Toniolo and Linder, Equation (13)

Percentage Accurate: 50.4% → 59.9%

Time: 24.4s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.4% 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: 59.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_1 \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_2 \leq 10^{-323}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \frac{2 \cdot {l\_m}^{2}}{Om}\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(U - U*, t\_1, 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \frac{n \cdot \sqrt{2}}{Om}\right) \cdot \left(\sqrt{U*} \cdot \sqrt{U}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* n (pow (/ l_m Om) 2.0)))
        (t_2
         (*
          (* (* 2.0 n) U)
          (+ (- t (* 2.0 (/ (* l_m l_m) Om))) (* t_1 (- U* U))))))
   (if (<= t_2 1e-323)
     (* (sqrt (* 2.0 n)) (sqrt (* U (- t (/ (* 2.0 (pow l_m 2.0)) Om)))))
     (if (<= t_2 INFINITY)
       (*
        (sqrt 2.0)
        (sqrt (* (* n U) (- t (fma (- U U*) t_1 (* 2.0 (* l_m (/ l_m Om))))))))
       (* (* l_m (/ (* n (sqrt 2.0)) Om)) (* (sqrt U*) (sqrt U)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = n * pow((l_m / Om), 2.0);
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) + (t_1 * (U_42_ - U)));
	double tmp;
	if (t_2 <= 1e-323) {
		tmp = sqrt((2.0 * n)) * sqrt((U * (t - ((2.0 * pow(l_m, 2.0)) / Om))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(2.0) * sqrt(((n * U) * (t - fma((U - U_42_), t_1, (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = (l_m * ((n * sqrt(2.0)) / Om)) * (sqrt(U_42_) * sqrt(U));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(n * (Float64(l_m / Om) ^ 2.0))
	t_2 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(t_1 * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 1e-323)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * Float64(t - Float64(Float64(2.0 * (l_m ^ 2.0)) / Om)))));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(2.0) * sqrt(Float64(Float64(n * U) * Float64(t - fma(Float64(U - U_42_), t_1, Float64(2.0 * Float64(l_m * Float64(l_m / Om))))))));
	else
		tmp = Float64(Float64(l_m * Float64(Float64(n * sqrt(2.0)) / Om)) * Float64(sqrt(U_42_) * sqrt(U)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 1e-323], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t - N[(N[(2.0 * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(t - N[(N[(U - U$42$), $MachinePrecision] * t$95$1 + N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * N[(N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[U$42$], $MachinePrecision] * N[Sqrt[U], $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*)))) < 9.88131e-324

