l, 2.0)) / Om))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= 6.6e-164:
		tmp = math.sqrt((2.0 * (U * (n * (t - (math.pow(l, 2.0) * (2.0 / Om)))))))
	else:
		tmp = math.sqrt(((2.0 * (n * U)) * (t - ((2.0 * math.pow(l, 2.0)) / Om))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= 6.6e-164)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64((l ^ 2.0) * Float64(2.0 / Om)))))));
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(Float64(2.0 * (l ^ 2.0)) / Om))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= 6.6e-164)
		tmp = sqrt((2.0 * (U * (n * (t - ((l ^ 2.0) * (2.0 / Om)))))));
	else
		tmp = sqrt(((2.0 * (n * U)) * (t - ((2.0 * (l ^ 2.0)) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, 6.6e-164], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 6.6e-164

   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 52.0%

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

      1. associate-*r* 52.0%

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

      2. fma-define 53.8%

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

      3. associate-*r* 51.4%

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

   6. Applied egg-rr 51.4%

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

   7. Taylor expanded in n around 0 45.1%

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

      1. associate-*r/ 45.1%

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

      2. associate-*l/ 45.1%

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

   9. Simplified 45.1%

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

   if 6.6e-164 < n

   1. Initial program 54.4%

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

   3. Simplified 57.8%

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

   5. Taylor expanded in Om around inf 47.7%

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

      1. associate-*r/ 47.7%

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

   7. Simplified 47.7%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 6.6 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - {\ell}^{2} \cdot \frac{2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.1% 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 48.6%

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

3. Simplified 53.3%

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

5. Taylor expanded in n around 0 41.5%

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

   1. pow1/2 47.7%

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

   2. associate-*r* 49.0%

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

   3. cancel-sign-sub-inv 49.0%

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

   4. metadata-eval 49.0%

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

7. Applied egg-rr 49.0%

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

Alternative 8: 45.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq 4.6 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U 4.6e+148)
   (sqrt (* (* 2.0 n) (* U (- t (* 2.0 (/ (* l l) Om))))))
   (* (sqrt (* 2.0 U)) (sqrt (* n t)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 4.6e+148) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * ((l * l) / Om))))));
	} else {
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= 4.6d+148) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * ((l * l) / om))))))
    else
        tmp = sqrt((2.0d0 * u)) * sqrt((n * t))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 4.6e+148) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * ((l * l) / Om))))));
	} else {
		tmp = Math.sqrt((2.0 * U)) * Math.sqrt((n * t));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= 4.6e+148:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * ((l * l) / Om))))))
	else:
		tmp = math.sqrt((2.0 * U)) * math.sqrt((n * t))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= 4.6e+148)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))))));
	else
		tmp = Float64(sqrt(Float64(2.0 * U)) * sqrt(Float64(n * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= 4.6e+148)
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * ((l * l) / Om))))));
	else
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, 4.6e+148], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 4.6000000000000001e148

