Toniolo and Linder, Equation (13)

Percentage Accurate: 49.9% → 64.8%

Time: 24.7s

Alternatives: 9

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: 49.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 64.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\\ t_2 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t\_2 \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(t - t\_1\right)\right)} \cdot \sqrt{2}\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+305}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(t\_2 \cdot \left(U - U*\right) + t\_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\sqrt{U* \cdot \left(2 \cdot U\right)} \cdot \left(n \cdot \ell\right)}{Om}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* l (/ l Om))))
        (t_2 (* n (pow (/ l Om) 2.0)))
        (t_3
         (*
          (* (* 2.0 n) U)
          (+ (- t (* 2.0 (/ (* l l) Om))) (* t_2 (- U* U))))))
   (if (<= t_3 0.0)
     (* (sqrt (* U (* n (- t t_1)))) (sqrt 2.0))
     (if (<= t_3 4e+305)
       (sqrt (* (* 2.0 (* n U)) (- t (+ (* t_2 (- U U*)) t_1))))
       (fabs (/ (* (sqrt (* U* (* 2.0 U))) (* n l)) Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = 2.0 * (l * (l / Om));
	double t_2 = n * pow((l / Om), 2.0);
	double t_3 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + (t_2 * (U_42_ - U)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((U * (n * (t - t_1)))) * sqrt(2.0);
	} else if (t_3 <= 4e+305) {
		tmp = sqrt(((2.0 * (n * U)) * (t - ((t_2 * (U - U_42_)) + t_1))));
	} else {
		tmp = fabs(((sqrt((U_42_ * (2.0 * U))) * (n * 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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = 2.0d0 * (l * (l / om))
    t_2 = n * ((l / om) ** 2.0d0)
    t_3 = ((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) + (t_2 * (u_42 - u)))
    if (t_3 <= 0.0d0) then
        tmp = sqrt((u * (n * (t - t_1)))) * sqrt(2.0d0)
    else if (t_3 <= 4d+305) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t - ((t_2 * (u - u_42)) + t_1))))
    else
        tmp = abs(((sqrt((u_42 * (2.0d0 * u))) * (n * 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 t_1 = 2.0 * (l * (l / Om));
	double t_2 = n * Math.pow((l / Om), 2.0);
	double t_3 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + (t_2 * (U_42_ - U)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = Math.sqrt((U * (n * (t - t_1)))) * Math.sqrt(2.0);
	} else if (t_3 <= 4e+305) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t - ((t_2 * (U - U_42_)) + t_1))));
	} else {
		tmp = Math.abs(((Math.sqrt((U_42_ * (2.0 * U))) * (n * l)) / Om));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = 2.0 * (l * (l / Om))
	t_2 = n * math.pow((l / Om), 2.0)
	t_3 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + (t_2 * (U_42_ - U)))
	tmp = 0
	if t_3 <= 0.0:
		tmp = math.sqrt((U * (n * (t - t_1)))) * math.sqrt(2.0)
	elif t_3 <= 4e+305:
		tmp = math.sqrt(((2.0 * (n * U)) * (t - ((t_2 * (U - U_42_)) + t_1))))
	else:
		tmp = math.fabs(((math.sqrt((U_42_ * (2.0 * U))) * (n * l)) / Om))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(2.0 * Float64(l * Float64(l / Om)))
	t_2 = Float64(n * (Float64(l / Om) ^ 2.0))
	t_3 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(t_2 * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(U * Float64(n * Float64(t - t_1)))) * sqrt(2.0));
	elseif (t_3 <= 4e+305)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(Float64(t_2 * Float64(U - U_42_)) + t_1))));
	else
		tmp = abs(Float64(Float64(sqrt(Float64(U_42_ * Float64(2.0 * U))) * Float64(n * l)) / Om));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = 2.0 * (l * (l / Om));
	t_2 = n * ((l / Om) ^ 2.0);
	t_3 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + (t_2 * (U_42_ - U)));
	tmp = 0.0;
	if (t_3 <= 0.0)
		tmp = sqrt((U * (n * (t - t_1)))) * sqrt(2.0);
	elseif (t_3 <= 4e+305)
		tmp = sqrt(((2.0 * (n * U)) * (t - ((t_2 * (U - U_42_)) + t_1))));
	else
		tmp = abs(((sqrt((U_42_ * (2.0 * U))) * (n * l)) / Om));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(U * N[(n * N[(t - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 4e+305], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(t$95$2 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Sqrt[N[(U$42$ * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.8%