   1. Initial program 11.1%

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

   3. Simplified 17.0%

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

   5. Taylor expanded in Om around inf 17.0%

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

      1. pow1/2 17.2%

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

      2. pow-to-exp 16.6%

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

      3. associate-*r* 16.6%

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

      4. associate-*r/ 16.6%

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

   7. Applied egg-rr 16.6%

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

      1. exp-to-pow 17.2%

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

      2. associate-*l* 42.4%

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

      3. unpow-prod-down 47.6%

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

      4. pow1/2 47.6%

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

      5. associate-/l* 47.6%

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

      6. unpow2 47.6%

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

      7. associate-*l/ 47.6%

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

      8. cancel-sign-sub-inv 47.6%

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

      9. metadata-eval 47.6%

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

      10. associate-*l/ 47.6%

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

      11. unpow2 47.6%

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

   9. Applied egg-rr 47.6%

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

       1. *-commutative 47.6%

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

       2. unpow1/2 47.5%

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

       3. metadata-eval 47.5%

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

       4. cancel-sign-sub-inv 47.5%

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

       5. associate-*r/ 47.5%

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

   11. Simplified 47.5%

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

   if 9.88131e-324 < (*.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 63.5%

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

   3. Simplified 59.4%

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

   6. Applied egg-rr 63.2%

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

      1. unpow2 63.2%

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

      2. associate-*l/ 69.2%

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

   8. Applied egg-rr 69.2%

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

   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 0.4%

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

   5. Taylor expanded in U* around inf 19.5%

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

      1. associate-/l* 19.7%

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

   7. Simplified 19.7%

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

      1. pow1/2 19.7%

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

      2. *-commutative 19.7%

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

      3. unpow-prod-down 7.4%

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

      4. pow1/2 7.4%

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

      5. pow1/2 7.4%

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

   9. Applied egg-rr 7.4%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{-323}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(U - U*, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{n \cdot \sqrt{2}}{Om}\right) \cdot \left(\sqrt{U*} \cdot \sqrt{U}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 60.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t\_2 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + t\_1\right)\\ \mathbf{if}\;t\_3 \leq 10^{-323}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \frac{2 \cdot {l\_m}^{2}}{Om}\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{t\_2 \cdot \left(\left(t - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right) + t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \frac{n \cdot \sqrt{2}}{Om}\right) \cdot \left(\sqrt{U*} \cdot \sqrt{U}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_2 (* (* 2.0 n) U))
        (t_3 (* t_2 (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_1))))
   (if (<= t_3 1e-323)
     (* (sqrt (* 2.0 n)) (sqrt (* U (- t (/ (* 2.0 (pow l_m 2.0)) Om)))))
     (if (<= t_3 INFINITY)
       (sqrt (* t_2 (+ (- t (* 2.0 (* l_m (/ l_m Om)))) t_1)))
       (* (* l_m (/ (* n (sqrt 2.0)) Om)) (* (sqrt U*) (sqrt U)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	double tmp;
	if (t_3 <= 1e-323) {
		tmp = sqrt((2.0 * n)) * sqrt((U * (t - ((2.0 * pow(l_m, 2.0)) / Om))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)));
	} else {
		tmp = (l_m * ((n * sqrt(2.0)) / Om)) * (sqrt(U_42_) * sqrt(U));
	}
	return tmp;
}

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U);
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	double tmp;
	if (t_3 <= 1e-323) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((U * (t - ((2.0 * Math.pow(l_m, 2.0)) / Om))));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)));
	} else {
		tmp = (l_m * ((n * Math.sqrt(2.0)) / Om)) * (Math.sqrt(U_42_) * Math.sqrt(U));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = (n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)
	t_2 = (2.0 * n) * U
	t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1)
	tmp = 0
	if t_3 <= 1e-323:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((U * (t - ((2.0 * math.pow(l_m, 2.0)) / Om))))
	elif t_3 <= math.inf:
		tmp = math.sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)))
	else:
		tmp = (l_m * ((n * math.sqrt(2.0)) / Om)) * (math.sqrt(U_42_) * math.sqrt(U))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + t_1))
	tmp = 0.0
	if (t_3 <= 1e-323)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * Float64(t - Float64(Float64(2.0 * (l_m ^ 2.0)) / Om)))));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))) + t_1)));
	else
		tmp = Float64(Float64(l_m * Float64(Float64(n * sqrt(2.0)) / Om)) * Float64(sqrt(U_42_) * sqrt(U)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = (n * ((l_m / Om) ^ 2.0)) * (U_42_ - U);
	t_2 = (2.0 * n) * U;
	t_3 = t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) + t_1);
	tmp = 0.0;
	if (t_3 <= 1e-323)
		tmp = sqrt((2.0 * n)) * sqrt((U * (t - ((2.0 * (l_m ^ 2.0)) / Om))));
	elseif (t_3 <= Inf)
		tmp = sqrt((t_2 * ((t - (2.0 * (l_m * (l_m / Om)))) + t_1)));
	else
		tmp = (l_m * ((n * sqrt(2.0)) / Om)) * (sqrt(U_42_) * sqrt(U));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 1e-323], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t - N[(N[(2.0 * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[(N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[U$42$], $MachinePrecision] * N[Sqrt[U], $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*)))) < 9.88131e-324

