Toniolo and Linder, Equation (13)

Percentage Accurate: 50.2% → 61.5%

Time: 23.4s

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: 50.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 61.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := n \cdot t\_2\\ t_4 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t\_3 \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_4 \leq 4 \cdot 10^{-292}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, t\_1, n \cdot \left(U \cdot t\_2\right)\right)\right)\right)}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(t\_3 \cdot \left(U - U*\right) + 2 \cdot t\_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(U \cdot {\ell}^{2}\right) \cdot \left(n \cdot \left(\frac{2}{Om} - \frac{n \cdot U*}{{Om}^{2}}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om)))
        (t_2 (pow (/ l Om) 2.0))
        (t_3 (* n t_2))
        (t_4
         (*
          (* (* 2.0 n) U)
          (+ (- t (* 2.0 (/ (* l l) Om))) (* t_3 (- U* U))))))
   (if (<= t_4 4e-292)
     (sqrt (* (* 2.0 n) (* U (- t (fma 2.0 t_1 (* n (* U t_2)))))))
     (if (<= t_4 INFINITY)
       (sqrt (* (* 2.0 (* n U)) (- t (+ (* t_3 (- U U*)) (* 2.0 t_1)))))
       (sqrt
        (*
         -2.0
         (*
          (* U (pow l 2.0))
          (* n (- (/ 2.0 Om) (/ (* n U*) (pow Om 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 = pow((l / Om), 2.0);
	double t_3 = n * t_2;
	double t_4 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + (t_3 * (U_42_ - U)));
	double tmp;
	if (t_4 <= 4e-292) {
		tmp = sqrt(((2.0 * n) * (U * (t - fma(2.0, t_1, (n * (U * t_2)))))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (n * U)) * (t - ((t_3 * (U - U_42_)) + (2.0 * t_1)))));
	} else {
		tmp = sqrt((-2.0 * ((U * pow(l, 2.0)) * (n * ((2.0 / Om) - ((n * U_42_) / pow(Om, 2.0)))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(l / Om))
	t_2 = Float64(l / Om) ^ 2.0
	t_3 = Float64(n * t_2)
	t_4 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(t_3 * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_4 <= 4e-292)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - fma(2.0, t_1, Float64(n * Float64(U * t_2)))))));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(Float64(t_3 * Float64(U - U_42_)) + Float64(2.0 * t_1)))));
	else
		tmp = sqrt(Float64(-2.0 * Float64(Float64(U * (l ^ 2.0)) * Float64(n * Float64(Float64(2.0 / Om) - Float64(Float64(n * U_42_) / (Om ^ 2.0)))))));
	end
	return 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[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(n * t$95$2), $MachinePrecision]}, Block[{t$95$4 = 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$3 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 4e-292], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * t$95$1 + N[(n * N[(U * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(t$95$3 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] + N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(U * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(n * N[(N[(2.0 / Om), $MachinePrecision] - N[(N[(n * U$42$), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.0000000000000002e-292

   1. Initial program 13.4%

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

   3. Simplified 40.4%

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

   5. Taylor expanded in U around inf 42.4%

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

      1. associate-/l* 42.4%

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

      2. unpow2 42.4%

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

      3. unpow2 42.4%

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

      4. times-frac 42.8%

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

      5. unpow2 42.8%

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

   7. Simplified 42.8%

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

   if 4.0000000000000002e-292 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 67.1%

