Toniolo and Linder, Equation (13)

Percentage Accurate: 49.6% → 62.9%

Time: 23.7s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 49.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 62.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(U \cdot U*\right)}^{0.25}\\ t_2 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := \left(n \cdot t_2\right) \cdot \left(U* - U\right)\\ t_4 := \left(2 \cdot n\right) \cdot U\\ t_5 := \sqrt{t_4 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_3\right)}\\ \mathbf{if}\;t_5 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + t_2 \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;t_5 \leq \infty:\\ \;\;\;\;\sqrt{t_4 \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + t_3\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\left|\left(n \cdot \sqrt{2}\right) \cdot \frac{\ell}{Om}\right| \cdot t_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (pow (* U U*) 0.25))
        (t_2 (pow (/ l Om) 2.0))
        (t_3 (* (* n t_2) (- U* U)))
        (t_4 (* (* 2.0 n) U))
        (t_5 (sqrt (* t_4 (+ (- t (* 2.0 (/ (* l l) Om))) t_3)))))
   (if (<= t_5 0.0)
     (sqrt
      (*
       (* 2.0 n)
       (* U (+ (- t (* 2.0 (/ l (/ Om l)))) (* t_2 (* n (- U* U)))))))
     (if (<= t_5 INFINITY)
       (sqrt (* t_4 (+ (- t (* 2.0 (* l (/ l Om)))) t_3)))
       (* t_1 (* (fabs (* (* n (sqrt 2.0)) (/ l Om))) t_1))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = pow((U * U_42_), 0.25);
	double t_2 = pow((l / Om), 2.0);
	double t_3 = (n * t_2) * (U_42_ - U);
	double t_4 = (2.0 * n) * U;
	double t_5 = sqrt((t_4 * ((t - (2.0 * ((l * l) / Om))) + t_3)));
	double tmp;
	if (t_5 <= 0.0) {
		tmp = sqrt(((2.0 * n) * (U * ((t - (2.0 * (l / (Om / l)))) + (t_2 * (n * (U_42_ - U)))))));
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = sqrt((t_4 * ((t - (2.0 * (l * (l / Om)))) + t_3)));
	} else {
		tmp = t_1 * (fabs(((n * sqrt(2.0)) * (l / Om))) * t_1);
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.pow((U * U_42_), 0.25);
	double t_2 = Math.pow((l / Om), 2.0);
	double t_3 = (n * t_2) * (U_42_ - U);
	double t_4 = (2.0 * n) * U;
	double t_5 = Math.sqrt((t_4 * ((t - (2.0 * ((l * l) / Om))) + t_3)));
	double tmp;
	if (t_5 <= 0.0) {
		tmp = Math.sqrt(((2.0 * n) * (U * ((t - (2.0 * (l / (Om / l)))) + (t_2 * (n * (U_42_ - U)))))));
	} else if (t_5 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_4 * ((t - (2.0 * (l * (l / Om)))) + t_3)));
	} else {
		tmp = t_1 * (Math.abs(((n * Math.sqrt(2.0)) * (l / Om))) * t_1);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.pow((U * U_42_), 0.25)
	t_2 = math.pow((l / Om), 2.0)
	t_3 = (n * t_2) * (U_42_ - U)
	t_4 = (2.0 * n) * U
	t_5 = math.sqrt((t_4 * ((t - (2.0 * ((l * l) / Om))) + t_3)))
	tmp = 0
	if t_5 <= 0.0:
		tmp = math.sqrt(((2.0 * n) * (U * ((t - (2.0 * (l / (Om / l)))) + (t_2 * (n * (U_42_ - U)))))))
	elif t_5 <= math.inf:
		tmp = math.sqrt((t_4 * ((t - (2.0 * (l * (l / Om)))) + t_3)))
	else:
		tmp = t_1 * (math.fabs(((n * math.sqrt(2.0)) * (l / Om))) * t_1)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(U * U_42_) ^ 0.25
	t_2 = Float64(l / Om) ^ 2.0
	t_3 = Float64(Float64(n * t_2) * Float64(U_42_ - U))
	t_4 = Float64(Float64(2.0 * n) * U)
	t_5 = sqrt(Float64(t_4 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_3)))
	tmp = 0.0
	if (t_5 <= 0.0)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(Float64(t - Float64(2.0 * Float64(l / Float64(Om / l)))) + Float64(t_2 * Float64(n * Float64(U_42_ - U)))))));
	elseif (t_5 <= Inf)
		tmp = sqrt(Float64(t_4 * Float64(Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))) + t_3)));
	else
		tmp = Float64(t_1 * Float64(abs(Float64(Float64(n * sqrt(2.0)) * Float64(l / Om))) * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (U * U_42_) ^ 0.25;
	t_2 = (l / Om) ^ 2.0;
	t_3 = (n * t_2) * (U_42_ - U);
	t_4 = (2.0 * n) * U;
	t_5 = sqrt((t_4 * ((t - (2.0 * ((l * l) / Om))) + t_3)));
	tmp = 0.0;
	if (t_5 <= 0.0)
		tmp = sqrt(((2.0 * n) * (U * ((t - (2.0 * (l / (Om / l)))) + (t_2 * (n * (U_42_ - U)))))));
	elseif (t_5 <= Inf)
		tmp = sqrt((t_4 * ((t - (2.0 * (l * (l / Om)))) + t_3)));
	else
		tmp = t_1 * (abs(((n * sqrt(2.0)) * (l / Om))) * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Power[N[(U * U$42$), $MachinePrecision], 0.25], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(n * t$95$2), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(t$95$4 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$5, 0.0], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(N[(t - N[(2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[Sqrt[N[(t$95$4 * N[(N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$1 * N[(N[Abs[N[(N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 13.2%