   1. Initial program 11.1%

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

   3. Simplified 17.0%

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

   5. Taylor expanded in Om around inf 17.0%

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

      1. pow1/2 17.2%

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

      2. pow-to-exp 16.6%

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

      3. associate-*r* 16.6%

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

      4. associate-*r/ 16.6%

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

   7. Applied egg-rr 16.6%

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

      1. exp-to-pow 17.2%

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

      2. associate-*l* 42.4%

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

      3. unpow-prod-down 47.6%

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

      4. pow1/2 47.6%

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

      5. associate-/l* 47.6%

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

      6. unpow2 47.6%

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

      7. associate-*l/ 47.6%

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

      8. cancel-sign-sub-inv 47.6%

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

      9. metadata-eval 47.6%

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

      10. associate-*l/ 47.6%

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

      11. unpow2 47.6%

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

   9. Applied egg-rr 47.6%

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

       1. *-commutative 47.6%

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

       2. unpow1/2 47.5%

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

       3. metadata-eval 47.5%

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

       4. cancel-sign-sub-inv 47.5%

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

       5. associate-*r/ 47.5%

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

   11. Simplified 47.5%

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

   if 9.88131e-324 < (*.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 63.5%

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

      1. associate-*r/ 69.1%

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

      2. *-commutative 69.1%

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

   4. Applied egg-rr 69.1%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.4%

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

   5. Taylor expanded in U* around inf 19.5%

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

      1. associate-/l* 19.7%

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

   7. Simplified 19.7%

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

      1. pow1/2 19.7%

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

      2. *-commutative 19.7%

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

      3. unpow-prod-down 7.4%

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

      4. pow1/2 7.4%

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

      5. pow1/2 7.4%

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

   9. Applied egg-rr 7.4%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{-323}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{n \cdot \sqrt{2}}{Om}\right) \cdot \left(\sqrt{U*} \cdot \sqrt{U}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.8% accurate, 0.4× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_2 (* (* 2.0 n) U))
        (t_3 (* t_2 (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_1))))
   (if (<= t_3 0.0)
     (sqrt
      (*
       (* 2.0 n)
       (-
        (* U t)
        (/ (* (pow l_m 2.0) (+ (/ (* U (* n (- U U*))) Om) (* 2.0 U))) Om))))
     (if (<= t_3 INFINITY)
       (sqrt (* t_2 (+ (- t (* 2.0 (* l_m (/ l_m Om)))) t_1)))
       (* (* l_m (/ (* n (sqrt 2.0)) Om)) (* (sqrt U*) (sqrt U)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 10.5%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in Om around -inf 42.6%

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

      1. +-commutative 42.6%

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

      2. mul-1-neg 42.6%

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

      3. unsub-neg 42.6%

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

      4. fma-define 42.6%

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

      5. associate-/l* 42.6%

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

      6. associate-/l* 42.6%

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

   7. Simplified 42.6%

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

   8. Taylor expanded in l around 0 42.7%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 63.3%

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

      1. associate-*r/ 68.9%

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

      2. *-commutative 68.9%

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

   4. Applied egg-rr 68.9%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.4%

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

   5. Taylor expanded in U* around inf 19.5%

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

      1. associate-/l* 19.7%

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

   7. Simplified 19.7%

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

      1. pow1/2 19.7%

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

      2. *-commutative 19.7%

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

      3. unpow-prod-down 7.4%

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

      4. pow1/2 7.4%

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

      5. pow1/2 7.4%

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

   9. Applied egg-rr 7.4%

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

4. Final simplification 58.5%

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

Alternative 4: 61.2% accurate, 0.5× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))
        (t_2 (* (* 2.0 n) U))
        (t_3 (* t_2 (+ (- t (* 2.0 (/ (* l_m l_m) Om))) t_1))))
   (if (<= t_3 0.0)
     (sqrt
      (*
       (* 2.0 n)
       (-
        (* U t)
        (/ (* (pow l_m 2.0) (+ (/ (* U (* n (- U U*))) Om) (* 2.0 U))) Om))))
     (if (<= t_3 INFINITY)
       (sqrt (* t_2 (+ (- t (* 2.0 (* l_m (/ l_m Om)))) t_1)))
       (* l_m (* (* n (/ (sqrt 2.0) Om)) (sqrt (* U U*))))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 10.5%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in Om around -inf 42.6%

