Toniolo and Linder, Equation (13)

Percentage Accurate: 49.3% → 67.3%

Time: 25.2s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 49.3% 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: 67.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := \left(n \cdot t\_2\right) \cdot \left(U* - U\right)\\ t_4 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t\_3\right)}\\ t_5 := t\_2 \cdot \left(n \cdot U\right)\\ \mathbf{if}\;t\_4 \leq 2 \cdot 10^{-155}:\\ \;\;\;\;\left(\sqrt{\left|2 \cdot U\right|} \cdot \sqrt{\left|n\right|}\right) \cdot \sqrt{\left|t - \mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, t\_5\right)\right|}\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(t\_3 - 2 \cdot t\_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|n \cdot \left(2 \cdot U\right)\right|} \cdot \sqrt{\left|t - \mathsf{fma}\left(2, t\_1, t\_5\right)\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om)))
        (t_2 (pow (/ l Om) 2.0))
        (t_3 (* (* n t_2) (- U* U)))
        (t_4 (sqrt (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l l) Om))) t_3))))
        (t_5 (* t_2 (* n U))))
   (if (<= t_4 2e-155)
     (*
      (* (sqrt (fabs (* 2.0 U))) (sqrt (fabs n)))
      (sqrt (fabs (- t (fma 2.0 (/ (pow l 2.0) Om) t_5)))))
     (if (<= t_4 5e+141)
       (sqrt (* (* 2.0 (* n U)) (+ t (- t_3 (* 2.0 t_1)))))
       (*
        (sqrt (fabs (* n (* 2.0 U))))
        (sqrt (fabs (- t (fma 2.0 t_1 t_5)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double t_2 = pow((l / Om), 2.0);
	double t_3 = (n * t_2) * (U_42_ - U);
	double t_4 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_3)));
	double t_5 = t_2 * (n * U);
	double tmp;
	if (t_4 <= 2e-155) {
		tmp = (sqrt(fabs((2.0 * U))) * sqrt(fabs(n))) * sqrt(fabs((t - fma(2.0, (pow(l, 2.0) / Om), t_5))));
	} else if (t_4 <= 5e+141) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (t_3 - (2.0 * t_1)))));
	} else {
		tmp = sqrt(fabs((n * (2.0 * U)))) * sqrt(fabs((t - fma(2.0, t_1, t_5))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(l / Om))
	t_2 = Float64(l / Om) ^ 2.0
	t_3 = Float64(Float64(n * t_2) * Float64(U_42_ - U))
	t_4 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_3)))
	t_5 = Float64(t_2 * Float64(n * U))
	tmp = 0.0
	if (t_4 <= 2e-155)
		tmp = Float64(Float64(sqrt(abs(Float64(2.0 * U))) * sqrt(abs(n))) * sqrt(abs(Float64(t - fma(2.0, Float64((l ^ 2.0) / Om), t_5)))));
	elseif (t_4 <= 5e+141)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(t_3 - Float64(2.0 * t_1)))));
	else
		tmp = Float64(sqrt(abs(Float64(n * Float64(2.0 * U)))) * sqrt(abs(Float64(t - fma(2.0, t_1, t_5)))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(n * t$95$2), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = 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] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 2e-155], N[(N[(N[Sqrt[N[Abs[N[(2.0 * U), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[n], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Abs[N[(t - N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 5e+141], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(t$95$3 - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[Abs[N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[N[(t - N[(2.0 * t$95$1 + t$95$5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2.00000000000000003e-155

   1. Initial program 16.7%

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

   3. Simplified 26.5%

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

   5. Taylor expanded in U around inf 26.4%

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

      1. associate-/l* 26.4%

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

      2. unpow2 26.4%

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

      3. unpow2 26.4%

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

      4. times-frac 26.6%

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

      5. unpow2 26.6%

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

   7. Simplified 26.6%

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

      1. add-sqr-sqrt 26.6%

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

      2. pow1/2 26.6%

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

      3. pow1/2 26.6%

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

      4. pow-prod-down 16.5%

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

   9. Applied egg-rr 9.9%

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

       1. unpow1/2 9.9%

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

       2. unpow2 9.9%

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

       3. rem-sqrt-square 16.7%

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

       4. associate-*l* 16.7%

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

       5. *-commutative 16.7%

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

   11. Simplified 16.7%

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

       1. pow1/2 16.7%

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

       2. fabs-mul 16.7%

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

       3. unpow-prod-down 53.2%

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

       4. associate-*r* 53.5%

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

       5. associate-*r* 53.5%

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

   13. Applied egg-rr 53.5%

       \[\leadsto \color{blue}{{\left(\left|\left(2 \cdot n\right) \cdot U\right|\right)}^{0.5} \cdot {\left(\left|t - \mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, \left(n \cdot U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right|\right)}^{0.5}} \]
   14. Step-by-step derivation

       1. unpow1/2 53.5%

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

       2. associate-*l* 53.2%

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

       3. *-commutative 53.2%

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

       4. associate-*r* 53.5%

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

       5. *-commutative 53.5%

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

       6. unpow1/2 53.5%

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

       7. *-commutative 53.5%

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

   15. Simplified 53.5%

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

       1. pow1/2 53.5%

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

       2. fabs-mul 53.5%

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

       3. unpow-prod-down 97.5%

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

   17. Applied egg-rr 97.5%

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

       1. unpow1/2 97.5%

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

       2. *-commutative 97.5%

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

       3. unpow1/2 97.5%

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

   19. Simplified 97.5%

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

   if 2.00000000000000003e-155 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5.00000000000000025e141

   1. Initial program 96.1%

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

   3. Simplified 96.1%

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

   if 5.00000000000000025e141 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 23.2%

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

   3. Simplified 32.6%

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

   5. Taylor expanded in U around inf 7.2%

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

      1. associate-/l* 7.3%

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

      2. unpow2 7.3%

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

      3. unpow2 7.3%

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

      4. times-frac 16.1%

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

      5. unpow2 16.1%

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

   7. Simplified 16.1%

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

      1. add-sqr-sqrt 16.1%

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

      2. pow1/2 16.1%

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

      3. pow1/2 35.8%

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

      4. pow-prod-down 31.1%

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

   9. Applied egg-rr 24.7%

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

       1. unpow1/2 24.7%

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

       2. unpow2 24.7%

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

       3. rem-sqrt-square 26.2%

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

       4. associate-*l* 26.2%

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

       5. *-commutative 26.2%

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

   11. Simplified 26.2%

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

       1. pow1/2 26.2%

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

       2. fabs-mul 26.2%

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

       3. unpow-prod-down 36.7%

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

       4. associate-*r* 36.7%

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

       5. associate-*r* 35.8%

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

   13. Applied egg-rr 35.8%

       \[\leadsto \color{blue}{{\left(\left|\left(2 \cdot n\right) \cdot U\right|\right)}^{0.5} \cdot {\left(\left|t - \mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, \left(n \cdot U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right|\right)}^{0.5}} \]
   14. Step-by-step derivation