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

   3. Simplified 70.2%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.1%

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

      1. associate-*r* 0.1%

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

      2. fma-define 14.3%

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

      3. associate-*r* 14.3%

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

   6. Applied egg-rr 14.3%

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

   7. Taylor expanded in U around 0 14.6%

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

      1. associate-/l* 14.4%

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

      2. unpow2 14.4%

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

      3. unpow2 14.4%

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

      4. times-frac 14.6%

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

      5. unpow2 14.6%

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

      6. neg-mul-1 14.6%

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

      7. distribute-lft-neg-out 14.6%

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

      8. *-commutative 14.6%

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

   9. Simplified 14.6%

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

   10. Taylor expanded in l around inf 32.9%

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

       1. associate-*r* 36.0%

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

       2. +-commutative 36.0%

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

       3. mul-1-neg 36.0%

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

       4. unsub-neg 36.0%

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

       5. associate-*r/ 36.0%

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

       6. metadata-eval 36.0%

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

       7. *-commutative 36.0%

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

   12. Simplified 36.0%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 4 \cdot 10^{-292}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left(U \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(U \cdot {\ell}^{2}\right) \cdot \left(n \cdot \left(\frac{2}{Om} - \frac{n \cdot U*}{{Om}^{2}}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t\_1 \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-146}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(-2 \cdot \frac{U \cdot {\ell}^{2}}{Om} + U \cdot t\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(t\_1 \cdot \left(U - U*\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) - \frac{-2 \cdot \left(U \cdot \left(U* \cdot \frac{{\left(n \cdot \ell\right)}^{2}}{Om}\right)\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (pow (/ l Om) 2.0)))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+ (- t (* 2.0 (/ (* l l) Om))) (* t_1 (- U* U)))))))
   (if (<= t_2 2e-146)
     (sqrt (* (* 2.0 n) (+ (* -2.0 (/ (* U (pow l 2.0)) Om)) (* U t))))
     (if (<= t_2 INFINITY)
       (sqrt
        (* (* 2.0 (* n U)) (- t (+ (* t_1 (- U U*)) (* 2.0 (* l (/ l Om)))))))
       (sqrt
        (-
         (* 2.0 (* U (* n t)))
         (/ (* -2.0 (* U (* U* (/ (pow (* n l) 2.0) Om)))) Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * pow((l / Om), 2.0);
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + (t_1 * (U_42_ - U)))));
	double tmp;
	if (t_2 <= 2e-146) {
		tmp = sqrt(((2.0 * n) * ((-2.0 * ((U * pow(l, 2.0)) / Om)) + (U * t))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (n * U)) * (t - ((t_1 * (U - U_42_)) + (2.0 * (l * (l / Om)))))));
	} else {
		tmp = sqrt(((2.0 * (U * (n * t))) - ((-2.0 * (U * (U_42_ * (pow((n * l), 2.0) / Om)))) / Om)));
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * Math.pow((l / Om), 2.0);
	double t_2 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + (t_1 * (U_42_ - U)))));
	double tmp;
	if (t_2 <= 2e-146) {
		tmp = Math.sqrt(((2.0 * n) * ((-2.0 * ((U * Math.pow(l, 2.0)) / Om)) + (U * t))));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t - ((t_1 * (U - U_42_)) + (2.0 * (l * (l / Om)))))));
	} else {
		tmp = Math.sqrt(((2.0 * (U * (n * t))) - ((-2.0 * (U * (U_42_ * (Math.pow((n * l), 2.0) / Om)))) / Om)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = n * math.pow((l / Om), 2.0)
	t_2 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + (t_1 * (U_42_ - U)))))
	tmp = 0
	if t_2 <= 2e-146:
		tmp = math.sqrt(((2.0 * n) * ((-2.0 * ((U * math.pow(l, 2.0)) / Om)) + (U * t))))
	elif t_2 <= math.inf:
		tmp = math.sqrt(((2.0 * (n * U)) * (t - ((t_1 * (U - U_42_)) + (2.0 * (l * (l / Om)))))))
	else:
		tmp = math.sqrt(((2.0 * (U * (n * t))) - ((-2.0 * (U * (U_42_ * (math.pow((n * l), 2.0) / Om)))) / Om)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * (Float64(l / Om) ^ 2.0))
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(t_1 * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_2 <= 2e-146)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(Float64(-2.0 * Float64(Float64(U * (l ^ 2.0)) / Om)) + Float64(U * t))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(Float64(t_1 * Float64(U - U_42_)) + Float64(2.0 * Float64(l * Float64(l / Om)))))));
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(U * Float64(n * t))) - Float64(Float64(-2.0 * Float64(U * Float64(U_42_ * Float64((Float64(n * l) ^ 2.0) / Om)))) / Om)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = n * ((l / Om) ^ 2.0);
	t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + (t_1 * (U_42_ - U)))));
	tmp = 0.0;
	if (t_2 <= 2e-146)
		tmp = sqrt(((2.0 * n) * ((-2.0 * ((U * (l ^ 2.0)) / Om)) + (U * t))));
	elseif (t_2 <= Inf)
		tmp = sqrt(((2.0 * (n * U)) * (t - ((t_1 * (U - U_42_)) + (2.0 * (l * (l / Om)))))));
	else
		tmp = sqrt(((2.0 * (U * (n * t))) - ((-2.0 * (U * (U_42_ * (((n * l) ^ 2.0) / Om)))) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 2e-146], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(N[(-2.0 * N[(N[(U * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(-2.0 * N[(U * N[(U$42$ * N[(N[Power[N[(n * l), $MachinePrecision], 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2.00000000000000005e-146