   1. Initial program 46.9%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in n around 0 40.5%

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

      1. unpow2 40.5%

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

   7. Applied egg-rr 40.5%

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

   if 4.6000000000000001e148 < U

   1. Initial program 70.0%

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

   3. Simplified 59.9%

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

   5. Taylor expanded in t around inf 70.0%

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

      1. pow1/2 70.0%

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

      2. associate-*r* 70.0%

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

      3. unpow-prod-down 85.6%

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

      4. pow1/2 85.6%

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

   7. Applied egg-rr 85.6%

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

      1. unpow1/2 85.6%

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

   9. Simplified 85.6%

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

Alternative 9: 45.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{+163}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left|n \cdot U\right|\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 7.8e+163)
   (sqrt (* (* 2.0 n) (* U (- t (* 2.0 (/ (* l l) Om))))))
   (sqrt (* t (* 2.0 (fabs (* n U)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 7.8e+163) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * ((l * l) / Om))))));
	} else {
		tmp = sqrt((t * (2.0 * fabs((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 (t <= 7.8d+163) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * ((l * l) / om))))))
    else
        tmp = sqrt((t * (2.0d0 * abs((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 (t <= 7.8e+163) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * ((l * l) / Om))))));
	} else {
		tmp = Math.sqrt((t * (2.0 * Math.abs((n * U)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= 7.8e+163:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * ((l * l) / Om))))))
	else:
		tmp = math.sqrt((t * (2.0 * math.fabs((n * U)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 7.8e+163)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))))));
	else
		tmp = sqrt(Float64(t * Float64(2.0 * abs(Float64(n * U)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= 7.8e+163)
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * ((l * l) / Om))))));
	else
		tmp = sqrt((t * (2.0 * abs((n * U)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 7.8e+163], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t * N[(2.0 * N[Abs[N[(n * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7.80000000000000047e163

   1. Initial program 48.5%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in n around 0 41.5%

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

      1. unpow2 41.5%

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

   7. Applied egg-rr 41.5%

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

   if 7.80000000000000047e163 < t

   1. Initial program 49.9%

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

   3. Simplified 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. Step-by-step derivation

      1. associate-*r* 53.7%

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

      2. fma-define 61.7%

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

      3. associate-*r* 53.9%

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

   6. Applied egg-rr 53.9%

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

   7. Taylor expanded in t around inf 54.3%

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

      1. add-sqr-sqrt 54.3%

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

      2. sqrt-unprod 51.0%

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

      3. pow2 51.0%

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

   9. Applied egg-rr 51.0%

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

       1. *-commutative 51.0%

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

       2. unpow2 51.0%

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

       3. rem-sqrt-square 58.7%

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

   11. Simplified 58.7%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{+163}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left|n \cdot U\right|\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 44.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq 1.15 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U 1.15e+33)
   (sqrt (* (* 2.0 n) (* U (- t (* 2.0 (/ (* l l) Om))))))
   (sqrt (fabs (* (* n t) (* 2.0 U))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 1.15e+33) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * ((l * l) / Om))))));
	} else {
		tmp = sqrt(fabs(((n * t) * (2.0 * U))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= 1.15d+33) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * ((l * l) / om))))))
    else
        tmp = sqrt(abs(((n * t) * (2.0d0 * u))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 1.15e+33) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * ((l * l) / Om))))));
	} else {
		tmp = Math.sqrt(Math.abs(((n * t) * (2.0 * U))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= 1.15e+33:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * ((l * l) / Om))))))
	else:
		tmp = math.sqrt(math.fabs(((n * t) * (2.0 * U))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= 1.15e+33)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))))));
	else
		tmp = sqrt(abs(Float64(Float64(n * t) * Float64(2.0 * U))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= 1.15e+33)
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * ((l * l) / Om))))));
	else
		tmp = sqrt(abs(((n * t) * (2.0 * U))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, 1.15e+33], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 1.15000000000000005e33

   1. Initial program 46.1%

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

   3. Simplified 52.9%

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

   5. Taylor expanded in n around 0 40.5%

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

      1. unpow2 40.5%

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

   7. Applied egg-rr 40.5%

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

   if 1.15000000000000005e33 < U

   1. Initial program 64.4%

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

   3. Simplified 67.1%

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

      1. associate-*r* 67.1%

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

      2. fma-define 67.1%

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

      3. associate-*r* 61.6%

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

   6. Applied egg-rr 61.6%

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

   7. Taylor expanded in t around inf 50.4%

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

      1. pow1 50.4%

         \[\leadsto \sqrt{\color{blue}{{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)}^{1}}} \]

      2. metadata-eval 50.4%

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

      3. metadata-eval 50.4%

         \[\leadsto \sqrt{{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)}^{\left(2 \cdot \color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}\right)}} \]

      4. pow-sqr 56.1%

         \[\leadsto \sqrt{\color{blue}{{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}} \]

      5. pow-prod-down 43.1%

         \[\leadsto \sqrt{\color{blue}{{\left(\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right) \cdot \left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}} \]