       1. unpow1/2 35.8%

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

       2. associate-*l* 35.8%

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

       3. *-commutative 35.8%

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

       4. associate-*r* 35.8%

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

       5. *-commutative 35.8%

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

       6. unpow1/2 35.8%

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

       7. *-commutative 35.8%

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

   15. Simplified 35.8%

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

       1. unpow2 35.8%

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

       2. associate-*l/ 46.5%

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

   17. Applied egg-rr 46.5%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{-155}:\\ \;\;\;\;\left(\sqrt{\left|2 \cdot U\right|} \cdot \sqrt{\left|n\right|}\right) \cdot \sqrt{\left|t - \mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot U\right)\right)\right|}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|n \cdot \left(2 \cdot U\right)\right|} \cdot \sqrt{\left|t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot U\right)\right)\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_3 := \left(2 \cdot n\right) \cdot U\\ t_4 := \sqrt{t\_3 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t\_2\right)}\\ \mathbf{if}\;t\_4 \leq 5 \cdot 10^{-158}:\\ \;\;\;\;\sqrt{U \cdot \left(t + t\_1 \cdot -2\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{t\_3 \cdot \left(\left(t - 2 \cdot t\_1\right) + t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{{\ell}^{2}}{Om} \cdot -2\right)\right)\right)\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om)))
        (t_2 (* (* n (pow (/ l Om) 2.0)) (- U* U)))
        (t_3 (* (* 2.0 n) U))
        (t_4 (sqrt (* t_3 (+ (- t (* 2.0 (/ (* l l) Om))) t_2)))))
   (if (<= t_4 5e-158)
     (* (sqrt (* U (+ t (* t_1 -2.0)))) (sqrt (* 2.0 n)))
     (if (<= t_4 INFINITY)
       (sqrt (* t_3 (+ (- t (* 2.0 t_1)) t_2)))
       (sqrt (fabs (* 2.0 (* n (* U (+ t (* (/ (pow l 2.0) Om) -2.0)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double t_2 = (n * pow((l / Om), 2.0)) * (U_42_ - U);
	double t_3 = (2.0 * n) * U;
	double t_4 = sqrt((t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2)));
	double tmp;
	if (t_4 <= 5e-158) {
		tmp = sqrt((U * (t + (t_1 * -2.0)))) * sqrt((2.0 * n));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)));
	} else {
		tmp = sqrt(fabs((2.0 * (n * (U * (t + ((pow(l, 2.0) / Om) * -2.0)))))));
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double t_2 = (n * Math.pow((l / Om), 2.0)) * (U_42_ - U);
	double t_3 = (2.0 * n) * U;
	double t_4 = Math.sqrt((t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2)));
	double tmp;
	if (t_4 <= 5e-158) {
		tmp = Math.sqrt((U * (t + (t_1 * -2.0)))) * Math.sqrt((2.0 * n));
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)));
	} else {
		tmp = Math.sqrt(Math.abs((2.0 * (n * (U * (t + ((Math.pow(l, 2.0) / Om) * -2.0)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = l * (l / Om)
	t_2 = (n * math.pow((l / Om), 2.0)) * (U_42_ - U)
	t_3 = (2.0 * n) * U
	t_4 = math.sqrt((t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2)))
	tmp = 0
	if t_4 <= 5e-158:
		tmp = math.sqrt((U * (t + (t_1 * -2.0)))) * math.sqrt((2.0 * n))
	elif t_4 <= math.inf:
		tmp = math.sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)))
	else:
		tmp = math.sqrt(math.fabs((2.0 * (n * (U * (t + ((math.pow(l, 2.0) / Om) * -2.0)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(l / Om))
	t_2 = Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_3 = Float64(Float64(2.0 * n) * U)
	t_4 = sqrt(Float64(t_3 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_2)))
	tmp = 0.0
	if (t_4 <= 5e-158)
		tmp = Float64(sqrt(Float64(U * Float64(t + Float64(t_1 * -2.0)))) * sqrt(Float64(2.0 * n)));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(t_3 * Float64(Float64(t - Float64(2.0 * t_1)) + t_2)));
	else
		tmp = sqrt(abs(Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(Float64((l ^ 2.0) / Om) * -2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l * (l / Om);
	t_2 = (n * ((l / Om) ^ 2.0)) * (U_42_ - U);
	t_3 = (2.0 * n) * U;
	t_4 = sqrt((t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2)));
	tmp = 0.0;
	if (t_4 <= 5e-158)
		tmp = sqrt((U * (t + (t_1 * -2.0)))) * sqrt((2.0 * n));
	elseif (t_4 <= Inf)
		tmp = sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)));
	else
		tmp = sqrt(abs((2.0 * (n * (U * (t + (((l ^ 2.0) / Om) * -2.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t$95$3 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 5e-158], N[(N[Sqrt[N[(U * N[(t + N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(t$95$3 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(2.0 * N[(n * N[(U * N[(t + N[(N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 4.99999999999999972e-158

   1. Initial program 14.0%

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

   3. Simplified 24.2%

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

   5. Taylor expanded in n around 0 24.3%

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

      1. pow1/2 24.3%

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

      2. *-commutative 24.3%

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

      3. unpow-prod-down 41.8%

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

      4. pow1/2 41.8%

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

      5. cancel-sign-sub-inv 41.8%

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

      6. metadata-eval 41.8%

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

      7. pow1/2 41.8%

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

   7. Applied egg-rr 41.8%

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

      1. unpow2 51.8%

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

      2. associate-*l/ 51.8%

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

   9. Applied egg-rr 41.8%

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

   if 4.99999999999999972e-158 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 67.7%