   1. Initial program 14.8%

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

   3. Simplified 44.4%

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

   5. Taylor expanded in Om around inf 47.0%

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

   if 2.00000000000000005e-146 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 67.1%

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

   3. Simplified 70.2%

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

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

   1. Initial program 0.0%

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

   3. Simplified 3.5%

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

   5. Taylor expanded in Om around -inf 9.6%

      \[\leadsto \sqrt{\color{blue}{-1 \cdot \frac{2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{Om} + 4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

   6. Taylor expanded in U* around inf 20.0%

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

      1. associate-/l* 20.0%

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

      2. associate-/l* 20.0%

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

      3. unpow2 20.0%

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

      4. unpow2 20.0%

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

      5. swap-sqr 31.9%

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

      6. unpow2 31.9%

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

      7. *-commutative 31.9%

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

   8. Simplified 31.9%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{-146}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(-2 \cdot \frac{U \cdot {\ell}^{2}}{Om} + U \cdot t\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) - \frac{-2 \cdot \left(U \cdot \left(U* \cdot \frac{{\left(n \cdot \ell\right)}^{2}}{Om}\right)\right)}{Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot U*\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right)}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+246}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) - \frac{-2 \cdot \left(U \cdot \left(U* \cdot \frac{{\left(n \cdot \ell\right)}^{2}}{Om}\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 1.5e+41)
   (sqrt
    (*
     (* 2.0 (* n U))
     (+ t (- (* n (* (pow (/ l Om) 2.0) U*)) (* (/ l Om) (* 2.0 l))))))
   (if (<= t 3.3e+246)
     (sqrt
      (-
       (* 2.0 (* U (* n t)))
       (/ (* -2.0 (* U (* U* (/ (pow (* n l) 2.0) Om)))) Om)))
     (* (sqrt (* (* 2.0 n) U)) (sqrt t)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 1.5e+41) {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * (pow((l / Om), 2.0) * U_42_)) - ((l / Om) * (2.0 * l))))));
	} else if (t <= 3.3e+246) {
		tmp = sqrt(((2.0 * (U * (n * t))) - ((-2.0 * (U * (U_42_ * (pow((n * l), 2.0) / Om)))) / Om)));
	} else {
		tmp = sqrt(((2.0 * n) * U)) * sqrt(t);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= 1.5d+41) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + ((n * (((l / om) ** 2.0d0) * u_42)) - ((l / om) * (2.0d0 * l))))))
    else if (t <= 3.3d+246) then
        tmp = sqrt(((2.0d0 * (u * (n * t))) - (((-2.0d0) * (u * (u_42 * (((n * l) ** 2.0d0) / om)))) / om)))
    else
        tmp = sqrt(((2.0d0 * n) * u)) * sqrt(t)
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 1.5e+41) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((n * (Math.pow((l / Om), 2.0) * U_42_)) - ((l / Om) * (2.0 * l))))));
	} else if (t <= 3.3e+246) {
		tmp = Math.sqrt(((2.0 * (U * (n * t))) - ((-2.0 * (U * (U_42_ * (Math.pow((n * l), 2.0) / Om)))) / Om)));
	} else {
		tmp = Math.sqrt(((2.0 * n) * U)) * Math.sqrt(t);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= 1.5e+41:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((n * (math.pow((l / Om), 2.0) * U_42_)) - ((l / Om) * (2.0 * l))))))
	elif t <= 3.3e+246:
		tmp = math.sqrt(((2.0 * (U * (n * t))) - ((-2.0 * (U * (U_42_ * (math.pow((n * l), 2.0) / Om)))) / Om)))
	else:
		tmp = math.sqrt(((2.0 * n) * U)) * math.sqrt(t)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 1.5e+41)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(n * Float64((Float64(l / Om) ^ 2.0) * U_42_)) - Float64(Float64(l / Om) * Float64(2.0 * l))))));
	elseif (t <= 3.3e+246)
		tmp = sqrt(Float64(Float64(2.0 * Float64(U * Float64(n * t))) - Float64(Float64(-2.0 * Float64(U * Float64(U_42_ * Float64((Float64(n * l) ^ 2.0) / Om)))) / Om)));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * U)) * sqrt(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= 1.5e+41)
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * (((l / Om) ^ 2.0) * U_42_)) - ((l / Om) * (2.0 * l))))));
	elseif (t <= 3.3e+246)
		tmp = sqrt(((2.0 * (U * (n * t))) - ((-2.0 * (U * (U_42_ * (((n * l) ^ 2.0) / Om)))) / Om)));
	else
		tmp = sqrt(((2.0 * n) * U)) * sqrt(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 1.5e+41], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(n * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * U$42$), $MachinePrecision]), $MachinePrecision] - N[(N[(l / Om), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 3.3e+246], N[Sqrt[N[(N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(-2.0 * N[(U * N[(U$42$ * N[(N[Power[N[(n * l), $MachinePrecision], 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.4999999999999999e41