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

   3. Simplified 58.0%

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

      1. sub-neg 58.0%

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

      2. distribute-lft-in 58.0%

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

   5. Applied egg-rr 58.0%

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

      1. distribute-lft-out 58.0%

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

      2. *-commutative 58.0%

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

      3. sub-neg 58.0%

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

      4. associate-*r* 59.9%

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

      5. *-commutative 59.9%

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

   7. Simplified 59.9%

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

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

   1. Initial program 70.3%

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

      1. associate-*l/ 72.9%

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

   3. Applied egg-rr 72.9%

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

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

   1. Initial program 0.0%

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

   3. Simplified 5.3%

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

   4. Taylor expanded in U* around inf 36.5%

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

      1. associate-/l* 36.6%

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

      2. associate-/r/ 36.3%

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

      3. *-commutative 36.3%

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

   6. Simplified 36.3%

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

   7. Taylor expanded in n around 0 28.9%

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

      1. associate-*l/ 29.0%

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

      2. associate-/l* 29.0%

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

   9. Simplified 29.0%

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

       1. associate-/r/ 28.9%

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

       2. add-sqr-sqrt 28.8%

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

       3. associate-*r* 28.9%

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

       4. *-commutative 28.9%

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

       5. associate-*r* 28.9%

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

       6. pow1/2 28.9%

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

       7. sqrt-pow1 28.9%

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

       8. metadata-eval 28.9%

          \[\leadsto \left(\frac{\left(\ell \cdot \sqrt{2}\right) \cdot n}{Om} \cdot {\left(U \cdot U*\right)}^{\color{blue}{0.25}}\right) \cdot \sqrt{\sqrt{U \cdot U*}} \]

       9. pow1/2 28.9%

          \[\leadsto \left(\frac{\left(\ell \cdot \sqrt{2}\right) \cdot n}{Om} \cdot {\left(U \cdot U*\right)}^{0.25}\right) \cdot \sqrt{\color{blue}{{\left(U \cdot U*\right)}^{0.5}}} \]

       10. sqrt-pow1 29.0%

           \[\leadsto \left(\frac{\left(\ell \cdot \sqrt{2}\right) \cdot n}{Om} \cdot {\left(U \cdot U*\right)}^{0.25}\right) \cdot \color{blue}{{\left(U \cdot U*\right)}^{\left(\frac{0.5}{2}\right)}} \]

       11. metadata-eval 29.0%

           \[\leadsto \left(\frac{\left(\ell \cdot \sqrt{2}\right) \cdot n}{Om} \cdot {\left(U \cdot U*\right)}^{0.25}\right) \cdot {\left(U \cdot U*\right)}^{\color{blue}{0.25}} \]

   11. Applied egg-rr 29.0%

       \[\leadsto \color{blue}{\left(\frac{\left(\ell \cdot \sqrt{2}\right) \cdot n}{Om} \cdot {\left(U \cdot U*\right)}^{0.25}\right) \cdot {\left(U \cdot U*\right)}^{0.25}} \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 28.5%

          \[\leadsto \left(\color{blue}{\left(\sqrt{\frac{\left(\ell \cdot \sqrt{2}\right) \cdot n}{Om}} \cdot \sqrt{\frac{\left(\ell \cdot \sqrt{2}\right) \cdot n}{Om}}\right)} \cdot {\left(U \cdot U*\right)}^{0.25}\right) \cdot {\left(U \cdot U*\right)}^{0.25} \]