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

   3. Simplified 33.3%

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

   5. Taylor expanded in n around 0 34.0%

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

      1. unpow2 34.0%

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

      2. associate-*r/ 39.0%

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

      3. *-commutative 39.0%

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

   7. Applied egg-rr 39.0%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 3.9999999999999998e305

   1. Initial program 98.2%

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

   3. Simplified 98.2%

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

   if 3.9999999999999998e305 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 20.6%

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

   3. Simplified 28.1%

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

      1. associate-*r* 28.1%

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

      2. sub-neg 28.1%

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

      3. distribute-lft-in 14.9%

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

   6. Applied egg-rr 14.9%

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

      1. distribute-lft-out 28.1%

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

      2. sub-neg 28.1%

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

   8. Simplified 28.1%

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

   9. Taylor expanded in U* around inf 25.2%

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

       1. associate-*r/ 25.2%

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

   11. Simplified 25.2%

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

       1. add-sqr-sqrt 25.2%

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

       2. rem-sqrt-square 25.2%

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

       3. sqrt-div 25.2%

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

   13. Applied egg-rr 25.9%

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

       1. pow1 25.9%

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

       2. associate-*r* 26.7%

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

       3. *-commutative 26.7%

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

       4. sqrt-unprod 49.5%

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

   15. Applied egg-rr 49.5%

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

       1. unpow1 49.5%

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

       2. *-commutative 49.5%

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

   17. Simplified 49.5%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)} \cdot \sqrt{2}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 4 \cdot 10^{+305}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\sqrt{U* \cdot \left(2 \cdot U\right)} \cdot \left(n \cdot \ell\right)}{Om}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 51.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \left|\frac{\sqrt{U* \cdot \left(2 \cdot U\right)} \cdot \left(n \cdot \ell\right)}{Om}\right|\\ \mathbf{if}\;Om \leq -1.5 \cdot 10^{-105}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + t\_1 \cdot -2\right)\right)}\\ \mathbf{elif}\;Om \leq -2.65 \cdot 10^{-143}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;Om \leq -2.8 \cdot 10^{-253}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 2.5 \cdot 10^{-47}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot t\_1\right)\right)} \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om)))
        (t_2 (fabs (/ (* (sqrt (* U* (* 2.0 U))) (* n l)) Om))))