   1. Initial program 54.4%

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

   3. Simplified 57.0%

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

      1. associate-*r* 57.0%

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

      2. fma-define 58.6%

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

      3. associate-*r* 58.6%

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

   6. Applied egg-rr 58.6%

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

   7. Taylor expanded in U around 0 48.1%

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

      1. associate-/l* 52.2%

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

      2. unpow2 52.2%

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

      3. unpow2 52.2%

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

      4. times-frac 58.5%

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

      5. unpow2 58.5%

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

      6. neg-mul-1 58.5%

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

      7. distribute-lft-neg-out 58.5%

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

      8. *-commutative 58.5%

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

   9. Simplified 58.5%

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

       1. fma-undefine 56.9%

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

   11. Applied egg-rr 56.9%

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

   if 1.4999999999999999e41 < t < 3.3e246

   1. Initial program 45.8%

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

   3. Simplified 57.0%

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

   5. Taylor expanded in Om around -inf 51.0%

      \[\leadsto \sqrt{\color{blue}{-1 \cdot \frac{2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{Om} + 4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

   6. Taylor expanded in U* around inf 57.7%

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

      1. associate-/l* 59.7%

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

      2. associate-/l* 59.6%

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

      3. unpow2 59.6%

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

      4. unpow2 59.6%

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

      5. swap-sqr 62.2%

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

      6. unpow2 62.2%

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

      7. *-commutative 62.2%

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

   8. Simplified 62.2%

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

   if 3.3e246 < t

   1. Initial program 21.0%

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

   3. Simplified 15.0%

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

   5. Taylor expanded in t around inf 15.0%

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

      1. pow1/2 15.4%

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

      2. associate-*r* 21.1%

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

      3. unpow-prod-down 53.0%

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

      4. pow1/2 53.0%

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

   7. Applied egg-rr 53.0%

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

      1. unpow1/2 53.0%

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

      2. *-commutative 53.0%

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

      3. *-commutative 53.0%

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

   9. Simplified 53.0%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot U*\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right)}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+246}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) - \frac{-2 \cdot \left(U \cdot \left(U* \cdot \frac{{\left(n \cdot \ell\right)}^{2}}{Om}\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 48.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.55 \cdot 10^{+16}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) - \frac{-2 \cdot \left(U \cdot \left(U* \cdot \frac{{\left(n \cdot \ell\right)}^{2}}{Om}\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.55e+16)
   (sqrt
    (-
     (* 2.0 (* U (* n t)))
     (/ (* -2.0 (* U (* U* (/ (pow (* n l) 2.0) Om)))) Om)))
   (pow (* 2.0 (* (* n U) (- t (/ (* 2.0 (pow l 2.0)) Om)))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.55e+16) {
		tmp = sqrt(((2.0 * (U * (n * t))) - ((-2.0 * (U * (U_42_ * (pow((n * l), 2.0) / Om)))) / Om)));
	} else {
		tmp = pow((2.0 * ((n * U) * (t - ((2.0 * pow(l, 2.0)) / Om)))), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 2.55d+16) then
        tmp = sqrt(((2.0d0 * (u * (n * t))) - (((-2.0d0) * (u * (u_42 * (((n * l) ** 2.0d0) / om)))) / om)))