       2. sqrt-unprod 50.4%

          \[\leadsto \left(\color{blue}{\sqrt{\frac{\left(\ell \cdot \sqrt{2}\right) \cdot n}{Om} \cdot \frac{\left(\ell \cdot \sqrt{2}\right) \cdot n}{Om}}} \cdot {\left(U \cdot U*\right)}^{0.25}\right) \cdot {\left(U \cdot U*\right)}^{0.25} \]

       3. pow1 50.4%

          \[\leadsto \left(\sqrt{\color{blue}{{\left(\frac{\left(\ell \cdot \sqrt{2}\right) \cdot n}{Om}\right)}^{1}} \cdot \frac{\left(\ell \cdot \sqrt{2}\right) \cdot n}{Om}} \cdot {\left(U \cdot U*\right)}^{0.25}\right) \cdot {\left(U \cdot U*\right)}^{0.25} \]

       4. pow1 50.4%

          \[\leadsto \left(\sqrt{{\left(\frac{\left(\ell \cdot \sqrt{2}\right) \cdot n}{Om}\right)}^{1} \cdot \color{blue}{{\left(\frac{\left(\ell \cdot \sqrt{2}\right) \cdot n}{Om}\right)}^{1}}} \cdot {\left(U \cdot U*\right)}^{0.25}\right) \cdot {\left(U \cdot U*\right)}^{0.25} \]

       5. pow-sqr 50.4%

          \[\leadsto \left(\sqrt{\color{blue}{{\left(\frac{\left(\ell \cdot \sqrt{2}\right) \cdot n}{Om}\right)}^{\left(2 \cdot 1\right)}}} \cdot {\left(U \cdot U*\right)}^{0.25}\right) \cdot {\left(U \cdot U*\right)}^{0.25} \]

       6. associate-/l* 50.4%

          \[\leadsto \left(\sqrt{{\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{\frac{Om}{n}}\right)}}^{\left(2 \cdot 1\right)}} \cdot {\left(U \cdot U*\right)}^{0.25}\right) \cdot {\left(U \cdot U*\right)}^{0.25} \]

       7. associate-/r/ 50.2%

          \[\leadsto \left(\sqrt{{\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{Om} \cdot n\right)}}^{\left(2 \cdot 1\right)}} \cdot {\left(U \cdot U*\right)}^{0.25}\right) \cdot {\left(U \cdot U*\right)}^{0.25} \]

       8. metadata-eval 50.2%

          \[\leadsto \left(\sqrt{{\left(\frac{\ell \cdot \sqrt{2}}{Om} \cdot n\right)}^{\color{blue}{2}}} \cdot {\left(U \cdot U*\right)}^{0.25}\right) \cdot {\left(U \cdot U*\right)}^{0.25} \]

   13. Applied egg-rr 50.2%

       \[\leadsto \left(\color{blue}{\sqrt{{\left(\frac{\ell \cdot \sqrt{2}}{Om} \cdot n\right)}^{2}}} \cdot {\left(U \cdot U*\right)}^{0.25}\right) \cdot {\left(U \cdot U*\right)}^{0.25} \]
   14. Step-by-step derivation

       1. unpow2 50.2%

          \[\leadsto \left(\sqrt{\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{Om} \cdot n\right) \cdot \left(\frac{\ell \cdot \sqrt{2}}{Om} \cdot n\right)}} \cdot {\left(U \cdot U*\right)}^{0.25}\right) \cdot {\left(U \cdot U*\right)}^{0.25} \]

       2. rem-sqrt-square 52.3%

          \[\leadsto \left(\color{blue}{\left|\frac{\ell \cdot \sqrt{2}}{Om} \cdot n\right|} \cdot {\left(U \cdot U*\right)}^{0.25}\right) \cdot {\left(U \cdot U*\right)}^{0.25} \]

       3. associate-/l* 52.3%

          \[\leadsto \left(\left|\color{blue}{\frac{\ell}{\frac{Om}{\sqrt{2}}}} \cdot n\right| \cdot {\left(U \cdot U*\right)}^{0.25}\right) \cdot {\left(U \cdot U*\right)}^{0.25} \]

       4. associate-*l/ 57.0%

          \[\leadsto \left(\left|\color{blue}{\frac{\ell \cdot n}{\frac{Om}{\sqrt{2}}}}\right| \cdot {\left(U \cdot U*\right)}^{0.25}\right) \cdot {\left(U \cdot U*\right)}^{0.25} \]