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

      1. +-commutative 42.6%

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

      2. mul-1-neg 42.6%

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

      3. unsub-neg 42.6%

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

      4. fma-define 42.6%

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

      5. associate-/l* 42.6%

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

      6. associate-/l* 42.6%

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

   7. Simplified 42.6%

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

   8. Taylor expanded in l around 0 42.7%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 63.3%

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

      1. associate-*r/ 68.9%

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

      2. *-commutative 68.9%

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

   4. Applied egg-rr 68.9%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.4%

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

   6. Applied egg-rr 0.0%

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

      1. unpow2 0.0%

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

      2. associate-*l/ 0.1%

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

   8. Applied egg-rr 0.1%

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

   9. Taylor expanded in U* around inf 19.5%

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

       1. associate-/l* 19.7%

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

       2. associate-*r/ 19.7%

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

       3. associate-*r* 19.8%

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

   11. Simplified 19.8%

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

4. Final simplification 59.9%

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

Alternative 5: 54.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;Om \leq -2.15 \cdot 10^{+111} \lor \neg \left(Om \leq 8.2 \cdot 10^{+152}\right):\\ \;\;\;\;\sqrt{\left(t - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \frac{{l\_m}^{2} \cdot \left(\frac{U \cdot \left(n \cdot \left(U - U*\right)\right)}{Om} + 2 \cdot U\right)}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (or (<= Om -2.15e+111) (not (<= Om 8.2e+152)))
   (sqrt (* (- t (* 2.0 (* l_m (/ l_m Om)))) (* 2.0 (* n U))))
   (sqrt
    (*
     (* 2.0 n)
     (-
      (* U t)
      (/ (* (pow l_m 2.0) (+ (/ (* U (* n (- U U*))) Om) (* 2.0 U))) Om))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((Om <= -2.15e+111) || !(Om <= 8.2e+152)) {
		tmp = sqrt(((t - (2.0 * (l_m * (l_m / Om)))) * (2.0 * (n * U))));
	} else {
		tmp = sqrt(((2.0 * n) * ((U * t) - ((pow(l_m, 2.0) * (((U * (n * (U - U_42_))) / Om) + (2.0 * U))) / Om))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((om <= (-2.15d+111)) .or. (.not. (om <= 8.2d+152))) then
        tmp = sqrt(((t - (2.0d0 * (l_m * (l_m / om)))) * (2.0d0 * (n * u))))
    else
        tmp = sqrt(((2.0d0 * n) * ((u * t) - (((l_m ** 2.0d0) * (((u * (n * (u - u_42))) / om) + (2.0d0 * u))) / om))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((Om <= -2.15e+111) || !(Om <= 8.2e+152)) {
		tmp = Math.sqrt(((t - (2.0 * (l_m * (l_m / Om)))) * (2.0 * (n * U))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * ((U * t) - ((Math.pow(l_m, 2.0) * (((U * (n * (U - U_42_))) / Om) + (2.0 * U))) / Om))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if (Om <= -2.15e+111) or not (Om <= 8.2e+152):
		tmp = math.sqrt(((t - (2.0 * (l_m * (l_m / Om)))) * (2.0 * (n * U))))
	else:
		tmp = math.sqrt(((2.0 * n) * ((U * t) - ((math.pow(l_m, 2.0) * (((U * (n * (U - U_42_))) / Om) + (2.0 * U))) / Om))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if ((Om <= -2.15e+111) || !(Om <= 8.2e+152))
		tmp = sqrt(Float64(Float64(t - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))) * Float64(2.0 * Float64(n * U))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(Float64(U * t) - Float64(Float64((l_m ^ 2.0) * Float64(Float64(Float64(U * Float64(n * Float64(U - U_42_))) / Om) + Float64(2.0 * U))) / Om))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if ((Om <= -2.15e+111) || ~((Om <= 8.2e+152)))
		tmp = sqrt(((t - (2.0 * (l_m * (l_m / Om)))) * (2.0 * (n * U))));
	else
		tmp = sqrt(((2.0 * n) * ((U * t) - (((l_m ^ 2.0) * (((U * (n * (U - U_42_))) / Om) + (2.0 * U))) / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[Or[LessEqual[Om, -2.15e+111], N[Not[LessEqual[Om, 8.2e+152]], $MachinePrecision]], N[Sqrt[N[(N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(N[(U * t), $MachinePrecision] - N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[(N[(U * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -2.14999999999999997e111 or 8.1999999999999996e152 < Om