   (if (<= Om -1.5e-105)
     (sqrt (* 2.0 (* (* n U) (+ t (* t_1 -2.0)))))
     (if (<= Om -2.65e-143)
       t_2
       (if (<= Om -2.8e-253)
         (sqrt (* (* 2.0 U) (* n (+ t (* -2.0 (/ (pow l 2.0) Om))))))
         (if (<= Om 2.5e-47)
           t_2
           (* (sqrt (* U (* n (- t (* 2.0 t_1))))) (sqrt 2.0))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double t_2 = fabs(((sqrt((U_42_ * (2.0 * U))) * (n * l)) / Om));
	double tmp;
	if (Om <= -1.5e-105) {
		tmp = sqrt((2.0 * ((n * U) * (t + (t_1 * -2.0)))));
	} else if (Om <= -2.65e-143) {
		tmp = t_2;
	} else if (Om <= -2.8e-253) {
		tmp = sqrt(((2.0 * U) * (n * (t + (-2.0 * (pow(l, 2.0) / Om))))));
	} else if (Om <= 2.5e-47) {
		tmp = t_2;
	} else {
		tmp = sqrt((U * (n * (t - (2.0 * t_1))))) * sqrt(2.0);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = l * (l / om)
    t_2 = abs(((sqrt((u_42 * (2.0d0 * u))) * (n * l)) / om))
    if (om <= (-1.5d-105)) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + (t_1 * (-2.0d0))))))
    else if (om <= (-2.65d-143)) then
        tmp = t_2
    else if (om <= (-2.8d-253)) then
        tmp = sqrt(((2.0d0 * u) * (n * (t + ((-2.0d0) * ((l ** 2.0d0) / om))))))
    else if (om <= 2.5d-47) then
        tmp = t_2
    else
        tmp = sqrt((u * (n * (t - (2.0d0 * t_1))))) * sqrt(2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double t_2 = Math.abs(((Math.sqrt((U_42_ * (2.0 * U))) * (n * l)) / Om));
	double tmp;
	if (Om <= -1.5e-105) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + (t_1 * -2.0)))));
	} else if (Om <= -2.65e-143) {
		tmp = t_2;
	} else if (Om <= -2.8e-253) {
		tmp = Math.sqrt(((2.0 * U) * (n * (t + (-2.0 * (Math.pow(l, 2.0) / Om))))));
	} else if (Om <= 2.5e-47) {
		tmp = t_2;
	} else {
		tmp = Math.sqrt((U * (n * (t - (2.0 * t_1))))) * Math.sqrt(2.0);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = l * (l / Om)
	t_2 = math.fabs(((math.sqrt((U_42_ * (2.0 * U))) * (n * l)) / Om))
	tmp = 0
	if Om <= -1.5e-105:
		tmp = math.sqrt((2.0 * ((n * U) * (t + (t_1 * -2.0)))))
	elif Om <= -2.65e-143:
		tmp = t_2
	elif Om <= -2.8e-253:
		tmp = math.sqrt(((2.0 * U) * (n * (t + (-2.0 * (math.pow(l, 2.0) / Om))))))
	elif Om <= 2.5e-47:
		tmp = t_2
	else:
		tmp = math.sqrt((U * (n * (t - (2.0 * t_1))))) * math.sqrt(2.0)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(l / Om))
	t_2 = abs(Float64(Float64(sqrt(Float64(U_42_ * Float64(2.0 * U))) * Float64(n * l)) / Om))
	tmp = 0.0
	if (Om <= -1.5e-105)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(t_1 * -2.0)))));
	elseif (Om <= -2.65e-143)
		tmp = t_2;
	elseif (Om <= -2.8e-253)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * Float64(t + Float64(-2.0 * Float64((l ^ 2.0) / Om))))));
	elseif (Om <= 2.5e-47)
		tmp = t_2;
	else
		tmp = Float64(sqrt(Float64(U * Float64(n * Float64(t - Float64(2.0 * t_1))))) * sqrt(2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l * (l / Om);
	t_2 = abs(((sqrt((U_42_ * (2.0 * U))) * (n * l)) / Om));
	tmp = 0.0;
	if (Om <= -1.5e-105)
		tmp = sqrt((2.0 * ((n * U) * (t + (t_1 * -2.0)))));
	elseif (Om <= -2.65e-143)
		tmp = t_2;
	elseif (Om <= -2.8e-253)
		tmp = sqrt(((2.0 * U) * (n * (t + (-2.0 * ((l ^ 2.0) / Om))))));
	elseif (Om <= 2.5e-47)
		tmp = t_2;
	else
		tmp = sqrt((U * (n * (t - (2.0 * t_1))))) * sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[(N[Sqrt[N[(U$42$ * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -1.5e-105], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, -2.65e-143], t$95$2, If[LessEqual[Om, -2.8e-253], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(t + N[(-2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 2.5e-47], t$95$2, N[(N[Sqrt[N[(U * N[(n * N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if Om < -1.5e-105