       5. associate-/l* 57.0%

          \[\leadsto \left(\left|\color{blue}{\frac{\left(\ell \cdot n\right) \cdot \sqrt{2}}{Om}}\right| \cdot {\left(U \cdot U*\right)}^{0.25}\right) \cdot {\left(U \cdot U*\right)}^{0.25} \]

       6. associate-*r* 56.9%

          \[\leadsto \left(\left|\frac{\color{blue}{\ell \cdot \left(n \cdot \sqrt{2}\right)}}{Om}\right| \cdot {\left(U \cdot U*\right)}^{0.25}\right) \cdot {\left(U \cdot U*\right)}^{0.25} \]

       7. associate-/l* 57.0%

          \[\leadsto \left(\left|\color{blue}{\frac{\ell}{\frac{Om}{n \cdot \sqrt{2}}}}\right| \cdot {\left(U \cdot U*\right)}^{0.25}\right) \cdot {\left(U \cdot U*\right)}^{0.25} \]

       8. associate-/r/ 52.3%

          \[\leadsto \left(\left|\color{blue}{\frac{\ell}{Om} \cdot \left(n \cdot \sqrt{2}\right)}\right| \cdot {\left(U \cdot U*\right)}^{0.25}\right) \cdot {\left(U \cdot U*\right)}^{0.25} \]

   15. Simplified 52.3%

       \[\leadsto \left(\color{blue}{\left|\frac{\ell}{Om} \cdot \left(n \cdot \sqrt{2}\right)\right|} \cdot {\left(U \cdot U*\right)}^{0.25}\right) \cdot {\left(U \cdot U*\right)}^{0.25} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(U \cdot U*\right)}^{0.25} \cdot \left(\left|\left(n \cdot \sqrt{2}\right) \cdot \frac{\ell}{Om}\right| \cdot {\left(U \cdot U*\right)}^{0.25}\right)\\ \end{array} \]

Alternative 2: 61.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 10.4%

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

   3. Simplified 46.2%

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

      1. sub-neg 46.2%

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

      2. distribute-lft-in 46.2%

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

   5. Applied egg-rr 46.2%

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

      1. distribute-lft-out 46.2%

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

      2. *-commutative 46.2%

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

      3. sub-neg 46.2%

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

      4. associate-*r* 50.2%

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

      5. *-commutative 50.2%

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

   7. Simplified 50.2%

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

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

   1. Initial program 70.3%

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

      1. associate-*l/ 72.9%

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

   3. Applied egg-rr 72.9%

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

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

   1. Initial program 0.0%

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

   3. Simplified 6.1%

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

   4. Taylor expanded in U* around inf 44.4%

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

      1. associate-/l* 44.4%

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

      2. associate-/r/ 44.0%

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

      3. *-commutative 44.0%

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

   6. Simplified 44.0%

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

      1. associate-*l/ 44.4%

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

      2. unpow2 44.4%

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

      3. associate-/r* 50.2%

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

      4. associate-*r* 49.7%

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

   8. Applied egg-rr 49.7%

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

4. Final simplification 66.3%

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

Alternative 3: 52.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq -9.5 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;U \leq 2.25 \cdot 10^{+164}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U -9.5e+65)
   (sqrt (* 2.0 (* U (* n (- t (* 2.0 (* l (/ l Om))))))))
   (if (<= U 2.25e+164)
     (sqrt
      (*
       (* 2.0 n)
       (*
        U
        (+
         (- t (* 2.0 (/ l (/ Om l))))
         (* (pow (/ l Om) 2.0) (* n (- U* U)))))))
     (* (sqrt (* 2.0 U)) (sqrt (* n t))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -9.5e+65) {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	} else if (U <= 2.25e+164) {
		tmp = sqrt(((2.0 * n) * (U * ((t - (2.0 * (l / (Om / l)))) + (pow((l / Om), 2.0) * (n * (U_42_ - U)))))));
	} else {
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= (-9.5d+65)) then
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * (l * (l / om))))))))
    else if (u <= 2.25d+164) then
        tmp = sqrt(((2.0d0 * n) * (u * ((t - (2.0d0 * (l / (om / l)))) + (((l / om) ** 2.0d0) * (n * (u_42 - u)))))))
    else
        tmp = sqrt((2.0d0 * u)) * sqrt((n * t))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -9.5e+65) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	} else if (U <= 2.25e+164) {
		tmp = Math.sqrt(((2.0 * n) * (U * ((t - (2.0 * (l / (Om / l)))) + (Math.pow((l / Om), 2.0) * (n * (U_42_ - U)))))));
	} else {
		tmp = Math.sqrt((2.0 * U)) * Math.sqrt((n * t));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= -9.5e+65:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))))
	elif U <= 2.25e+164:
		tmp = math.sqrt(((2.0 * n) * (U * ((t - (2.0 * (l / (Om / l)))) + (math.pow((l / Om), 2.0) * (n * (U_42_ - U)))))))
	else:
		tmp = math.sqrt((2.0 * U)) * math.sqrt((n * t))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= -9.5e+65)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om))))))));
	elseif (U <= 2.25e+164)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(Float64(t - Float64(2.0 * Float64(l / Float64(Om / l)))) + Float64((Float64(l / Om) ^ 2.0) * Float64(n * Float64(U_42_ - U)))))));
	else
		tmp = Float64(sqrt(Float64(2.0 * U)) * sqrt(Float64(n * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= -9.5e+65)
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	elseif (U <= 2.25e+164)
		tmp = sqrt(((2.0 * n) * (U * ((t - (2.0 * (l / (Om / l)))) + (((l / Om) ^ 2.0) * (n * (U_42_ - U)))))));
	else
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, -9.5e+65], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[U, 2.25e+164], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(N[(t - N[(2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(n * N[(U$42$ - U), $MachinePrecision]), $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 3 regimes

