Toniolo and Linder, Equation (13)

Percentage Accurate: 49.4% → 62.0%

Time: 26.6s

Alternatives: 14

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: 49.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 62.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{Om}\\ t_2 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := t\_2 \cdot \left(U* - U\right)\\ t_4 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t\_3\right)\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, t\_1, U \cdot t\_2 - t\_2 \cdot U*\right)\right)\right)}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{\left(t + \left(t\_3 - 2 \cdot t\_1\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\left(2 \cdot \left(U \cdot -2\right)\right) \cdot \left(n \cdot \frac{{\ell}^{2}}{Om}\right)\right)}^{0.16666666666666666}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* l (/ l Om)))
        (t_2 (* n (pow (/ l Om) 2.0)))
        (t_3 (* t_2 (- U* U)))
        (t_4 (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l l) Om))) t_3))))
   (if (<= t_4 0.0)
     (sqrt (* (* 2.0 n) (* U (- t (fma 2.0 t_1 (- (* U t_2) (* t_2 U*)))))))
     (if (<= t_4 INFINITY)
       (sqrt (* (+ t (- t_3 (* 2.0 t_1))) (* 2.0 (* n U))))
       (pow
        (pow
         (* (* 2.0 (* U -2.0)) (* n (/ (pow l 2.0) Om)))
         0.16666666666666666)
        3.0)))))

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);
	double t_3 = t_2 * (U_42_ - U);
	double t_4 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_3);
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt(((2.0 * n) * (U * (t - fma(2.0, t_1, ((U * t_2) - (t_2 * U_42_)))))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt(((t + (t_3 - (2.0 * t_1))) * (2.0 * (n * U))));
	} else {
		tmp = pow(pow(((2.0 * (U * -2.0)) * (n * (pow(l, 2.0) / Om))), 0.16666666666666666), 3.0);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l * Float64(l / Om))
	t_2 = Float64(n * (Float64(l / Om) ^ 2.0))
	t_3 = Float64(t_2 * Float64(U_42_ - U))
	t_4 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_3))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - fma(2.0, t_1, Float64(Float64(U * t_2) - Float64(t_2 * U_42_)))))));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(Float64(t + Float64(t_3 - Float64(2.0 * t_1))) * Float64(2.0 * Float64(n * U))));
	else
		tmp = (Float64(Float64(2.0 * Float64(U * -2.0)) * Float64(n * Float64((l ^ 2.0) / Om))) ^ 0.16666666666666666) ^ 3.0;
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * t$95$1 + N[(N[(U * t$95$2), $MachinePrecision] - N[(t$95$2 * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(N[(t + N[(t$95$3 - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[Power[N[(N[(2.0 * N[(U * -2.0), $MachinePrecision]), $MachinePrecision] * N[(n * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.16666666666666666], $MachinePrecision], 3.0], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 15.4%

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

   3. Simplified 45.2%

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

      1. associate-*r* 48.5%

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

      2. sub-neg 48.5%

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

      3. distribute-lft-in 48.5%

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

   6. Applied egg-rr 48.5%

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

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

   1. Initial program 68.3%

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

   3. Simplified 72.8%

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

   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 3.0%

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

   5. Taylor expanded in n around 0 2.1%

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

      1. add-cube-cbrt 2.1%

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

      2. pow3 2.1%

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

      3. associate-*r* 1.3%

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

      4. *-commutative 1.3%

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

      5. cancel-sign-sub-inv 1.3%

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

      6. metadata-eval 1.3%

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

   7. Applied egg-rr 1.3%

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

   8. Taylor expanded in t around 0 2.3%

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

      1. associate-*r/ 2.3%

         \[\leadsto {\left(\sqrt[3]{\sqrt{2 \cdot \color{blue}{\frac{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}}}\right)}^{3} \]

      2. associate-*r* 2.3%

         \[\leadsto {\left(\sqrt[3]{\sqrt{2 \cdot \frac{\color{blue}{\left(-2 \cdot U\right) \cdot \left({\ell}^{2} \cdot n\right)}}{Om}}}\right)}^{3} \]

      3. *-commutative 2.3%

         \[\leadsto {\left(\sqrt[3]{\sqrt{2 \cdot \frac{\left(-2 \cdot U\right) \cdot \color{blue}{\left(n \cdot {\ell}^{2}\right)}}{Om}}}\right)}^{3} \]

   10. Simplified 2.3%

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

       1. *-un-lft-identity 2.3%

          \[\leadsto {\color{blue}{\left(1 \cdot \sqrt[3]{\sqrt{2 \cdot \frac{\left(-2 \cdot U\right) \cdot \left(n \cdot {\ell}^{2}\right)}{Om}}}\right)}}^{3} \]