   1. Initial program 55.0%

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

   3. Simplified 52.3%

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

   5. Taylor expanded in n around 0 48.5%

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

      1. associate-*r* 53.8%

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

      2. sub-neg 53.8%

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

      3. *-commutative 53.8%

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

      4. distribute-rgt-neg-in 53.8%

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

      5. metadata-eval 53.8%

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

   7. Simplified 53.8%

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

      1. unpow2 48.3%

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

      2. associate-*r/ 53.6%

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

      3. *-commutative 53.6%

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

   9. Applied egg-rr 60.3%

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

   if -1.5e-105 < Om < -2.64999999999999998e-143 or -2.80000000000000006e-253 < Om < 2.50000000000000006e-47

   1. Initial program 37.3%

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

   3. Simplified 35.7%

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

      1. associate-*r* 40.8%

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

      2. sub-neg 40.8%

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

      3. distribute-lft-in 26.5%

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

   6. Applied egg-rr 26.5%

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

      1. distribute-lft-out 40.8%

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

      2. sub-neg 40.8%

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

   8. Simplified 40.8%

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

   9. Taylor expanded in U* around inf 34.2%

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

       1. associate-*r/ 34.2%

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

   11. Simplified 34.2%

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

       1. add-sqr-sqrt 34.2%

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

       2. rem-sqrt-square 34.2%

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

       3. sqrt-div 34.2%

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

   13. Applied egg-rr 26.3%

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

       1. pow1 26.3%

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

       2. associate-*r* 27.8%

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

       3. *-commutative 27.8%

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

       4. sqrt-unprod 61.3%

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

   15. Applied egg-rr 61.3%

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

       1. unpow1 61.3%

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

       2. *-commutative 61.3%

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

   17. Simplified 61.3%

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

   if -2.64999999999999998e-143 < Om < -2.80000000000000006e-253

   1. Initial program 42.1%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in n around 0 54.9%

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

      1. associate-*r* 54.9%

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

      2. *-commutative 54.9%

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

      3. cancel-sign-sub-inv 54.9%

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

      4. metadata-eval 54.9%

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

   7. Simplified 54.9%

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

   if 2.50000000000000006e-47 < Om

   1. Initial program 59.1%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in n around 0 58.3%

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

      1. unpow2 58.3%

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

      2. associate-*r/ 60.5%

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

      3. *-commutative 60.5%

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

   7. Applied egg-rr 60.5%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1.5 \cdot 10^{-105}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{elif}\;Om \leq -2.65 \cdot 10^{-143}:\\ \;\;\;\;\left|\frac{\sqrt{U* \cdot \left(2 \cdot U\right)} \cdot \left(n \cdot \ell\right)}{Om}\right|\\ \mathbf{elif}\;Om \leq -2.8 \cdot 10^{-253}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 2.5 \cdot 10^{-47}:\\ \;\;\;\;\left|\frac{\sqrt{U* \cdot \left(2 \cdot U\right)} \cdot \left(n \cdot \ell\right)}{Om}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)} \cdot \sqrt{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq -6.6 \cdot 10^{-100}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{elif}\;Om \leq -2.7 \cdot 10^{-143} \lor \neg \left(Om \leq -6.8 \cdot 10^{-252}\right) \land Om \leq 2 \cdot 10^{-47}:\\ \;\;\;\;\left|\frac{\sqrt{U* \cdot \left(2 \cdot U\right)} \cdot \left(n \cdot \ell\right)}{Om}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om -6.6e-100)
   (sqrt (* 2.0 (* (* n U) (+ t (* (* l (/ l Om)) -2.0)))))
   (if (or (<= Om -2.7e-143) (and (not (<= Om -6.8e-252)) (<= Om 2e-47)))
     (fabs (/ (* (sqrt (* U* (* 2.0 U))) (* n l)) Om))
     (sqrt (* (* 2.0 U) (* n (+ t (* -2.0 (/ (pow l 2.0) Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= -6.6e-100) {
		tmp = sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
	} else if ((Om <= -2.7e-143) || (!(Om <= -6.8e-252) && (Om <= 2e-47))) {
		tmp = fabs(((sqrt((U_42_ * (2.0 * U))) * (n * l)) / Om));
	} else {
		tmp = sqrt(((2.0 * U) * (n * (t + (-2.0 * (pow(l, 2.0) / Om))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (om <= (-6.6d-100)) then
        tmp = sqrt((2.0d0 * ((n * u) * (t + ((l * (l / om)) * (-2.0d0))))))
    else if ((om <= (-2.7d-143)) .or. (.not. (om <= (-6.8d-252))) .and. (om <= 2d-47)) then
        tmp = abs(((sqrt((u_42 * (2.0d0 * u))) * (n * l)) / om))
    else
        tmp = sqrt(((2.0d0 * u) * (n * (t + ((-2.0d0) * ((l ** 2.0d0) / om))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= -6.6e-100) {
		tmp = Math.sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
	} else if ((Om <= -2.7e-143) || (!(Om <= -6.8e-252) && (Om <= 2e-47))) {
		tmp = Math.abs(((Math.sqrt((U_42_ * (2.0 * U))) * (n * l)) / Om));
	} else {
		tmp = Math.sqrt(((2.0 * U) * (n * (t + (-2.0 * (Math.pow(l, 2.0) / Om))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if Om <= -6.6e-100:
		tmp = math.sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))))
	elif (Om <= -2.7e-143) or (not (Om <= -6.8e-252) and (Om <= 2e-47)):
		tmp = math.fabs(((math.sqrt((U_42_ * (2.0 * U))) * (n * l)) / Om))
	else:
		tmp = math.sqrt(((2.0 * U) * (n * (t + (-2.0 * (math.pow(l, 2.0) / Om))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (Om <= -6.6e-100)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(Float64(l * Float64(l / Om)) * -2.0)))));
	elseif ((Om <= -2.7e-143) || (!(Om <= -6.8e-252) && (Om <= 2e-47)))
		tmp = abs(Float64(Float64(sqrt(Float64(U_42_ * Float64(2.0 * U))) * Float64(n * l)) / Om));
	else
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * Float64(t + Float64(-2.0 * Float64((l ^ 2.0) / Om))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (Om <= -6.6e-100)
		tmp = sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
	elseif ((Om <= -2.7e-143) || (~((Om <= -6.8e-252)) && (Om <= 2e-47)))
		tmp = abs(((sqrt((U_42_ * (2.0 * U))) * (n * l)) / Om));
	else
		tmp = sqrt(((2.0 * U) * (n * (t + (-2.0 * ((l ^ 2.0) / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[Om, -6.6e-100], N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[Om, -2.7e-143], And[N[Not[LessEqual[Om, -6.8e-252]], $MachinePrecision], LessEqual[Om, 2e-47]]], N[Abs[N[(N[(N[Sqrt[N[(U$42$ * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(t + N[(-2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if Om < -6.59999999999999993e-100