2. if U < -9.5000000000000005e65

   1. Initial program 53.7%

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

   3. Simplified 48.9%

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

   4. Taylor expanded in n around 0 56.8%

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

      1. unpow2 56.8%

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

      2. associate-*l/ 59.1%

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

   6. Applied egg-rr 59.1%

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

   if -9.5000000000000005e65 < U < 2.24999999999999988e164

   1. Initial program 51.5%

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

   3. Simplified 59.4%

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

      1. sub-neg 59.4%

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

      2. distribute-lft-in 51.9%

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

   5. Applied egg-rr 51.9%

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

      1. distribute-lft-out 59.4%

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

      2. *-commutative 59.4%

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

      3. sub-neg 59.4%

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

      4. associate-*r* 58.3%

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

      5. *-commutative 58.3%

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

   7. Simplified 58.3%

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

   if 2.24999999999999988e164 < U

   1. Initial program 50.9%

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

   3. Simplified 42.5%

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

   4. Taylor expanded in t around inf 44.3%

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

      1. associate-*r* 44.3%

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

      2. sqrt-prod 63.1%

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

   6. Applied egg-rr 63.1%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -9.5 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;U \leq 2.25 \cdot 10^{+164}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \end{array} \]

Alternative 4: 53.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.45 \cdot 10^{+204}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \frac{\frac{\left(n \cdot U*\right) \cdot {\ell}^{2}}{Om}}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.45e+204)
   (sqrt
    (*
     (* 2.0 n)
     (*
      U
      (+ (- t (* 2.0 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2.0)) (- U* U))))))
   (sqrt (* (* 2.0 n) (* U (/ (/ (* (* n U*) (pow l 2.0)) Om) Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.45e+204) {
		tmp = sqrt(((2.0 * n) * (U * ((t - (2.0 * (l / (Om / l)))) + ((n * pow((l / Om), 2.0)) * (U_42_ - U))))));
	} else {
		tmp = sqrt(((2.0 * n) * (U * ((((n * U_42_) * pow(l, 2.0)) / Om) / 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 <= 1.45d+204) then
        tmp = sqrt(((2.0d0 * n) * (u * ((t - (2.0d0 * (l / (om / l)))) + ((n * ((l / om) ** 2.0d0)) * (u_42 - u))))))
    else
        tmp = sqrt(((2.0d0 * n) * (u * ((((n * u_42) * (l ** 2.0d0)) / om) / 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 <= 1.45e+204) {
		tmp = Math.sqrt(((2.0 * n) * (U * ((t - (2.0 * (l / (Om / l)))) + ((n * Math.pow((l / Om), 2.0)) * (U_42_ - U))))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * ((((n * U_42_) * Math.pow(l, 2.0)) / Om) / Om))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.45e+204:
		tmp = math.sqrt(((2.0 * n) * (U * ((t - (2.0 * (l / (Om / l)))) + ((n * math.pow((l / Om), 2.0)) * (U_42_ - U))))))
	else:
		tmp = math.sqrt(((2.0 * n) * (U * ((((n * U_42_) * math.pow(l, 2.0)) / Om) / Om))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.45e+204)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(Float64(t - Float64(2.0 * Float64(l / Float64(Om / l)))) + Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(Float64(Float64(Float64(n * U_42_) * (l ^ 2.0)) / Om) / Om))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 1.45e+204)
		tmp = sqrt(((2.0 * n) * (U * ((t - (2.0 * (l / (Om / l)))) + ((n * ((l / Om) ^ 2.0)) * (U_42_ - U))))));
	else
		tmp = sqrt(((2.0 * n) * (U * ((((n * U_42_) * (l ^ 2.0)) / Om) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.45e+204], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(N[(t - N[(2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(N[(N[(N[(n * U$42$), $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.45000000000000002e204