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

      1. associate-*r/ 73.1%

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

      2. *-commutative 73.1%

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

   4. Applied egg-rr 73.1%

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

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

   1. Initial program 0.0%

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

   3. Simplified 8.2%

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

   5. Taylor expanded in n around 0 4.3%

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

      1. add-sqr-sqrt 4.3%

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

      2. pow1/2 4.3%

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

      3. pow1/2 40.1%

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

      4. pow-prod-down 40.4%

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

      5. pow2 40.4%

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

      6. associate-*l* 40.4%

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

      7. cancel-sign-sub-inv 40.4%

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

      8. metadata-eval 40.4%

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

   7. Applied egg-rr 40.4%

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

      1. unpow1/2 40.4%

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

      2. unpow2 40.4%

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

      3. rem-sqrt-square 40.4%

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

   9. Simplified 40.4%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{-158}:\\ \;\;\;\;\sqrt{U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{{\ell}^{2}}{Om} \cdot -2\right)\right)\right)\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_3 := \left(2 \cdot n\right) \cdot U\\ t_4 := \sqrt{t\_3 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t\_2\right)}\\ \mathbf{if}\;t\_4 \leq 5 \cdot 10^{-158}:\\ \;\;\;\;\sqrt{U \cdot \left(t + t\_1 \cdot -2\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{t\_3 \cdot \left(\left(t - 2 \cdot t\_1\right) + t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{{\ell}^{2}}{Om} \cdot -2\right)\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om)))
        (t_2 (* (* n (pow (/ l Om) 2.0)) (- U* U)))
        (t_3 (* (* 2.0 n) U))
        (t_4 (sqrt (* t_3 (+ (- t (* 2.0 (/ (* l l) Om))) t_2)))))
   (if (<= t_4 5e-158)
     (* (sqrt (* U (+ t (* t_1 -2.0)))) (sqrt (* 2.0 n)))
     (if (<= t_4 INFINITY)
       (sqrt (* t_3 (+ (- t (* 2.0 t_1)) t_2)))
       (pow (* 2.0 (* n (* U (+ t (* (/ (pow l 2.0) Om) -2.0))))) 0.5)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double t_2 = (n * pow((l / Om), 2.0)) * (U_42_ - U);
	double t_3 = (2.0 * n) * U;
	double t_4 = sqrt((t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2)));
	double tmp;
	if (t_4 <= 5e-158) {
		tmp = sqrt((U * (t + (t_1 * -2.0)))) * sqrt((2.0 * n));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)));
	} else {
		tmp = pow((2.0 * (n * (U * (t + ((pow(l, 2.0) / Om) * -2.0))))), 0.5);
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double t_2 = (n * Math.pow((l / Om), 2.0)) * (U_42_ - U);
	double t_3 = (2.0 * n) * U;
	double t_4 = Math.sqrt((t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2)));
	double tmp;
	if (t_4 <= 5e-158) {
		tmp = Math.sqrt((U * (t + (t_1 * -2.0)))) * Math.sqrt((2.0 * n));
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)));
	} else {
		tmp = Math.pow((2.0 * (n * (U * (t + ((Math.pow(l, 2.0) / Om) * -2.0))))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = l * (l / Om)
	t_2 = (n * math.pow((l / Om), 2.0)) * (U_42_ - U)
	t_3 = (2.0 * n) * U
	t_4 = math.sqrt((t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2)))
	tmp = 0
	if t_4 <= 5e-158:
		tmp = math.sqrt((U * (t + (t_1 * -2.0)))) * math.sqrt((2.0 * n))
	elif t_4 <= math.inf:
		tmp = math.sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)))
	else:
		tmp = math.pow((2.0 * (n * (U * (t + ((math.pow(l, 2.0) / Om) * -2.0))))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(l / Om))
	t_2 = Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_3 = Float64(Float64(2.0 * n) * U)
	t_4 = sqrt(Float64(t_3 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_2)))
	tmp = 0.0
	if (t_4 <= 5e-158)
		tmp = Float64(sqrt(Float64(U * Float64(t + Float64(t_1 * -2.0)))) * sqrt(Float64(2.0 * n)));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(t_3 * Float64(Float64(t - Float64(2.0 * t_1)) + t_2)));
	else
		tmp = Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(Float64((l ^ 2.0) / Om) * -2.0))))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l * (l / Om);
	t_2 = (n * ((l / Om) ^ 2.0)) * (U_42_ - U);
	t_3 = (2.0 * n) * U;
	t_4 = sqrt((t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2)));
	tmp = 0.0;
	if (t_4 <= 5e-158)
		tmp = sqrt((U * (t + (t_1 * -2.0)))) * sqrt((2.0 * n));
	elseif (t_4 <= Inf)
		tmp = sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)));
	else
		tmp = (2.0 * (n * (U * (t + (((l ^ 2.0) / Om) * -2.0))))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t$95$3 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 5e-158], N[(N[Sqrt[N[(U * N[(t + N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(t$95$3 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(2.0 * N[(n * N[(U * N[(t + N[(N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 4.99999999999999972e-158

   1. Initial program 14.0%

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

   3. Simplified 24.2%

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

   5. Taylor expanded in n around 0 24.3%

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

      1. pow1/2 24.3%

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

      2. *-commutative 24.3%

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

      3. unpow-prod-down 41.8%

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

      4. pow1/2 41.8%

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

      5. cancel-sign-sub-inv 41.8%

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

      6. metadata-eval 41.8%

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

      7. pow1/2 41.8%

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

   7. Applied egg-rr 41.8%

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

      1. unpow2 51.8%

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

      2. associate-*l/ 51.8%

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

   9. Applied egg-rr 41.8%

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

   if 4.99999999999999972e-158 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 67.7%

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

      1. associate-*r/ 73.1%

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

      2. *-commutative 73.1%

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

   4. Applied egg-rr 73.1%

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

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

   1. Initial program 0.0%

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

   3. Simplified 8.2%

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

   5. Taylor expanded in n around 0 4.3%

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

      1. pow1/2 40.1%

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

      2. associate-*l* 40.1%

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

      3. cancel-sign-sub-inv 40.1%

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

      4. metadata-eval 40.1%

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

   7. Applied egg-rr 40.1%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{-158}:\\ \;\;\;\;\sqrt{U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{{\ell}^{2}}{Om} \cdot -2\right)\right)\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq -2.6 \cdot 10^{+46} \lor \neg \left(Om \leq 0.00094\right):\\ \;\;\;\;\sqrt{\left|n \cdot \left(2 \cdot U\right)\right|} \cdot \sqrt{\left|t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot U\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{{\ell}^{2} \cdot \left(2 + \frac{n \cdot \left(U - U*\right)}{Om}\right)}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= Om -2.6e+46) (not (<= Om 0.00094)))
   (*
    (sqrt (fabs (* n (* 2.0 U))))
    (sqrt
     (fabs (- t (fma 2.0 (* l (/ l Om)) (* (pow (/ l Om) 2.0) (* n U)))))))
   (sqrt
    (*
     (* 2.0 (* n U))
     (- t (/ (* (pow l 2.0) (+ 2.0 (/ (* n (- U U*)) Om))) Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((Om <= -2.6e+46) || !(Om <= 0.00094)) {
		tmp = sqrt(fabs((n * (2.0 * U)))) * sqrt(fabs((t - fma(2.0, (l * (l / Om)), (pow((l / Om), 2.0) * (n * U))))));
	} else {
		tmp = sqrt(((2.0 * (n * U)) * (t - ((pow(l, 2.0) * (2.0 + ((n * (U - U_42_)) / Om))) / Om))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((Om <= -2.6e+46) || !(Om <= 0.00094))
		tmp = Float64(sqrt(abs(Float64(n * Float64(2.0 * U)))) * sqrt(abs(Float64(t - fma(2.0, Float64(l * Float64(l / Om)), Float64((Float64(l / Om) ^ 2.0) * Float64(n * U)))))));
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(Float64((l ^ 2.0) * Float64(2.0 + Float64(Float64(n * Float64(U - U_42_)) / Om))) / Om))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[Om, -2.6e+46], N[Not[LessEqual[Om, 0.00094]], $MachinePrecision]], N[(N[Sqrt[N[Abs[N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -2.60000000000000013e46 or 9.39999999999999972e-4 < Om