   1. Initial program 55.0%

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

   3. Simplified 52.3%

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

   5. Taylor expanded in n around 0 48.5%

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

      1. associate-*r* 53.8%

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

      2. sub-neg 53.8%

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

      3. *-commutative 53.8%

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

      4. distribute-rgt-neg-in 53.8%

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

      5. metadata-eval 53.8%

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

   7. Simplified 53.8%

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

      1. unpow2 48.3%

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

      2. associate-*r/ 53.6%

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

      3. *-commutative 53.6%

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

   9. Applied egg-rr 60.3%

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

   if -6.59999999999999993e-100 < Om < -2.70000000000000009e-143 or -6.7999999999999999e-252 < Om < 1.9999999999999999e-47

   1. Initial program 37.3%

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

   3. Simplified 35.7%

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

      1. associate-*r* 40.8%

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

      2. sub-neg 40.8%

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

      3. distribute-lft-in 26.5%

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

   6. Applied egg-rr 26.5%

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

      1. distribute-lft-out 40.8%

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

      2. sub-neg 40.8%

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

   8. Simplified 40.8%

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

   9. Taylor expanded in U* around inf 34.2%

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

       1. associate-*r/ 34.2%

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

   11. Simplified 34.2%

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

       1. add-sqr-sqrt 34.2%

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

       2. rem-sqrt-square 34.2%

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

       3. sqrt-div 34.2%

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

   13. Applied egg-rr 26.3%

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

       1. pow1 26.3%

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

       2. associate-*r* 27.8%

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

       3. *-commutative 27.8%

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

       4. sqrt-unprod 61.3%

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

   15. Applied egg-rr 61.3%

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

       1. unpow1 61.3%

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

       2. *-commutative 61.3%

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

   17. Simplified 61.3%

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

   if -2.70000000000000009e-143 < Om < -6.7999999999999999e-252 or 1.9999999999999999e-47 < Om