       2. pow1/3 2.3%

          \[\leadsto {\left(1 \cdot \color{blue}{{\left(\sqrt{2 \cdot \frac{\left(-2 \cdot U\right) \cdot \left(n \cdot {\ell}^{2}\right)}{Om}}\right)}^{0.3333333333333333}}\right)}^{3} \]

       3. pow1/2 30.2%

          \[\leadsto {\left(1 \cdot {\color{blue}{\left({\left(2 \cdot \frac{\left(-2 \cdot U\right) \cdot \left(n \cdot {\ell}^{2}\right)}{Om}\right)}^{0.5}\right)}}^{0.3333333333333333}\right)}^{3} \]

       4. metadata-eval 30.2%

          \[\leadsto {\left(1 \cdot {\left({\left(2 \cdot \frac{\left(-2 \cdot U\right) \cdot \left(n \cdot {\ell}^{2}\right)}{Om}\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}\right)}^{0.3333333333333333}\right)}^{3} \]

       5. pow-pow 30.2%

          \[\leadsto {\left(1 \cdot \color{blue}{{\left(2 \cdot \frac{\left(-2 \cdot U\right) \cdot \left(n \cdot {\ell}^{2}\right)}{Om}\right)}^{\left(\left(1.5 \cdot 0.3333333333333333\right) \cdot 0.3333333333333333\right)}}\right)}^{3} \]

       6. associate-/l* 33.1%

          \[\leadsto {\left(1 \cdot {\left(2 \cdot \color{blue}{\left(\left(-2 \cdot U\right) \cdot \frac{n \cdot {\ell}^{2}}{Om}\right)}\right)}^{\left(\left(1.5 \cdot 0.3333333333333333\right) \cdot 0.3333333333333333\right)}\right)}^{3} \]

       7. *-commutative 33.1%

          \[\leadsto {\left(1 \cdot {\left(2 \cdot \left(\color{blue}{\left(U \cdot -2\right)} \cdot \frac{n \cdot {\ell}^{2}}{Om}\right)\right)}^{\left(\left(1.5 \cdot 0.3333333333333333\right) \cdot 0.3333333333333333\right)}\right)}^{3} \]

       8. metadata-eval 33.1%

          \[\leadsto {\left(1 \cdot {\left(2 \cdot \left(\left(U \cdot -2\right) \cdot \frac{n \cdot {\ell}^{2}}{Om}\right)\right)}^{\left(\color{blue}{0.5} \cdot 0.3333333333333333\right)}\right)}^{3} \]

       9. metadata-eval 33.1%

          \[\leadsto {\left(1 \cdot {\left(2 \cdot \left(\left(U \cdot -2\right) \cdot \frac{n \cdot {\ell}^{2}}{Om}\right)\right)}^{\color{blue}{0.16666666666666666}}\right)}^{3} \]

   12. Applied egg-rr 33.1%

       \[\leadsto {\color{blue}{\left(1 \cdot {\left(2 \cdot \left(\left(U \cdot -2\right) \cdot \frac{n \cdot {\ell}^{2}}{Om}\right)\right)}^{0.16666666666666666}\right)}}^{3} \]
   13. Step-by-step derivation

       1. *-lft-identity 33.1%

          \[\leadsto {\color{blue}{\left({\left(2 \cdot \left(\left(U \cdot -2\right) \cdot \frac{n \cdot {\ell}^{2}}{Om}\right)\right)}^{0.16666666666666666}\right)}}^{3} \]

       2. associate-*r* 33.1%

          \[\leadsto {\left({\color{blue}{\left(\left(2 \cdot \left(U \cdot -2\right)\right) \cdot \frac{n \cdot {\ell}^{2}}{Om}\right)}}^{0.16666666666666666}\right)}^{3} \]

       3. associate-/l* 33.1%

          \[\leadsto {\left({\left(\left(2 \cdot \left(U \cdot -2\right)\right) \cdot \color{blue}{\left(n \cdot \frac{{\ell}^{2}}{Om}\right)}\right)}^{0.16666666666666666}\right)}^{3} \]