   1. Initial program 54.3%

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

   3. Simplified 57.8%

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

   if 1.45000000000000002e204 < l

   1. Initial program 24.6%

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

   3. Simplified 29.3%

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

   4. Taylor expanded in U* around inf 43.2%

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

      1. associate-/l* 43.2%

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

      2. associate-/r/ 42.6%

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

      3. *-commutative 42.6%

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

   6. Simplified 42.6%

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

      1. associate-*l/ 43.2%

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

      2. unpow2 43.2%

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

      3. associate-/r* 52.3%

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

      4. associate-*r* 51.9%

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

   8. Applied egg-rr 51.9%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.45 \cdot 10^{+204}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \frac{\frac{\left(n \cdot U*\right) \cdot {\ell}^{2}}{Om}}{Om}\right)}\\ \end{array} \]

Alternative 5: 47.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+138} \lor \neg \left(t \leq 4.6 \cdot 10^{+136}\right):\\ \;\;\;\;\sqrt{\left|2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= t -8e+138) (not (<= t 4.6e+136)))
   (sqrt (fabs (* 2.0 (* n (* U t)))))
   (sqrt (* 2.0 (* U (* n (- t (* 2.0 (* l (/ l Om))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((t <= -8e+138) || !(t <= 4.6e+136)) {
		tmp = sqrt(fabs((2.0 * (n * (U * t)))));
	} else {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / 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 ((t <= (-8d+138)) .or. (.not. (t <= 4.6d+136))) then
        tmp = sqrt(abs((2.0d0 * (n * (u * t)))))
    else
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * (l * (l / 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 ((t <= -8e+138) || !(t <= 4.6e+136)) {
		tmp = Math.sqrt(Math.abs((2.0 * (n * (U * t)))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (t <= -8e+138) or not (t <= 4.6e+136):
		tmp = math.sqrt(math.fabs((2.0 * (n * (U * t)))))
	else:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((t <= -8e+138) || !(t <= 4.6e+136))
		tmp = sqrt(abs(Float64(2.0 * Float64(n * Float64(U * t)))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((t <= -8e+138) || ~((t <= 4.6e+136)))
		tmp = sqrt(abs((2.0 * (n * (U * t)))));
	else
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[t, -8e+138], N[Not[LessEqual[t, 4.6e+136]], $MachinePrecision]], N[Sqrt[N[Abs[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -8.0000000000000003e138 or 4.6e136 < t

   1. Initial program 44.8%

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

   3. Simplified 50.5%

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

   4. Taylor expanded in t around inf 50.3%

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

      1. *-commutative 50.3%

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

      2. associate-*r* 50.3%

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

      3. add-sqr-sqrt 50.3%

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

      4. sqrt-unprod 41.1%

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

      5. associate-*r* 41.1%

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

      6. associate-*r* 41.1%

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

      7. pow1 41.1%

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

      8. associate-*r* 41.1%

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

      9. pow1 41.1%

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

      10. associate-*r* 41.1%

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

      11. pow-sqr 41.1%

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

      12. associate-*r* 41.1%

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

      13. *-commutative 41.1%

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

      14. associate-*l* 41.1%

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

      15. *-commutative 41.1%

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

      16. metadata-eval 41.1%

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

   6. Applied egg-rr 41.1%

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

      1. unpow2 41.1%

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

      2. rem-sqrt-square 55.5%

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

      3. associate-*r* 55.5%

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

      4. *-commutative 55.5%

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

      5. associate-*l* 55.5%

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

   8. Simplified 55.5%

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

   if -8.0000000000000003e138 < t < 4.6e136

   1. Initial program 55.4%

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

   3. Simplified 56.1%

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

   4. Taylor expanded in n around 0 51.9%

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

      1. unpow2 51.9%

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

      2. associate-*l/ 53.6%

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

   6. Applied egg-rr 53.6%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+138} \lor \neg \left(t \leq 4.6 \cdot 10^{+136}\right):\\ \;\;\;\;\sqrt{\left|2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \]