   1. Initial program 56.4%

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

   3. Simplified 60.2%

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

   5. Taylor expanded in n around 0 58.2%

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

      1. associate-*r* 58.2%

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

      2. *-commutative 58.2%

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

      3. cancel-sign-sub-inv 58.2%

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

      4. metadata-eval 58.2%

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

   7. Simplified 58.2%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -6.6 \cdot 10^{-100}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{elif}\;Om \leq -2.7 \cdot 10^{-143} \lor \neg \left(Om \leq -6.8 \cdot 10^{-252}\right) \land Om \leq 2 \cdot 10^{-47}:\\ \;\;\;\;\left|\frac{\sqrt{U* \cdot \left(2 \cdot U\right)} \cdot \left(n \cdot \ell\right)}{Om}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{if}\;Om \leq -5.2 \cdot 10^{-106}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Om \leq 2.5 \cdot 10^{-47}:\\ \;\;\;\;\left|\sqrt{U \cdot \left(2 \cdot U*\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right|\\ \mathbf{elif}\;Om \leq 2.9 \cdot 10^{+133} \lor \neg \left(Om \leq 5.8 \cdot 10^{+221}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 (* (* n U) (+ t (* (* l (/ l Om)) -2.0)))))))
   (if (<= Om -5.2e-106)
     t_1
     (if (<= Om 2.5e-47)
       (fabs (* (sqrt (* U (* 2.0 U*))) (* n (/ l Om))))
       (if (or (<= Om 2.9e+133) (not (<= Om 5.8e+221)))
         t_1
         (sqrt (* 2.0 (* U (* n t)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
	double tmp;
	if (Om <= -5.2e-106) {
		tmp = t_1;
	} else if (Om <= 2.5e-47) {
		tmp = fabs((sqrt((U * (2.0 * U_42_))) * (n * (l / Om))));
	} else if ((Om <= 2.9e+133) || !(Om <= 5.8e+221)) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = sqrt((2.0d0 * ((n * u) * (t + ((l * (l / om)) * (-2.0d0))))))
    if (om <= (-5.2d-106)) then
        tmp = t_1
    else if (om <= 2.5d-47) then
        tmp = abs((sqrt((u * (2.0d0 * u_42))) * (n * (l / om))))
    else if ((om <= 2.9d+133) .or. (.not. (om <= 5.8d+221))) then
        tmp = t_1
    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 t_1 = Math.sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
	double tmp;
	if (Om <= -5.2e-106) {
		tmp = t_1;
	} else if (Om <= 2.5e-47) {
		tmp = Math.abs((Math.sqrt((U * (2.0 * U_42_))) * (n * (l / Om))));
	} else if ((Om <= 2.9e+133) || !(Om <= 5.8e+221)) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))))
	tmp = 0
	if Om <= -5.2e-106:
		tmp = t_1
	elif Om <= 2.5e-47:
		tmp = math.fabs((math.sqrt((U * (2.0 * U_42_))) * (n * (l / Om))))
	elif (Om <= 2.9e+133) or not (Om <= 5.8e+221):
		tmp = t_1
	else:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(Float64(l * Float64(l / Om)) * -2.0)))))
	tmp = 0.0
	if (Om <= -5.2e-106)
		tmp = t_1;
	elseif (Om <= 2.5e-47)
		tmp = abs(Float64(sqrt(Float64(U * Float64(2.0 * U_42_))) * Float64(n * Float64(l / Om))));
	elseif ((Om <= 2.9e+133) || !(Om <= 5.8e+221))
		tmp = t_1;
	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_)
	t_1 = sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
	tmp = 0.0;
	if (Om <= -5.2e-106)
		tmp = t_1;
	elseif (Om <= 2.5e-47)
		tmp = abs((sqrt((U * (2.0 * U_42_))) * (n * (l / Om))));
	elseif ((Om <= 2.9e+133) || ~((Om <= 5.8e+221)))
		tmp = t_1;
	else
		tmp = sqrt((2.0 * (U * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -5.2e-106], t$95$1, If[LessEqual[Om, 2.5e-47], N[Abs[N[(N[Sqrt[N[(U * N[(2.0 * U$42$), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(n * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[Om, 2.9e+133], N[Not[LessEqual[Om, 5.8e+221]], $MachinePrecision]], t$95$1, N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if Om < -5.2000000000000001e-106 or 2.50000000000000006e-47 < Om < 2.9000000000000001e133 or 5.7999999999999996e221 < Om