    else
        tmp = (2.0d0 * ((n * u) * (t - ((2.0d0 * (l ** 2.0d0)) / om)))) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.55e+16) {
		tmp = Math.sqrt(((2.0 * (U * (n * t))) - ((-2.0 * (U * (U_42_ * (Math.pow((n * l), 2.0) / Om)))) / Om)));
	} else {
		tmp = Math.pow((2.0 * ((n * U) * (t - ((2.0 * Math.pow(l, 2.0)) / Om)))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.55e+16:
		tmp = math.sqrt(((2.0 * (U * (n * t))) - ((-2.0 * (U * (U_42_ * (math.pow((n * l), 2.0) / Om)))) / Om)))
	else:
		tmp = math.pow((2.0 * ((n * U) * (t - ((2.0 * math.pow(l, 2.0)) / Om)))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.55e+16)
		tmp = sqrt(Float64(Float64(2.0 * Float64(U * Float64(n * t))) - Float64(Float64(-2.0 * Float64(U * Float64(U_42_ * Float64((Float64(n * l) ^ 2.0) / Om)))) / Om)));
	else
		tmp = Float64(2.0 * Float64(Float64(n * U) * Float64(t - Float64(Float64(2.0 * (l ^ 2.0)) / Om)))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2.55e+16)
		tmp = sqrt(((2.0 * (U * (n * t))) - ((-2.0 * (U * (U_42_ * (((n * l) ^ 2.0) / Om)))) / Om)));
	else
		tmp = (2.0 * ((n * U) * (t - ((2.0 * (l ^ 2.0)) / Om)))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.55e+16], N[Sqrt[N[(N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(-2.0 * N[(U * N[(U$42$ * N[(N[Power[N[(n * l), $MachinePrecision], 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t - N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.55e16

   1. Initial program 53.8%

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

   3. Simplified 57.0%

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

   5. Taylor expanded in Om around -inf 43.7%

      \[\leadsto \sqrt{\color{blue}{-1 \cdot \frac{2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{Om} + 4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

   6. Taylor expanded in U* around inf 44.4%

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

      1. associate-/l* 45.3%

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

      2. associate-/l* 45.3%

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

      3. unpow2 45.3%

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

      4. unpow2 45.3%

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

      5. swap-sqr 53.6%

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

      6. unpow2 53.6%

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

      7. *-commutative 53.6%

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

   8. Simplified 53.6%

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

   if 2.55e16 < l

   1. Initial program 36.8%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in n around 0 27.8%

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

      1. pow1/2 42.5%

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

      2. associate-*r* 45.9%

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

      3. *-commutative 45.9%

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

      4. associate-*r/ 45.9%

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

   7. Applied egg-rr 45.9%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.55 \cdot 10^{+16}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) - \frac{-2 \cdot \left(U \cdot \left(U* \cdot \frac{{\left(n \cdot \ell\right)}^{2}}{Om}\right)\right)}{Om}}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq -7.5 \cdot 10^{-139}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U -7.5e-139)
   (pow (* 2.0 (* (* n U) (- t (/ (* 2.0 (pow l 2.0)) Om)))) 0.5)
   (sqrt (* 2.0 (* U (* n (- t (* 2.0 (* l (/ l Om))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -7.5e-139) {
		tmp = pow((2.0 * ((n * U) * (t - ((2.0 * pow(l, 2.0)) / Om)))), 0.5);
	} else {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= (-7.5d-139)) then
        tmp = (2.0d0 * ((n * u) * (t - ((2.0d0 * (l ** 2.0d0)) / om)))) ** 0.5d0
    else
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * (l * (l / om))))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -7.5e-139) {
		tmp = Math.pow((2.0 * ((n * U) * (t - ((2.0 * Math.pow(l, 2.0)) / Om)))), 0.5);
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= -7.5e-139:
		tmp = math.pow((2.0 * ((n * U) * (t - ((2.0 * math.pow(l, 2.0)) / Om)))), 0.5)
	else:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= -7.5e-139)
		tmp = Float64(2.0 * Float64(Float64(n * U) * Float64(t - Float64(Float64(2.0 * (l ^ 2.0)) / Om)))) ^ 0.5;
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= -7.5e-139)
		tmp = (2.0 * ((n * U) * (t - ((2.0 * (l ^ 2.0)) / Om)))) ^ 0.5;
	else
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, -7.5e-139], N[Power[N[(2.0 * N[(N[(n * U), $MachinePrecision] * N[(t - N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -7.5000000000000001e-139