Alternative 6: 48.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right|}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+123}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t -1.55e+138)
   (sqrt (fabs (* 2.0 (* n (* U t)))))
   (if (<= t 5.5e+123)
     (sqrt (* 2.0 (* U (* n (- t (* 2.0 (* l (/ l Om))))))))
     (* (sqrt (* (* 2.0 n) U)) (sqrt t)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -1.55e+138) {
		tmp = sqrt(fabs((2.0 * (n * (U * t)))));
	} else if (t <= 5.5e+123) {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	} else {
		tmp = sqrt(((2.0 * n) * U)) * sqrt(t);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= (-1.55d+138)) then
        tmp = sqrt(abs((2.0d0 * (n * (u * t)))))
    else if (t <= 5.5d+123) then
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * (l * (l / om))))))))
    else
        tmp = sqrt(((2.0d0 * n) * u)) * sqrt(t)
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -1.55e+138) {
		tmp = Math.sqrt(Math.abs((2.0 * (n * (U * t)))));
	} else if (t <= 5.5e+123) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * U)) * Math.sqrt(t);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= -1.55e+138:
		tmp = math.sqrt(math.fabs((2.0 * (n * (U * t)))))
	elif t <= 5.5e+123:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))))
	else:
		tmp = math.sqrt(((2.0 * n) * U)) * math.sqrt(t)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= -1.55e+138)
		tmp = sqrt(abs(Float64(2.0 * Float64(n * Float64(U * t)))));
	elseif (t <= 5.5e+123)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om))))))));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * U)) * sqrt(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= -1.55e+138)
		tmp = sqrt(abs((2.0 * (n * (U * t)))));
	elseif (t <= 5.5e+123)
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	else
		tmp = sqrt(((2.0 * n) * U)) * sqrt(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, -1.55e+138], N[Sqrt[N[Abs[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 5.5e+123], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.5499999999999999e138

   1. Initial program 40.6%

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

   3. Simplified 45.4%

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

   4. Taylor expanded in t around inf 42.5%

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

      1. *-commutative 42.5%

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

      2. associate-*r* 42.5%

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

      3. add-sqr-sqrt 42.5%

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

      4. sqrt-unprod 36.8%

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

      5. associate-*r* 36.8%

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

      6. associate-*r* 36.8%

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

      7. pow1 36.8%

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

      8. associate-*r* 36.8%

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

      9. pow1 36.8%

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

      10. associate-*r* 36.8%

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

      11. pow-sqr 36.8%

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

      12. associate-*r* 36.8%

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

      13. *-commutative 36.8%

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

      14. associate-*l* 36.8%

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

      15. *-commutative 36.8%

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

      16. metadata-eval 36.8%

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

   6. Applied egg-rr 36.8%

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

      1. unpow2 36.8%

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

      2. rem-sqrt-square 50.7%

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

      3. associate-*r* 50.7%

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

      4. *-commutative 50.7%

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

      5. associate-*l* 50.7%

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

   8. Simplified 50.7%

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

   if -1.5499999999999999e138 < t < 5.5000000000000002e123

   1. Initial program 55.8%

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

   3. Simplified 55.9%

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

   4. Taylor expanded in n around 0 52.3%

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

      1. unpow2 52.3%

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

      2. associate-*l/ 54.0%

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

   6. Applied egg-rr 54.0%

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

   if 5.5000000000000002e123 < t

   1. Initial program 47.7%

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

   3. Simplified 45.5%

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

   4. Taylor expanded in t around inf 54.1%

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

      1. associate-*r* 52.2%

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

      2. *-commutative 52.2%

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

   6. Simplified 52.2%

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

      1. associate-*l* 52.2%

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

      2. associate-*r* 52.2%

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

      3. sqrt-prod 68.2%

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

      4. *-commutative 68.2%

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

      5. associate-*l* 68.2%

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

   8. Applied egg-rr 68.2%

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

      1. associate-*r* 68.2%

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

   10. Simplified 68.2%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right|}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+123}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t}\\ \end{array} \]

Alternative 7: 47.8% accurate, 2.0× speedup

Math

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

FPCore

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.8%

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

3. Simplified 51.3%

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

4. Taylor expanded in n around 0 48.8%

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

   1. unpow2 48.8%

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

   2. associate-*l/ 50.7%

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

6. Applied egg-rr 50.7%

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

7. Final simplification 50.7%

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

Alternative 8: 36.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U* \leq 3.5 \cdot 10^{-54}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U* 3.5e-54)
   (sqrt (* 2.0 (* t (* n U))))
   (pow (* 2.0 (* n (* U t))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= 3.5e-54) {
		tmp = sqrt((2.0 * (t * (n * U))));
	} else {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u_42 <= 3.5d-54) then
        tmp = sqrt((2.0d0 * (t * (n * u))))
    else
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= 3.5e-54) {
		tmp = Math.sqrt((2.0 * (t * (n * U))));
	} else {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U_42_ <= 3.5e-54:
		tmp = math.sqrt((2.0 * (t * (n * U))))
	else:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= 3.5e-54)
		tmp = sqrt(Float64(2.0 * Float64(t * Float64(n * U))));
	else
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U_42_ <= 3.5e-54)
		tmp = sqrt((2.0 * (t * (n * U))));
	else
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U$42$, 3.5e-54], N[Sqrt[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < 3.49999999999999982e-54