   1. Initial program 48.9%

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

   3. Simplified 51.4%

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

   5. Taylor expanded in U around inf 42.2%

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

      1. associate-/l* 43.8%

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

      2. unpow2 43.8%

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

      3. unpow2 43.8%

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

      4. times-frac 51.5%

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

      5. unpow2 51.5%

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

   7. Simplified 51.5%

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

      1. add-sqr-sqrt 51.5%

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

      2. pow1/2 51.5%

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

      3. pow1/2 52.3%

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

      4. pow-prod-down 36.2%

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

   9. Applied egg-rr 30.1%

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

       1. unpow1/2 30.1%

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

       2. unpow2 30.1%

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

       3. rem-sqrt-square 49.5%

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

       4. associate-*l* 49.5%

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

       5. *-commutative 49.5%

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

   11. Simplified 49.5%

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

       1. pow1/2 49.5%

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

       2. fabs-mul 49.5%

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

       3. unpow-prod-down 63.3%

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

       4. associate-*r* 63.4%

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

       5. associate-*r* 62.5%

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

   13. Applied egg-rr 62.5%

       \[\leadsto \color{blue}{{\left(\left|\left(2 \cdot n\right) \cdot U\right|\right)}^{0.5} \cdot {\left(\left|t - \mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, \left(n \cdot U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right|\right)}^{0.5}} \]
   14. Step-by-step derivation

       1. unpow1/2 62.5%

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

       2. associate-*l* 62.5%

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

       3. *-commutative 62.5%

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

       4. associate-*r* 62.5%

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

       5. *-commutative 62.5%

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

       6. unpow1/2 62.5%

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

       7. *-commutative 62.5%

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

   15. Simplified 62.5%

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

       1. unpow2 62.5%

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

       2. associate-*l/ 72.0%

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

   17. Applied egg-rr 72.0%

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

   if -2.60000000000000013e46 < Om < 9.39999999999999972e-4

   1. Initial program 54.2%

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

   3. Simplified 54.2%

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

   5. Taylor expanded in Om around inf 59.5%

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

   6. Taylor expanded in l around 0 66.7%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -2.6 \cdot 10^{+46} \lor \neg \left(Om \leq 0.00094\right):\\ \;\;\;\;\sqrt{\left|n \cdot \left(2 \cdot U\right)\right|} \cdot \sqrt{\left|t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot U\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{{\ell}^{2} \cdot \left(2 + \frac{n \cdot \left(U - U*\right)}{Om}\right)}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_3 := \left(2 \cdot n\right) \cdot U\\ t_4 := t\_3 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t\_2\right)\\ \mathbf{if}\;t\_4 \leq 10^{-315}:\\ \;\;\;\;\sqrt{U \cdot \left(t + t\_1 \cdot -2\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{t\_3 \cdot \left(\left(t - 2 \cdot t\_1\right) + t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(U \cdot {\ell}^{2}\right) \cdot \left(n \cdot \left(\frac{2}{Om} + n \cdot \frac{U - U*}{{Om}^{2}}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om)))
        (t_2 (* (* n (pow (/ l Om) 2.0)) (- U* U)))
        (t_3 (* (* 2.0 n) U))
        (t_4 (* t_3 (+ (- t (* 2.0 (/ (* l l) Om))) t_2))))
   (if (<= t_4 1e-315)
     (* (sqrt (* U (+ t (* t_1 -2.0)))) (sqrt (* 2.0 n)))
     (if (<= t_4 INFINITY)
       (sqrt (* t_3 (+ (- t (* 2.0 t_1)) t_2)))
       (sqrt
        (*
         -2.0
         (*
          (* U (pow l 2.0))
          (* n (+ (/ 2.0 Om) (* n (/ (- U U*) (pow Om 2.0))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double t_2 = (n * pow((l / Om), 2.0)) * (U_42_ - U);
	double t_3 = (2.0 * n) * U;
	double t_4 = t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2);
	double tmp;
	if (t_4 <= 1e-315) {
		tmp = sqrt((U * (t + (t_1 * -2.0)))) * sqrt((2.0 * n));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)));
	} else {
		tmp = sqrt((-2.0 * ((U * pow(l, 2.0)) * (n * ((2.0 / Om) + (n * ((U - U_42_) / pow(Om, 2.0))))))));
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double t_2 = (n * Math.pow((l / Om), 2.0)) * (U_42_ - U);
	double t_3 = (2.0 * n) * U;
	double t_4 = t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2);
	double tmp;
	if (t_4 <= 1e-315) {
		tmp = Math.sqrt((U * (t + (t_1 * -2.0)))) * Math.sqrt((2.0 * n));
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)));
	} else {
		tmp = Math.sqrt((-2.0 * ((U * Math.pow(l, 2.0)) * (n * ((2.0 / Om) + (n * ((U - U_42_) / Math.pow(Om, 2.0))))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = l * (l / Om)
	t_2 = (n * math.pow((l / Om), 2.0)) * (U_42_ - U)
	t_3 = (2.0 * n) * U
	t_4 = t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2)
	tmp = 0
	if t_4 <= 1e-315:
		tmp = math.sqrt((U * (t + (t_1 * -2.0)))) * math.sqrt((2.0 * n))
	elif t_4 <= math.inf:
		tmp = math.sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)))
	else:
		tmp = math.sqrt((-2.0 * ((U * math.pow(l, 2.0)) * (n * ((2.0 / Om) + (n * ((U - U_42_) / math.pow(Om, 2.0))))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(l / Om))
	t_2 = Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_3 = Float64(Float64(2.0 * n) * U)
	t_4 = Float64(t_3 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_2))
	tmp = 0.0
	if (t_4 <= 1e-315)
		tmp = Float64(sqrt(Float64(U * Float64(t + Float64(t_1 * -2.0)))) * sqrt(Float64(2.0 * n)));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(t_3 * Float64(Float64(t - Float64(2.0 * t_1)) + t_2)));
	else
		tmp = sqrt(Float64(-2.0 * Float64(Float64(U * (l ^ 2.0)) * Float64(n * Float64(Float64(2.0 / Om) + Float64(n * Float64(Float64(U - U_42_) / (Om ^ 2.0))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l * (l / Om);
	t_2 = (n * ((l / Om) ^ 2.0)) * (U_42_ - U);
	t_3 = (2.0 * n) * U;
	t_4 = t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2);
	tmp = 0.0;
	if (t_4 <= 1e-315)
		tmp = sqrt((U * (t + (t_1 * -2.0)))) * sqrt((2.0 * n));
	elseif (t_4 <= Inf)
		tmp = sqrt((t_3 * ((t - (2.0 * t_1)) + t_2)));
	else
		tmp = sqrt((-2.0 * ((U * (l ^ 2.0)) * (n * ((2.0 / Om) + (n * ((U - U_42_) / (Om ^ 2.0))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 1e-315], N[(N[Sqrt[N[(U * N[(t + N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(t$95$3 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(U * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(n * N[(N[(2.0 / Om), $MachinePrecision] + N[(n * N[(N[(U - U$42$), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.999999985e-316