   1. Initial program 54.9%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in Om around inf 52.0%

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

      1. unpow2 54.6%

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

      2. associate-*l/ 67.5%

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

   7. Applied egg-rr 62.8%

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

   if -2.14999999999999997e111 < Om < 8.1999999999999996e152

   1. Initial program 46.2%

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

   3. Simplified 47.7%

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

   5. Taylor expanded in Om around -inf 46.1%

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

      1. +-commutative 46.1%

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

      2. mul-1-neg 46.1%

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

      3. unsub-neg 46.1%

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

      4. fma-define 46.1%

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

      5. associate-/l* 47.3%

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

      6. associate-/l* 46.0%

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

   7. Simplified 46.0%

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

   8. Taylor expanded in l around 0 47.9%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -2.15 \cdot 10^{+111} \lor \neg \left(Om \leq 8.2 \cdot 10^{+152}\right):\\ \;\;\;\;\sqrt{\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot t - \frac{{\ell}^{2} \cdot \left(\frac{U \cdot \left(n \cdot \left(U - U*\right)\right)}{Om} + 2 \cdot U\right)}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := 2 \cdot \left(n \cdot U\right)\\ \mathbf{if}\;t \leq 3.5 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t + \left(\left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* n U))))
   (if (<= t 3.5e+145)
     (sqrt
      (*
       t_1
       (+
        t
        (-
         (* (* n (pow (/ l_m Om) 2.0)) (- U* U))
         (* 2.0 (* l_m (/ l_m Om)))))))
     (* (sqrt t_1) (sqrt t)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double tmp;
	if (t <= 3.5e+145) {
		tmp = sqrt((t_1 * (t + (((n * pow((l_m / Om), 2.0)) * (U_42_ - U)) - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = sqrt(t_1) * sqrt(t);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 * (n * u)
    if (t <= 3.5d+145) then
        tmp = sqrt((t_1 * (t + (((n * ((l_m / om) ** 2.0d0)) * (u_42 - u)) - (2.0d0 * (l_m * (l_m / om)))))))
    else
        tmp = sqrt(t_1) * sqrt(t)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double tmp;
	if (t <= 3.5e+145) {
		tmp = Math.sqrt((t_1 * (t + (((n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U)) - (2.0 * (l_m * (l_m / Om)))))));
	} else {
		tmp = Math.sqrt(t_1) * Math.sqrt(t);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = 2.0 * (n * U)
	tmp = 0
	if t <= 3.5e+145:
		tmp = math.sqrt((t_1 * (t + (((n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)) - (2.0 * (l_m * (l_m / Om)))))))
	else:
		tmp = math.sqrt(t_1) * math.sqrt(t)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(2.0 * Float64(n * U))
	tmp = 0.0
	if (t <= 3.5e+145)
		tmp = sqrt(Float64(t_1 * Float64(t + Float64(Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U)) - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))))));
	else
		tmp = Float64(sqrt(t_1) * sqrt(t));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = 2.0 * (n * U);
	tmp = 0.0;
	if (t <= 3.5e+145)
		tmp = sqrt((t_1 * (t + (((n * ((l_m / Om) ^ 2.0)) * (U_42_ - U)) - (2.0 * (l_m * (l_m / Om)))))));
	else
		tmp = sqrt(t_1) * sqrt(t);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 3.5e+145], N[Sqrt[N[(t$95$1 * N[(t + N[(N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 3.5000000000000001e145