   1. Initial program 55.5%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in n around 0 48.6%

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

      1. associate-*r* 53.8%

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

      2. sub-neg 53.8%

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

      3. *-commutative 53.8%

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

      4. distribute-rgt-neg-in 53.8%

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

      5. metadata-eval 53.8%

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

   7. Simplified 53.8%

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

      1. unpow2 48.3%

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

      2. associate-*r/ 52.2%

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

      3. *-commutative 52.2%

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

   9. Applied egg-rr 58.3%

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

   if -5.2000000000000001e-106 < Om < 2.50000000000000006e-47

   1. Initial program 38.4%

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

   3. Simplified 39.8%

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

      1. associate-*r* 43.7%

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

      2. sub-neg 43.7%

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

      3. distribute-lft-in 29.9%

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

   6. Applied egg-rr 29.9%

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

      1. distribute-lft-out 43.7%

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

      2. sub-neg 43.7%

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

   8. Simplified 43.7%

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

   9. Taylor expanded in U* around inf 29.5%

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

       1. associate-*r/ 29.5%

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

   11. Simplified 29.5%

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

       1. add-sqr-sqrt 29.5%

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

       2. rem-sqrt-square 29.5%

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

       3. sqrt-div 29.5%

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

   13. Applied egg-rr 25.8%

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

       1. div-inv 25.8%

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

       2. associate-*r* 26.9%

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

       3. *-commutative 26.9%

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

       4. sqrt-unprod 54.5%

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

   15. Applied egg-rr 54.5%

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

       1. associate-*r/ 54.5%

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

       2. *-rgt-identity 54.5%

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

       3. associate-*r/ 50.8%

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

       4. associate-*l* 50.8%

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

       5. associate-/l* 51.0%

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

   17. Simplified 51.0%

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

   if 2.9000000000000001e133 < Om < 5.7999999999999996e221

   1. Initial program 65.5%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in t around inf 80.7%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -5.2 \cdot 10^{-106}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{elif}\;Om \leq 2.5 \cdot 10^{-47}:\\ \;\;\;\;\left|\sqrt{U \cdot \left(2 \cdot U*\right)} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right|\\ \mathbf{elif}\;Om \leq 2.9 \cdot 10^{+133} \lor \neg \left(Om \leq 5.8 \cdot 10^{+221}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{if}\;Om \leq -7.2 \cdot 10^{-101}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;Om \leq 2 \cdot 10^{-47}:\\ \;\;\;\;\left|\frac{\sqrt{U* \cdot \left(2 \cdot U\right)} \cdot \left(n \cdot \ell\right)}{Om}\right|\\ \mathbf{elif}\;Om \leq 1.65 \cdot 10^{+133} \lor \neg \left(Om \leq 4.4 \cdot 10^{+221}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 (* (* n U) (+ t (* (* l (/ l Om)) -2.0)))))))
   (if (<= Om -7.2e-101)
     t_1
     (if (<= Om 2e-47)
       (fabs (/ (* (sqrt (* U* (* 2.0 U))) (* n l)) Om))
       (if (or (<= Om 1.65e+133) (not (<= Om 4.4e+221)))
         t_1
         (sqrt (* 2.0 (* U (* n t)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
	double tmp;
	if (Om <= -7.2e-101) {
		tmp = t_1;
	} else if (Om <= 2e-47) {
		tmp = fabs(((sqrt((U_42_ * (2.0 * U))) * (n * l)) / Om));
	} else if ((Om <= 1.65e+133) || !(Om <= 4.4e+221)) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = sqrt((2.0d0 * ((n * u) * (t + ((l * (l / om)) * (-2.0d0))))))
    if (om <= (-7.2d-101)) then
        tmp = t_1
    else if (om <= 2d-47) then
        tmp = abs(((sqrt((u_42 * (2.0d0 * u))) * (n * l)) / om))
    else if ((om <= 1.65d+133) .or. (.not. (om <= 4.4d+221))) then
        tmp = t_1
    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 t_1 = Math.sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
	double tmp;
	if (Om <= -7.2e-101) {
		tmp = t_1;
	} else if (Om <= 2e-47) {
		tmp = Math.abs(((Math.sqrt((U_42_ * (2.0 * U))) * (n * l)) / Om));
	} else if ((Om <= 1.65e+133) || !(Om <= 4.4e+221)) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))))
	tmp = 0
	if Om <= -7.2e-101:
		tmp = t_1
	elif Om <= 2e-47:
		tmp = math.fabs(((math.sqrt((U_42_ * (2.0 * U))) * (n * l)) / Om))
	elif (Om <= 1.65e+133) or not (Om <= 4.4e+221):
		tmp = t_1
	else:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(2.0 * Float64(Float64(n * U) * Float64(t + Float64(Float64(l * Float64(l / Om)) * -2.0)))))
	tmp = 0.0
	if (Om <= -7.2e-101)
		tmp = t_1;
	elseif (Om <= 2e-47)
		tmp = abs(Float64(Float64(sqrt(Float64(U_42_ * Float64(2.0 * U))) * Float64(n * l)) / Om));
	elseif ((Om <= 1.65e+133) || !(Om <= 4.4e+221))
		tmp = t_1;
	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_)
	t_1 = sqrt((2.0 * ((n * U) * (t + ((l * (l / Om)) * -2.0)))));
	tmp = 0.0;
	if (Om <= -7.2e-101)
		tmp = t_1;
	elseif (Om <= 2e-47)
		tmp = abs(((sqrt((U_42_ * (2.0 * U))) * (n * l)) / Om));
	elseif ((Om <= 1.65e+133) || ~((Om <= 4.4e+221)))
		tmp = t_1;
	else
		tmp = sqrt((2.0 * (U * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t + N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, -7.2e-101], t$95$1, If[LessEqual[Om, 2e-47], N[Abs[N[(N[(N[Sqrt[N[(U$42$ * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[Om, 1.65e+133], N[Not[LessEqual[Om, 4.4e+221]], $MachinePrecision]], t$95$1, N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if Om < -7.19999999999999999e-101 or 1.9999999999999999e-47 < Om < 1.65e133 or 4.3999999999999999e221 < Om