   1. Initial program 12.7%

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

   3. Simplified 22.0%

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

   5. Taylor expanded in n around 0 25.1%

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

      1. pow1/2 25.3%

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

      2. *-commutative 25.3%

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

      3. unpow-prod-down 38.0%

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

      4. pow1/2 37.8%

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

      5. cancel-sign-sub-inv 37.8%

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

      6. metadata-eval 37.8%

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

      7. pow1/2 37.8%

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

   7. Applied egg-rr 37.8%

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

      1. unpow2 47.3%

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

      2. associate-*l/ 50.2%

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

   9. Applied egg-rr 38.0%

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

   if 9.999999985e-316 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 67.7%

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

      1. associate-*r/ 73.1%

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

      2. *-commutative 73.1%

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

   4. Applied egg-rr 73.1%

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

   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 8.8%

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

   5. Taylor expanded in l around inf 44.9%

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

      1. associate-*r* 44.9%

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

      2. associate-*r/ 44.9%

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

      3. metadata-eval 44.9%

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

      4. associate-/l* 44.9%

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

   7. Simplified 44.9%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 10^{-315}:\\ \;\;\;\;\sqrt{U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(U \cdot {\ell}^{2}\right) \cdot \left(n \cdot \left(\frac{2}{Om} + n \cdot \frac{U - U*}{{Om}^{2}}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;n \leq -6 \cdot 10^{-174}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;n \leq -5.5 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot t\_1\right)\right)}\\ \mathbf{elif}\;n \leq 2.35 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{U \cdot \left(t + t\_1 \cdot -2\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om)))
        (t_2
         (sqrt
          (* (* (* 2.0 n) U) (+ t (* (* n (pow (/ l Om) 2.0)) (- U* U)))))))
   (if (<= n -6e-174)
     t_2
     (if (<= n -5.5e-308)
       (sqrt (* (* 2.0 n) (* U (- t (* 2.0 t_1)))))
       (if (<= n 2.35e+45)
         (* (sqrt (* U (+ t (* t_1 -2.0)))) (sqrt (* 2.0 n)))
         t_2)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double t_2 = sqrt((((2.0 * n) * U) * (t + ((n * pow((l / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (n <= -6e-174) {
		tmp = t_2;
	} else if (n <= -5.5e-308) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	} else if (n <= 2.35e+45) {
		tmp = sqrt((U * (t + (t_1 * -2.0)))) * sqrt((2.0 * n));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = l * (l / om)
    t_2 = sqrt((((2.0d0 * n) * u) * (t + ((n * ((l / om) ** 2.0d0)) * (u_42 - u)))))
    if (n <= (-6d-174)) then
        tmp = t_2
    else if (n <= (-5.5d-308)) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * t_1)))))
    else if (n <= 2.35d+45) then
        tmp = sqrt((u * (t + (t_1 * (-2.0d0))))) * sqrt((2.0d0 * n))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = l * (l / Om);
	double t_2 = Math.sqrt((((2.0 * n) * U) * (t + ((n * Math.pow((l / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (n <= -6e-174) {
		tmp = t_2;
	} else if (n <= -5.5e-308) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	} else if (n <= 2.35e+45) {
		tmp = Math.sqrt((U * (t + (t_1 * -2.0)))) * Math.sqrt((2.0 * n));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = l * (l / Om)
	t_2 = math.sqrt((((2.0 * n) * U) * (t + ((n * math.pow((l / Om), 2.0)) * (U_42_ - U)))))
	tmp = 0
	if n <= -6e-174:
		tmp = t_2
	elif n <= -5.5e-308:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))))
	elif n <= 2.35e+45:
		tmp = math.sqrt((U * (t + (t_1 * -2.0)))) * math.sqrt((2.0 * n))
	else:
		tmp = t_2
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(l / Om))
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (n <= -6e-174)
		tmp = t_2;
	elseif (n <= -5.5e-308)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * t_1)))));
	elseif (n <= 2.35e+45)
		tmp = Float64(sqrt(Float64(U * Float64(t + Float64(t_1 * -2.0)))) * sqrt(Float64(2.0 * n)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = l * (l / Om);
	t_2 = sqrt((((2.0 * n) * U) * (t + ((n * ((l / Om) ^ 2.0)) * (U_42_ - U)))));
	tmp = 0.0;
	if (n <= -6e-174)
		tmp = t_2;
	elseif (n <= -5.5e-308)
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * t_1)))));
	elseif (n <= 2.35e+45)
		tmp = sqrt((U * (t + (t_1 * -2.0)))) * sqrt((2.0 * n));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -6e-174], t$95$2, If[LessEqual[n, -5.5e-308], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 2.35e+45], N[(N[Sqrt[N[(U * N[(t + N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.00000000000000042e-174 or 2.35000000000000001e45 < n