   1. Initial program 49.6%

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

   3. Simplified 55.3%

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

   if 3.5000000000000001e145 < t

   1. Initial program 47.8%

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

   3. Simplified 47.9%

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

   5. Taylor expanded in t around inf 44.9%

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

      1. associate-*r* 47.8%

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

   7. Simplified 47.8%

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

      1. pow1/2 50.7%

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

      2. associate-*r* 50.7%

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

      3. unpow-prod-down 69.1%

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

      4. *-commutative 69.1%

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

      5. pow1/2 69.1%

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

   9. Applied egg-rr 69.1%

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

       1. unpow1/2 69.1%

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

   11. Simplified 69.1%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := 2 \cdot \left(n \cdot U\right)\\ \mathbf{if}\;t \leq 1.56 \cdot 10^{+50}:\\ \;\;\;\;\sqrt{\left(t - 2 \cdot \left(l\_m \cdot \frac{l\_m}{Om}\right)\right) \cdot t\_1}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+159}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* n U))))
   (if (<= t 1.56e+50)
     (sqrt (* (- t (* 2.0 (* l_m (/ l_m Om)))) t_1))
     (if (<= t 4.5e+159)
       (pow (* 2.0 (* U (* n t))) 0.5)
       (* (sqrt t_1) (sqrt t))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double tmp;
	if (t <= 1.56e+50) {
		tmp = sqrt(((t - (2.0 * (l_m * (l_m / Om)))) * t_1));
	} else if (t <= 4.5e+159) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = sqrt(t_1) * sqrt(t);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 * (n * u)
    if (t <= 1.56d+50) then
        tmp = sqrt(((t - (2.0d0 * (l_m * (l_m / om)))) * t_1))
    else if (t <= 4.5d+159) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else
        tmp = sqrt(t_1) * sqrt(t)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double tmp;
	if (t <= 1.56e+50) {
		tmp = Math.sqrt(((t - (2.0 * (l_m * (l_m / Om)))) * t_1));
	} else if (t <= 4.5e+159) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = Math.sqrt(t_1) * Math.sqrt(t);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = 2.0 * (n * U)
	tmp = 0
	if t <= 1.56e+50:
		tmp = math.sqrt(((t - (2.0 * (l_m * (l_m / Om)))) * t_1))
	elif t <= 4.5e+159:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	else:
		tmp = math.sqrt(t_1) * math.sqrt(t)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(2.0 * Float64(n * U))
	tmp = 0.0
	if (t <= 1.56e+50)
		tmp = sqrt(Float64(Float64(t - Float64(2.0 * Float64(l_m * Float64(l_m / Om)))) * t_1));
	elseif (t <= 4.5e+159)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	else
		tmp = Float64(sqrt(t_1) * sqrt(t));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = 2.0 * (n * U);
	tmp = 0.0;
	if (t <= 1.56e+50)
		tmp = sqrt(((t - (2.0 * (l_m * (l_m / Om)))) * t_1));
	elseif (t <= 4.5e+159)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	else
		tmp = sqrt(t_1) * sqrt(t);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 1.56e+50], N[Sqrt[N[(N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 4.5e+159], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.56e50

   1. Initial program 50.7%

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

   3. Simplified 57.2%

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

   5. Taylor expanded in Om around inf 43.8%

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

      1. unpow2 50.4%

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

      2. associate-*l/ 56.9%

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

   7. Applied egg-rr 49.8%

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

   if 1.56e50 < t < 4.50000000000000026e159

   1. Initial program 43.9%

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

   3. Simplified 48.6%

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

   5. Taylor expanded in t around inf 44.3%

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

      1. pow1/2 47.8%

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

   7. Applied egg-rr 47.8%

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

   if 4.50000000000000026e159 < t

   1. Initial program 45.9%

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

   3. Simplified 43.0%

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

   5. Taylor expanded in t around inf 39.8%

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

      1. associate-*r* 45.9%

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

   7. Simplified 45.9%

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

      1. pow1/2 49.0%

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

      2. associate-*r* 49.0%

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

      3. unpow-prod-down 69.2%

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

      4. *-commutative 69.2%

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

      5. pow1/2 69.2%

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

   9. Applied egg-rr 69.2%

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

       1. unpow1/2 69.2%

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

   11. Simplified 69.2%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.56 \cdot 10^{+50}:\\ \;\;\;\;\sqrt{\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+159}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.1% accurate, 2.0× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (sqrt (* (- t (* 2.0 (* l_m (/ l_m Om)))) (* 2.0 (* n U)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(N[(t - N[(2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 49.3%

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

3. Simplified 54.3%

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

5. Taylor expanded in Om around inf 42.9%

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

   1. unpow2 49.1%

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

   2. associate-*l/ 54.4%

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

7. Applied egg-rr 47.5%

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

8. Final simplification 47.5%

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

Alternative 9: 38.1% accurate, 2.1× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (pow (* 2.0 (* U (* n t))) 0.5))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]

Derivation

1. Initial program 49.3%

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

3. Simplified 50.5%

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

5. Taylor expanded in t around inf 31.3%

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

   1. pow1/2 34.0%

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

7. Applied egg-rr 34.0%

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

Alternative 10: 35.9% accurate, 2.1× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* 2.0 (* t (* n U)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 49.3%

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

3. Simplified 50.5%

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

5. Taylor expanded in t around inf 31.3%

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

   1. associate-*r* 32.0%

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

7. Simplified 32.0%

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

8. Final simplification 32.0%

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

Alternative 11: 36.0% accurate, 2.1× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* 2.0 (* U (* n t)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 49.3%

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

3. Simplified 50.5%

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

5. Taylor expanded in t around inf 31.3%

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

Reproduce

herbie shell --seed 2024136 
(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*))))))