   1. Initial program 55.5%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in n around 0 48.6%

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

      1. associate-*r* 53.8%

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

      2. sub-neg 53.8%

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

      3. *-commutative 53.8%

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

      4. distribute-rgt-neg-in 53.8%

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

      5. metadata-eval 53.8%

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

   7. Simplified 53.8%

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

      1. unpow2 48.3%

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

      2. associate-*r/ 52.2%

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

      3. *-commutative 52.2%

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

   9. Applied egg-rr 58.3%

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

   if -7.19999999999999999e-101 < Om < 1.9999999999999999e-47

   1. Initial program 38.4%

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

   3. Simplified 39.8%

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

      1. associate-*r* 43.7%

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

      2. sub-neg 43.7%

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

      3. distribute-lft-in 29.9%

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

   6. Applied egg-rr 29.9%

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

      1. distribute-lft-out 43.7%

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

      2. sub-neg 43.7%

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

   8. Simplified 43.7%

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

   9. Taylor expanded in U* around inf 29.5%

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

       1. associate-*r/ 29.5%

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

   11. Simplified 29.5%

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

       1. add-sqr-sqrt 29.5%

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

       2. rem-sqrt-square 29.5%

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

       3. sqrt-div 29.5%

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

   13. Applied egg-rr 25.8%

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

       1. pow1 25.8%

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

       2. associate-*r* 26.9%

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

       3. *-commutative 26.9%

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

       4. sqrt-unprod 54.5%

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

   15. Applied egg-rr 54.5%

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

       1. unpow1 54.5%

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

       2. *-commutative 54.5%

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

   17. Simplified 54.5%

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

   if 1.65e133 < Om < 4.3999999999999999e221

   1. Initial program 65.5%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in t around inf 80.7%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -7.2 \cdot 10^{-101}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{elif}\;Om \leq 2 \cdot 10^{-47}:\\ \;\;\;\;\left|\frac{\sqrt{U* \cdot \left(2 \cdot U\right)} \cdot \left(n \cdot \ell\right)}{Om}\right|\\ \mathbf{elif}\;Om \leq 1.65 \cdot 10^{+133} \lor \neg \left(Om \leq 4.4 \cdot 10^{+221}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 47.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 51.7%

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

5. Taylor expanded in n around 0 47.3%

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

   1. associate-*r* 46.9%

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

   2. sub-neg 46.9%

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

   3. *-commutative 46.9%

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

   4. distribute-rgt-neg-in 46.9%

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

   5. metadata-eval 46.9%

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

7. Simplified 46.9%

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

   1. unpow2 47.1%

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

   2. associate-*r/ 49.7%

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

   3. *-commutative 49.7%

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

9. Applied egg-rr 49.6%

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

10. Final simplification 49.6%

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

Alternative 7: 37.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 5 \cdot 10^{+132}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om 5e+132)
   (pow (* 2.0 (* t (* n U))) 0.5)
   (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 (Om <= 5e+132) {
		tmp = pow((2.0 * (t * (n * U))), 0.5);
	} 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 (om <= 5d+132) then
        tmp = (2.0d0 * (t * (n * u))) ** 0.5d0
    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 (Om <= 5e+132) {
		tmp = Math.pow((2.0 * (t * (n * U))), 0.5);
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if Om <= 5e+132:
		tmp = math.pow((2.0 * (t * (n * U))), 0.5)
	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 (Om <= 5e+132)
		tmp = Float64(2.0 * Float64(t * Float64(n * U))) ^ 0.5;
	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 (Om <= 5e+132)
		tmp = (2.0 * (t * (n * U))) ^ 0.5;
	else
		tmp = sqrt((2.0 * (U * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[Om, 5e+132], N[Power[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < 5.0000000000000001e132

   1. Initial program 48.8%

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

   3. Simplified 47.3%

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

   5. Taylor expanded in t around inf 36.7%

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

      1. pow1/2 37.7%

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

      2. associate-*r* 40.6%

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

      3. *-commutative 40.6%

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

      4. *-commutative 40.6%

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

   7. Applied egg-rr 40.6%

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

   if 5.0000000000000001e132 < Om

   1. Initial program 63.3%

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

   3. Simplified 71.5%

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

   5. Taylor expanded in t around inf 72.3%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 5 \cdot 10^{+132}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om 5e+132) (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 (Om <= 5e+132) {
		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 (om <= 5d+132) 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 (Om <= 5e+132) {
		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 Om <= 5e+132:
		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 (Om <= 5e+132)
		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 (Om <= 5e+132)
		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[Om, 5e+132], 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 Om < 5.0000000000000001e132

   1. Initial program 48.8%

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

   3. Simplified 47.3%

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

   5. Taylor expanded in t around inf 36.7%

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

      1. associate-*r* 39.6%

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

   7. Simplified 39.6%

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

   if 5.0000000000000001e132 < Om

   1. Initial program 63.3%

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

   3. Simplified 71.5%

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

   5. Taylor expanded in t around inf 72.3%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.5% 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 51.5%

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

3. Simplified 51.7%

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

5. Taylor expanded in t around inf 43.3%

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

6. Final simplification 43.3%

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

Reproduce

herbie shell --seed 2024071 
(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*))))))