   1. Initial program 56.2%

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

      1. add-sqr-sqrt 25.7%

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

      2. times-frac 28.8%

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

   4. Applied egg-rr 28.8%

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

      1. unpow2 28.8%

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

   6. Simplified 28.8%

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

   7. Taylor expanded in t around inf 64.1%

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

   if -6.00000000000000042e-174 < n < -5.5e-308

   1. Initial program 52.4%

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

   3. Simplified 52.1%

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

   5. Taylor expanded in n around 0 45.5%

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

      1. unpow2 56.2%

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

      2. associate-*l/ 69.7%

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

   7. Applied egg-rr 60.4%

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

   if -5.5e-308 < n < 2.35000000000000001e45

   1. Initial program 41.4%

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

   3. Simplified 39.4%

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

   5. Taylor expanded in n around 0 38.5%

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

      1. pow1/2 38.6%

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

      2. *-commutative 38.6%

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

      3. unpow-prod-down 48.3%

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

      4. pow1/2 48.1%

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

      5. cancel-sign-sub-inv 48.1%

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

      6. metadata-eval 48.1%

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

      7. pow1/2 48.1%

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

   7. Applied egg-rr 48.1%

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

      1. unpow2 48.6%

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

      2. associate-*l/ 51.1%

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

   9. Applied egg-rr 51.8%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6 \cdot 10^{-174}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;n \leq -5.5 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;n \leq 2.35 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{U \cdot \left(t + \left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(n \cdot U\right)\\ \mathbf{if}\;t \leq -4.2 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 4.65 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t - \frac{{\ell}^{2} \cdot \left(2 + \frac{n \cdot \left(U - U*\right)}{Om}\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* n U))))
   (if (<= t -4.2e-112)
     (sqrt
      (*
       t_1
       (+ t (- (* (* n (pow (/ l Om) 2.0)) (- U* U)) (* 2.0 (* l (/ l Om)))))))
     (if (<= t 4.65e+148)
       (sqrt
        (* t_1 (- t (/ (* (pow l 2.0) (+ 2.0 (/ (* n (- U U*)) Om))) Om))))
       (* (sqrt (* n (* 2.0 U))) (sqrt t))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double tmp;
	if (t <= -4.2e-112) {
		tmp = sqrt((t_1 * (t + (((n * pow((l / Om), 2.0)) * (U_42_ - U)) - (2.0 * (l * (l / Om)))))));
	} else if (t <= 4.65e+148) {
		tmp = sqrt((t_1 * (t - ((pow(l, 2.0) * (2.0 + ((n * (U - U_42_)) / Om))) / Om))));
	} else {
		tmp = sqrt((n * (2.0 * U))) * sqrt(t);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 * (n * u)
    if (t <= (-4.2d-112)) then
        tmp = sqrt((t_1 * (t + (((n * ((l / om) ** 2.0d0)) * (u_42 - u)) - (2.0d0 * (l * (l / om)))))))
    else if (t <= 4.65d+148) then
        tmp = sqrt((t_1 * (t - (((l ** 2.0d0) * (2.0d0 + ((n * (u - u_42)) / om))) / om))))
    else
        tmp = sqrt((n * (2.0d0 * u))) * sqrt(t)
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = 2.0 * (n * U);
	double tmp;
	if (t <= -4.2e-112) {
		tmp = Math.sqrt((t_1 * (t + (((n * Math.pow((l / Om), 2.0)) * (U_42_ - U)) - (2.0 * (l * (l / Om)))))));
	} else if (t <= 4.65e+148) {
		tmp = Math.sqrt((t_1 * (t - ((Math.pow(l, 2.0) * (2.0 + ((n * (U - U_42_)) / Om))) / Om))));
	} else {
		tmp = Math.sqrt((n * (2.0 * U))) * Math.sqrt(t);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = 2.0 * (n * U)
	tmp = 0
	if t <= -4.2e-112:
		tmp = math.sqrt((t_1 * (t + (((n * math.pow((l / Om), 2.0)) * (U_42_ - U)) - (2.0 * (l * (l / Om)))))))
	elif t <= 4.65e+148:
		tmp = math.sqrt((t_1 * (t - ((math.pow(l, 2.0) * (2.0 + ((n * (U - U_42_)) / Om))) / Om))))
	else:
		tmp = math.sqrt((n * (2.0 * U))) * math.sqrt(t)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(2.0 * Float64(n * U))
	tmp = 0.0
	if (t <= -4.2e-112)
		tmp = sqrt(Float64(t_1 * Float64(t + Float64(Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U)) - Float64(2.0 * Float64(l * Float64(l / Om)))))));
	elseif (t <= 4.65e+148)
		tmp = sqrt(Float64(t_1 * Float64(t - Float64(Float64((l ^ 2.0) * Float64(2.0 + Float64(Float64(n * Float64(U - U_42_)) / Om))) / Om))));
	else
		tmp = Float64(sqrt(Float64(n * Float64(2.0 * U))) * sqrt(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = 2.0 * (n * U);
	tmp = 0.0;
	if (t <= -4.2e-112)
		tmp = sqrt((t_1 * (t + (((n * ((l / Om) ^ 2.0)) * (U_42_ - U)) - (2.0 * (l * (l / Om)))))));
	elseif (t <= 4.65e+148)
		tmp = sqrt((t_1 * (t - (((l ^ 2.0) * (2.0 + ((n * (U - U_42_)) / Om))) / Om))));
	else
		tmp = sqrt((n * (2.0 * U))) * sqrt(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.2e-112], N[Sqrt[N[(t$95$1 * N[(t + N[(N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 4.65e+148], N[Sqrt[N[(t$95$1 * N[(t - N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.2000000000000001e-112

   1. Initial program 55.8%

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

   3. Simplified 63.3%

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

   if -4.2000000000000001e-112 < t < 4.64999999999999992e148

   1. Initial program 50.4%

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

   3. Simplified 52.5%

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

   5. Taylor expanded in Om around inf 50.7%

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

   6. Taylor expanded in l around 0 58.8%

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

   if 4.64999999999999992e148 < t

   1. Initial program 43.1%

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

   3. Simplified 34.4%

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

   5. Taylor expanded in t around inf 43.5%

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

      1. associate-*r* 43.3%

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

   7. Simplified 43.3%

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

      1. pow1/2 46.5%

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

      2. associate-*r* 46.5%

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

      3. *-commutative 46.5%

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

      4. unpow-prod-down 72.2%

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

      5. pow1/2 69.1%

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

      6. associate-*r* 69.2%

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

      7. pow1/2 69.2%

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

   9. Applied egg-rr 69.2%

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

       1. associate-*l* 69.1%

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

       2. *-commutative 69.1%

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

       3. associate-*r* 69.2%

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

       4. *-commutative 69.2%

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

   11. Simplified 69.2%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 4.65 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{{\ell}^{2} \cdot \left(2 + \frac{n \cdot \left(U - U*\right)}{Om}\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -700:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{{\ell}^{2} \cdot \left(2 + \frac{n \cdot \left(U - U*\right)}{Om}\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t -700.0)
   (sqrt (* (* (* 2.0 n) U) (+ t (* (* n (pow (/ l Om) 2.0)) (- U* U)))))
   (if (<= t 5.8e+148)
     (sqrt
      (*
       (* 2.0 (* n U))
       (- t (/ (* (pow l 2.0) (+ 2.0 (/ (* n (- U U*)) Om))) Om))))
     (* (sqrt (* n (* 2.0 U))) (sqrt t)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -700.0) {
		tmp = sqrt((((2.0 * n) * U) * (t + ((n * pow((l / Om), 2.0)) * (U_42_ - U)))));
	} else if (t <= 5.8e+148) {
		tmp = sqrt(((2.0 * (n * U)) * (t - ((pow(l, 2.0) * (2.0 + ((n * (U - U_42_)) / Om))) / Om))));
	} else {
		tmp = sqrt((n * (2.0 * U))) * sqrt(t);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= (-700.0d0)) then
        tmp = sqrt((((2.0d0 * n) * u) * (t + ((n * ((l / om) ** 2.0d0)) * (u_42 - u)))))
    else if (t <= 5.8d+148) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t - (((l ** 2.0d0) * (2.0d0 + ((n * (u - u_42)) / om))) / om))))
    else
        tmp = sqrt((n * (2.0d0 * u))) * sqrt(t)
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -700.0) {
		tmp = Math.sqrt((((2.0 * n) * U) * (t + ((n * Math.pow((l / Om), 2.0)) * (U_42_ - U)))));
	} else if (t <= 5.8e+148) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t - ((Math.pow(l, 2.0) * (2.0 + ((n * (U - U_42_)) / Om))) / Om))));
	} else {
		tmp = Math.sqrt((n * (2.0 * U))) * Math.sqrt(t);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= -700.0:
		tmp = math.sqrt((((2.0 * n) * U) * (t + ((n * math.pow((l / Om), 2.0)) * (U_42_ - U)))))
	elif t <= 5.8e+148:
		tmp = math.sqrt(((2.0 * (n * U)) * (t - ((math.pow(l, 2.0) * (2.0 + ((n * (U - U_42_)) / Om))) / Om))))
	else:
		tmp = math.sqrt((n * (2.0 * U))) * math.sqrt(t)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= -700.0)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t + Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U)))));
	elseif (t <= 5.8e+148)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(Float64((l ^ 2.0) * Float64(2.0 + Float64(Float64(n * Float64(U - U_42_)) / Om))) / Om))));
	else
		tmp = Float64(sqrt(Float64(n * Float64(2.0 * U))) * sqrt(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= -700.0)
		tmp = sqrt((((2.0 * n) * U) * (t + ((n * ((l / Om) ^ 2.0)) * (U_42_ - U)))));
	elseif (t <= 5.8e+148)
		tmp = sqrt(((2.0 * (n * U)) * (t - (((l ^ 2.0) * (2.0 + ((n * (U - U_42_)) / Om))) / Om))));
	else
		tmp = sqrt((n * (2.0 * U))) * sqrt(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, -700.0], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 5.8e+148], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -700