   1. Initial program 58.7%

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

   3. Simplified 54.4%

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

   5. Taylor expanded in n around 0 48.4%

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

      1. pow1/2 56.9%

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

      2. associate-*r* 60.1%

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

      3. *-commutative 60.1%

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

      4. associate-*r/ 60.1%

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

   7. Applied egg-rr 60.1%

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

   if -7.5000000000000001e-139 < U

   1. Initial program 46.5%

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

   3. Simplified 53.4%

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

   5. Taylor expanded in n around 0 48.4%

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

      1. unpow2 48.4%

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

      2. associate-*r/ 49.5%

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

      3. *-commutative 49.5%

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

   7. Applied egg-rr 49.5%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -7.5 \cdot 10^{-139}:\\ \;\;\;\;{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4.1 \cdot 10^{+254}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 4.1e+254)
   (sqrt (* 2.0 (* U (* n (- t (* 2.0 (* l (/ l Om))))))))
   (* (sqrt (* (* 2.0 n) U)) (sqrt t))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 4.1e+254) {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	} else {
		tmp = sqrt(((2.0 * n) * U)) * sqrt(t);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= 4.1d+254) then
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * (l * (l / om))))))))
    else
        tmp = sqrt(((2.0d0 * n) * u)) * sqrt(t)
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 4.1e+254) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * U)) * Math.sqrt(t);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= 4.1e+254:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))))
	else:
		tmp = math.sqrt(((2.0 * n) * U)) * math.sqrt(t)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 4.1e+254)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om))))))));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * U)) * sqrt(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= 4.1e+254)
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (l * (l / Om))))))));
	else
		tmp = sqrt(((2.0 * n) * U)) * sqrt(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 4.1e+254], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.09999999999999987e254

   1. Initial program 52.1%

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

   3. Simplified 56.0%

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

   5. Taylor expanded in n around 0 49.5%

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

      1. unpow2 49.5%

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

      2. associate-*r/ 51.5%

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

      3. *-commutative 51.5%

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

   7. Applied egg-rr 51.5%

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

   if 4.09999999999999987e254 < t

   1. Initial program 24.3%

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

   3. Simplified 17.4%

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

   5. Taylor expanded in t around inf 18.0%

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

      1. pow1/2 18.0%

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

      2. associate-*r* 24.9%

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

      3. unpow-prod-down 63.6%

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

      4. pow1/2 63.6%

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

   7. Applied egg-rr 63.6%

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

      1. unpow1/2 63.6%

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

      2. *-commutative 63.6%

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

      3. *-commutative 63.6%

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

   9. Simplified 63.6%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.1 \cdot 10^{+254}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.4%

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

3. Simplified 53.7%

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

5. Taylor expanded in n around 0 48.4%

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

   1. unpow2 48.4%

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

   2. associate-*r/ 50.3%

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

   3. *-commutative 50.3%

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

7. Applied egg-rr 50.3%

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

8. Final simplification 50.3%

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

Alternative 8: 38.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.4%

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

3. Simplified 53.7%

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

5. Taylor expanded in t around inf 38.8%

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

   1. add-cbrt-cube 24.1%

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

   2. pow3 24.1%

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

7. Applied egg-rr 24.1%

   \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\sqrt[3]{{\left(n \cdot t\right)}^{3}}}\right)} \]
8. Step-by-step derivation

   1. pow1/2 26.3%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \sqrt[3]{{\left(n \cdot t\right)}^{3}}\right)\right)}^{0.5}} \]

   2. rem-cbrt-cube 39.7%

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

9. Applied egg-rr 39.7%

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

Alternative 9: 36.2% 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 50.4%

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

3. Simplified 53.7%

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

5. Taylor expanded in t around inf 38.8%

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

Reproduce

herbie shell --seed 2024152 
(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*))))))