   1. Initial program 52.3%

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

   3. Simplified 53.8%

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

   4. Taylor expanded in t around inf 42.2%

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

      1. associate-*r* 44.9%

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

      2. *-commutative 44.9%

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

   6. Simplified 44.9%

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

   if 3.49999999999999982e-54 < U*

   1. Initial program 50.8%

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

   3. Simplified 46.8%

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

   4. Taylor expanded in t around inf 39.1%

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

      1. pow1/2 44.8%

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

      2. *-commutative 44.8%

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

      3. associate-*l* 42.8%

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

   6. Applied egg-rr 42.8%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq 3.5 \cdot 10^{-54}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 9: 35.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U* \leq 8.4 \cdot 10^{-54}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U* 8.4e-54)
   (sqrt (* 2.0 (* t (* n U))))
   (sqrt (* 2.0 (* U (* n t))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= 8.4e-54) {
		tmp = sqrt((2.0 * (t * (n * U))));
	} else {
		tmp = sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u_42 <= 8.4d-54) then
        tmp = sqrt((2.0d0 * (t * (n * u))))
    else
        tmp = sqrt((2.0d0 * (u * (n * t))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= 8.4e-54) {
		tmp = Math.sqrt((2.0 * (t * (n * U))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U_42_ <= 8.4e-54:
		tmp = math.sqrt((2.0 * (t * (n * U))))
	else:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= 8.4e-54)
		tmp = sqrt(Float64(2.0 * Float64(t * Float64(n * U))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U_42_ <= 8.4e-54)
		tmp = sqrt((2.0 * (t * (n * U))));
	else
		tmp = sqrt((2.0 * (U * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U$42$, 8.4e-54], N[Sqrt[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < 8.4e-54

   1. Initial program 52.3%

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

   3. Simplified 53.8%

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

   4. Taylor expanded in t around inf 42.2%

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

      1. associate-*r* 44.9%

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

      2. *-commutative 44.9%

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

   6. Simplified 44.9%

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

   if 8.4e-54 < U*

   1. Initial program 50.8%

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

   3. Simplified 46.8%

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

   4. Taylor expanded in t around inf 39.1%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq 8.4 \cdot 10^{-54}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 10: 35.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U* \leq 8 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U* 8e-55) (sqrt (* 2.0 (* t (* n U)))) (sqrt (* (* 2.0 n) (* U t)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= 8e-55) {
		tmp = sqrt((2.0 * (t * (n * U))));
	} else {
		tmp = sqrt(((2.0 * n) * (U * t)));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u_42 <= 8d-55) then
        tmp = sqrt((2.0d0 * (t * (n * u))))
    else
        tmp = sqrt(((2.0d0 * n) * (u * t)))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= 8e-55) {
		tmp = Math.sqrt((2.0 * (t * (n * U))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * t)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U_42_ <= 8e-55:
		tmp = math.sqrt((2.0 * (t * (n * U))))
	else:
		tmp = math.sqrt(((2.0 * n) * (U * t)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= 8e-55)
		tmp = sqrt(Float64(2.0 * Float64(t * Float64(n * U))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U_42_ <= 8e-55)
		tmp = sqrt((2.0 * (t * (n * U))));
	else
		tmp = sqrt(((2.0 * n) * (U * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U$42$, 8e-55], N[Sqrt[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < 7.99999999999999996e-55

   1. Initial program 52.3%

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

   3. Simplified 53.8%

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

   4. Taylor expanded in t around inf 42.2%

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

      1. associate-*r* 44.9%

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

      2. *-commutative 44.9%

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

   6. Simplified 44.9%

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

   if 7.99999999999999996e-55 < U*

   1. Initial program 50.8%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in t around inf 39.4%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq 8 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \]

Alternative 11: 35.8% 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 51.8%

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

3. Simplified 51.3%

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

4. Taylor expanded in t around inf 41.1%

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

5. Final simplification 41.1%

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

Reproduce

herbie shell --seed 2023336 
(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*))))))