   1. Initial program 53.5%

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

      1. add-sqr-sqrt 26.8%

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

      2. times-frac 30.4%

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

   4. Applied egg-rr 30.4%

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

      1. unpow2 30.4%

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

   6. Simplified 30.4%

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

   7. Taylor expanded in t around inf 56.9%

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

   if -700 < t < 5.7999999999999999e148

   1. Initial program 52.1%

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

   3. Simplified 55.8%

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

   5. Taylor expanded in Om around inf 51.6%

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

   6. Taylor expanded in l around 0 59.5%

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

   if 5.7999999999999999e148 < t

   1. Initial program 43.1%

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

   3. Simplified 34.4%

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

   5. Taylor expanded in t around inf 43.5%

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

      1. associate-*r* 43.3%

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

   7. Simplified 43.3%

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

      1. pow1/2 46.5%

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

      2. associate-*r* 46.5%

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

      3. *-commutative 46.5%

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

      4. unpow-prod-down 72.2%

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

      5. pow1/2 69.1%

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

      6. associate-*r* 69.2%

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

      7. pow1/2 69.2%

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

   9. Applied egg-rr 69.2%

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

       1. associate-*l* 69.1%

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

       2. *-commutative 69.1%

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

       3. associate-*r* 69.2%

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

       4. *-commutative 69.2%

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

   11. Simplified 69.2%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -700:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{{\ell}^{2} \cdot \left(2 + \frac{n \cdot \left(U - U*\right)}{Om}\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{+137}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{{\ell}^{2}}{Om} \cdot -2\right)\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 2.9e+137)
   (pow (* 2.0 (* n (* U (+ t (* (/ (pow l 2.0) Om) -2.0))))) 0.5)
   (* (sqrt (* n (* 2.0 U))) (sqrt t))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 2.9e+137) {
		tmp = pow((2.0 * (n * (U * (t + ((pow(l, 2.0) / Om) * -2.0))))), 0.5);
	} else {
		tmp = sqrt((n * (2.0 * U))) * sqrt(t);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= 2.9d+137) then
        tmp = (2.0d0 * (n * (u * (t + (((l ** 2.0d0) / om) * (-2.0d0)))))) ** 0.5d0
    else
        tmp = sqrt((n * (2.0d0 * u))) * sqrt(t)
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 2.9e+137) {
		tmp = Math.pow((2.0 * (n * (U * (t + ((Math.pow(l, 2.0) / Om) * -2.0))))), 0.5);
	} else {
		tmp = Math.sqrt((n * (2.0 * U))) * Math.sqrt(t);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= 2.9e+137:
		tmp = math.pow((2.0 * (n * (U * (t + ((math.pow(l, 2.0) / Om) * -2.0))))), 0.5)
	else:
		tmp = math.sqrt((n * (2.0 * U))) * math.sqrt(t)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 2.9e+137)
		tmp = Float64(2.0 * Float64(n * Float64(U * Float64(t + Float64(Float64((l ^ 2.0) / Om) * -2.0))))) ^ 0.5;
	else
		tmp = Float64(sqrt(Float64(n * Float64(2.0 * U))) * sqrt(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= 2.9e+137)
		tmp = (2.0 * (n * (U * (t + (((l ^ 2.0) / Om) * -2.0))))) ^ 0.5;
	else
		tmp = sqrt((n * (2.0 * U))) * sqrt(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 2.9e+137], N[Power[N[(2.0 * N[(n * N[(U * N[(t + N[(N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[(N[Sqrt[N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.89999999999999985e137

   1. Initial program 52.2%

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

   3. Simplified 51.2%

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

   5. Taylor expanded in n around 0 42.8%

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

      1. pow1/2 49.7%

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

      2. associate-*l* 49.7%

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

      3. cancel-sign-sub-inv 49.7%

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

      4. metadata-eval 49.7%

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

   7. Applied egg-rr 49.7%

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

   if 2.89999999999999985e137 < t

   1. Initial program 45.2%

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

   3. Simplified 37.1%

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

   5. Taylor expanded in t around inf 43.0%

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

      1. associate-*r* 42.8%

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

   7. Simplified 42.8%

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

      1. pow1/2 45.6%

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

      2. associate-*r* 45.6%

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

      3. *-commutative 45.6%

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

      4. unpow-prod-down 69.2%

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

      5. pow1/2 66.3%

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

      6. associate-*r* 66.3%

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

      7. pow1/2 66.3%

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

   9. Applied egg-rr 66.3%

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

       1. associate-*l* 66.3%

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

       2. *-commutative 66.3%

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

       3. associate-*r* 66.3%

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

       4. *-commutative 66.3%

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

   11. Simplified 66.3%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{+137}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot \left(t + \frac{{\ell}^{2}}{Om} \cdot -2\right)\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 1.05e+80)
   (sqrt (* (* 2.0 n) (* U (- t (* 2.0 (* l (/ l Om)))))))
   (* (sqrt (* n (* 2.0 U))) (sqrt t))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 1.05e+80) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = sqrt((n * (2.0 * U))) * sqrt(t);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= 1.05d+80) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * (l * (l / om)))))))
    else
        tmp = sqrt((n * (2.0d0 * u))) * sqrt(t)
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 1.05e+80) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = Math.sqrt((n * (2.0 * U))) * Math.sqrt(t);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= 1.05e+80:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))))
	else:
		tmp = math.sqrt((n * (2.0 * U))) * math.sqrt(t)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 1.05e+80)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))))));
	else
		tmp = Float64(sqrt(Float64(n * Float64(2.0 * U))) * sqrt(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= 1.05e+80)
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	else
		tmp = sqrt((n * (2.0 * U))) * sqrt(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 1.05e+80], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.05000000000000001e80