      6. swap-sqr 28.1%

         \[\leadsto \sqrt{{\color{blue}{\left(\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right) \cdot \left(t \cdot t\right)\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      7. *-commutative 28.1%

         \[\leadsto \sqrt{{\left(\left(\left(2 \cdot \color{blue}{\left(U \cdot n\right)}\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right) \cdot \left(t \cdot t\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      8. associate-*r* 28.1%

         \[\leadsto \sqrt{{\left(\left(\color{blue}{\left(\left(2 \cdot U\right) \cdot n\right)} \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right) \cdot \left(t \cdot t\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      9. *-commutative 28.1%

         \[\leadsto \sqrt{{\left(\left(\left(\left(2 \cdot U\right) \cdot n\right) \cdot \left(2 \cdot \color{blue}{\left(U \cdot n\right)}\right)\right) \cdot \left(t \cdot t\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      10. associate-*r* 28.1%

          \[\leadsto \sqrt{{\left(\left(\left(\left(2 \cdot U\right) \cdot n\right) \cdot \color{blue}{\left(\left(2 \cdot U\right) \cdot n\right)}\right) \cdot \left(t \cdot t\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      11. swap-sqr 43.1%

          \[\leadsto \sqrt{{\color{blue}{\left(\left(\left(\left(2 \cdot U\right) \cdot n\right) \cdot t\right) \cdot \left(\left(\left(2 \cdot U\right) \cdot n\right) \cdot t\right)\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      12. associate-*r* 43.0%

          \[\leadsto \sqrt{{\left(\color{blue}{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)} \cdot \left(\left(\left(2 \cdot U\right) \cdot n\right) \cdot t\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      13. associate-*r* 45.8%

          \[\leadsto \sqrt{{\left(\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right) \cdot \color{blue}{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      14. pow2 45.8%

          \[\leadsto \sqrt{{\color{blue}{\left({\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{2}\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      15. associate-*l* 45.8%

          \[\leadsto \sqrt{{\left({\color{blue}{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}}^{2}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      16. metadata-eval 45.8%

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

   9. Applied egg-rr 45.8%

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

       1. unpow1/2 45.8%

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

       2. unpow2 45.8%

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

       3. rem-sqrt-square 59.7%

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

       4. associate-*r* 59.7%

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

       5. *-commutative 59.7%

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

   11. Simplified 59.7%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.15 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\left(n \cdot t\right) \cdot \left(2 \cdot U\right)\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 44.4% accurate, 2.0× speedup

Math

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

FPCore

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

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))))));
}

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))))))
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))))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.6%

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

3. Simplified 53.3%

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

5. Taylor expanded in n around 0 41.5%

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

   1. unpow2 41.5%

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

7. Applied egg-rr 41.5%

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

Alternative 12: 38.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = ((n * t) * (2.0d0 * u)) ** 0.5d0
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.pow(((n * t) * (2.0 * U)), 0.5);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.6%

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

3. Simplified 53.3%

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

5. Taylor expanded in t around inf 34.6%

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

   1. pow1/2 36.6%

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

   2. associate-*r* 36.6%

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

7. Applied egg-rr 36.6%

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

8. Final simplification 36.6%

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

Alternative 13: 36.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.6%

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

3. Simplified 54.0%

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

   1. associate-*r* 54.0%

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

   2. fma-define 56.4%

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

   3. associate-*r* 56.0%

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

6. Applied egg-rr 56.0%

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

7. Taylor expanded in t around inf 36.1%

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

8. Final simplification 36.1%

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

Alternative 14: 36.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.6%

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

3. Simplified 53.3%

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

5. Taylor expanded in t around inf 34.6%

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

   1. associate-*r* 36.1%

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

7. Simplified 36.1%

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

8. Final simplification 36.1%

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

Alternative 15: 36.6% 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 48.6%

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

3. Simplified 53.3%

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

5. Taylor expanded in t around inf 34.6%

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

Reproduce

herbie shell --seed 2024125 
(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*))))))