   1. Initial program 51.5%

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

   3. Simplified 51.4%

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

   5. Taylor expanded in n around 0 42.8%

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

      1. unpow2 54.6%

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

      2. associate-*l/ 59.3%

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

   7. Applied egg-rr 46.4%

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

   if 1.05000000000000001e80 < t

   1. Initial program 50.1%

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

   3. Simplified 41.0%

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

   5. Taylor expanded in t around inf 48.7%

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

      1. associate-*r* 48.6%

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

   7. Simplified 48.6%

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

      1. pow1/2 50.6%

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

      2. associate-*r* 50.6%

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

      3. *-commutative 50.6%

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

      4. unpow-prod-down 64.6%

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

      5. pow1/2 62.5%

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

      6. associate-*r* 62.5%

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

      7. pow1/2 62.5%

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

   9. Applied egg-rr 62.5%

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

       1. associate-*l* 62.5%

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

       2. *-commutative 62.5%

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

       3. associate-*r* 62.5%

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

       4. *-commutative 62.5%

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

   11. Simplified 62.5%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq 7 \cdot 10^{+54}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U 7e+54)
   (sqrt (* (* 2.0 n) (* U (- t (* 2.0 (* l (/ l Om)))))))
   (* (sqrt (* 2.0 U)) (sqrt (* n t)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 7e+54) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= 7d+54) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * (l * (l / om)))))))
    else
        tmp = sqrt((2.0d0 * u)) * sqrt((n * t))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 7e+54) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = Math.sqrt((2.0 * U)) * Math.sqrt((n * t));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= 7e+54:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))))
	else:
		tmp = math.sqrt((2.0 * U)) * math.sqrt((n * t))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= 7e+54)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))))));
	else
		tmp = Float64(sqrt(Float64(2.0 * U)) * sqrt(Float64(n * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= 7e+54)
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	else
		tmp = sqrt((2.0 * U)) * sqrt((n * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, 7e+54], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 7.0000000000000002e54

   1. Initial program 50.6%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in n around 0 42.5%

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

      1. unpow2 55.7%

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

      2. associate-*l/ 60.2%

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

   7. Applied egg-rr 46.7%

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

   if 7.0000000000000002e54 < U

   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 44.7%

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

   5. Taylor expanded in t around inf 41.8%

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

      1. pow1/2 42.2%

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

      2. associate-*r* 42.2%

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

      3. unpow-prod-down 54.1%

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

      4. pow1/2 54.1%

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

   7. Applied egg-rr 54.1%

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

      1. unpow1/2 54.1%

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

   9. Simplified 54.1%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 7 \cdot 10^{+54}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot U} \cdot \sqrt{n \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5.7 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 5.7e+79)
   (sqrt (* (* 2.0 n) (* U (- t (* 2.0 (* l (/ l Om)))))))
   (sqrt (* 2.0 (fabs (* U (* n t)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 5.7e+79) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = sqrt((2.0 * fabs((U * (n * t)))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= 5.7d+79) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * (l * (l / om)))))))
    else
        tmp = sqrt((2.0d0 * abs((u * (n * t)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 5.7e+79) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = Math.sqrt((2.0 * Math.abs((U * (n * t)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= 5.7e+79:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))))
	else:
		tmp = math.sqrt((2.0 * math.fabs((U * (n * t)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 5.7e+79)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))))));
	else
		tmp = sqrt(Float64(2.0 * abs(Float64(U * Float64(n * t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= 5.7e+79)
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	else
		tmp = sqrt((2.0 * abs((U * (n * t)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 5.7e+79], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[Abs[N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.6999999999999997e79

   1. Initial program 51.5%

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

   3. Simplified 51.4%

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

   5. Taylor expanded in n around 0 42.8%

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

      1. unpow2 54.6%

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

      2. associate-*l/ 59.3%

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

   7. Applied egg-rr 46.4%

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

   if 5.6999999999999997e79 < t

   1. Initial program 50.1%

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

   3. Simplified 41.0%

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

   5. Taylor expanded in t around inf 48.7%

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

      1. associate-*r* 48.6%

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

      2. add-sqr-sqrt 48.5%

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

      3. pow1/2 48.5%

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

      4. pow1/2 50.6%

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

      5. pow-prod-down 39.3%

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

      6. pow2 39.3%

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

      7. associate-*r* 41.3%

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

   7. Applied egg-rr 41.3%

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

      1. unpow1/2 41.3%

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

      2. unpow2 41.3%

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

      3. rem-sqrt-square 53.1%

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

   9. Simplified 53.1%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.7 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot t\right)\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 1.5e+79)
   (sqrt (* (* 2.0 n) (* U (- t (* 2.0 (* l (/ l Om)))))))
   (pow (* (* 2.0 U) (* n t)) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 1.5e+79) {
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = pow(((2.0 * U) * (n * t)), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= 1.5d+79) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - (2.0d0 * (l * (l / om)))))))
    else
        tmp = ((2.0d0 * u) * (n * t)) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 1.5e+79) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	} else {
		tmp = Math.pow(((2.0 * U) * (n * t)), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= 1.5e+79:
		tmp = math.sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))))
	else:
		tmp = math.pow(((2.0 * U) * (n * t)), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 1.5e+79)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))))));
	else
		tmp = Float64(Float64(2.0 * U) * Float64(n * t)) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= 1.5e+79)
		tmp = sqrt(((2.0 * n) * (U * (t - (2.0 * (l * (l / Om)))))));
	else
		tmp = ((2.0 * U) * (n * t)) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 1.5e+79], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.49999999999999987e79

   1. Initial program 51.5%

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

   3. Simplified 51.4%

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

   5. Taylor expanded in n around 0 42.8%

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

      1. unpow2 54.6%

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

      2. associate-*l/ 59.3%

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

   7. Applied egg-rr 46.4%

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

   if 1.49999999999999987e79 < t

   1. Initial program 50.1%

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

   3. Simplified 41.0%

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

   5. Taylor expanded in t around inf 48.7%

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

      1. pow1/2 52.7%

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

      2. pow-to-exp 46.2%

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

   7. Applied egg-rr 46.2%

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

      1. associate-*r* 46.2%

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

      2. log-prod 31.2%

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

   9. Applied egg-rr 31.2%

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

       1. *-commutative 31.2%

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

   11. Simplified 31.2%

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

       1. sum-log 46.2%

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

       2. exp-to-pow 52.7%

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

   13. Applied egg-rr 52.7%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 37.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.2%

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

3. Simplified 49.3%

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

5. Taylor expanded in t around inf 34.2%

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

   1. pow1/2 35.9%

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

   2. pow-to-exp 32.2%

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

7. Applied egg-rr 32.2%

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

   1. associate-*r* 32.2%

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

   2. log-prod 19.5%

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

9. Applied egg-rr 19.5%

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

    1. *-commutative 19.5%

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

11. Simplified 19.5%

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

    1. sum-log 32.2%

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

    2. exp-to-pow 35.9%

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

13. Applied egg-rr 35.9%

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

14. Final simplification 35.9%

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

Alternative 15: 37.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.2%

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

3. Simplified 49.3%

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

5. Taylor expanded in t around inf 34.2%

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

   1. pow1/2 35.9%

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

7. Applied egg-rr 35.9%

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

Alternative 16: 35.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.2%

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

3. Simplified 49.3%

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

5. Taylor expanded in t around inf 34.2%

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

Reproduce

herbie shell --seed 2024172 
(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*))))))