   14. Simplified 33.1%

       \[\leadsto {\color{blue}{\left({\left(\left(2 \cdot \left(U \cdot -2\right)\right) \cdot \left(n \cdot \frac{{\ell}^{2}}{Om}\right)\right)}^{0.16666666666666666}\right)}}^{3} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, U \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot U*\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\left(2 \cdot \left(U \cdot -2\right)\right) \cdot \left(n \cdot \frac{{\ell}^{2}}{Om}\right)\right)}^{0.16666666666666666}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{{\ell}^{2}}{Om}\\ t_2 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_3 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t\_2\right)}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot t\_1\right)\right)\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\left(t + \left(t\_2 - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\left(2 \cdot \left(U \cdot -2\right)\right) \cdot \left(n \cdot t\_1\right)\right)}^{0.16666666666666666}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (pow l 2.0) Om))
        (t_2 (* (* n (pow (/ l Om) 2.0)) (- U* U)))
        (t_3 (sqrt (* (* (* 2.0 n) U) (+ (- t (* 2.0 (/ (* l l) Om))) t_2)))))
   (if (<= t_3 0.0)
     (sqrt (* 2.0 (* U (* n (- t (* 2.0 t_1))))))
     (if (<= t_3 INFINITY)
       (sqrt (* (+ t (- t_2 (* 2.0 (* l (/ l Om))))) (* 2.0 (* n U))))
       (pow (pow (* (* 2.0 (* U -2.0)) (* n t_1)) 0.16666666666666666) 3.0)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = pow(l, 2.0) / Om;
	double t_2 = (n * pow((l / Om), 2.0)) * (U_42_ - U);
	double t_3 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_2)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * t_1))))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((t + (t_2 - (2.0 * (l * (l / Om))))) * (2.0 * (n * U))));
	} else {
		tmp = pow(pow(((2.0 * (U * -2.0)) * (n * t_1)), 0.16666666666666666), 3.0);
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.pow(l, 2.0) / Om;
	double t_2 = (n * Math.pow((l / Om), 2.0)) * (U_42_ - U);
	double t_3 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_2)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * t_1))))));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((t + (t_2 - (2.0 * (l * (l / Om))))) * (2.0 * (n * U))));
	} else {
		tmp = Math.pow(Math.pow(((2.0 * (U * -2.0)) * (n * t_1)), 0.16666666666666666), 3.0);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.pow(l, 2.0) / Om
	t_2 = (n * math.pow((l / Om), 2.0)) * (U_42_ - U)
	t_3 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_2)))
	tmp = 0
	if t_3 <= 0.0:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * t_1))))))
	elif t_3 <= math.inf:
		tmp = math.sqrt(((t + (t_2 - (2.0 * (l * (l / Om))))) * (2.0 * (n * U))))
	else:
		tmp = math.pow(math.pow(((2.0 * (U * -2.0)) * (n * t_1)), 0.16666666666666666), 3.0)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64((l ^ 2.0) / Om)
	t_2 = Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_3 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_2)))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * t_1))))));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(Float64(t + Float64(t_2 - Float64(2.0 * Float64(l * Float64(l / Om))))) * Float64(2.0 * Float64(n * U))));
	else
		tmp = (Float64(Float64(2.0 * Float64(U * -2.0)) * Float64(n * t_1)) ^ 0.16666666666666666) ^ 3.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (l ^ 2.0) / Om;
	t_2 = (n * ((l / Om) ^ 2.0)) * (U_42_ - U);
	t_3 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_2)));
	tmp = 0.0;
	if (t_3 <= 0.0)
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * t_1))))));
	elseif (t_3 <= Inf)
		tmp = sqrt(((t + (t_2 - (2.0 * (l * (l / Om))))) * (2.0 * (n * U))));
	else
		tmp = (((2.0 * (U * -2.0)) * (n * t_1)) ^ 0.16666666666666666) ^ 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[Power[l, 2.0], $MachinePrecision] / Om), $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[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$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(N[(t + N[(t$95$2 - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[Power[N[(N[(2.0 * N[(U * -2.0), $MachinePrecision]), $MachinePrecision] * N[(n * t$95$1), $MachinePrecision]), $MachinePrecision], 0.16666666666666666], $MachinePrecision], 3.0], $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*))))) < 0.0

   1. Initial program 15.9%

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

   3. Simplified 46.6%

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

   5. Taylor expanded in n around 0 46.7%

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

   if 0.0 < (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*))))) < +inf.0

   1. Initial program 68.3%

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

   3. Simplified 72.8%

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

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

   1. Initial program 0.0%

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

   3. Simplified 3.0%

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

   5. Taylor expanded in n around 0 2.0%

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

      1. add-cube-cbrt 2.0%

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

      2. pow3 2.0%

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

      3. associate-*r* 1.2%

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

      4. *-commutative 1.2%

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

      5. cancel-sign-sub-inv 1.2%

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

      6. metadata-eval 1.2%

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

   7. Applied egg-rr 1.2%

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

   8. Taylor expanded in t around 0 2.2%

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

      1. associate-*r/ 2.2%

         \[\leadsto {\left(\sqrt[3]{\sqrt{2 \cdot \color{blue}{\frac{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}}}\right)}^{3} \]

      2. associate-*r* 2.2%

         \[\leadsto {\left(\sqrt[3]{\sqrt{2 \cdot \frac{\color{blue}{\left(-2 \cdot U\right) \cdot \left({\ell}^{2} \cdot n\right)}}{Om}}}\right)}^{3} \]

      3. *-commutative 2.2%

         \[\leadsto {\left(\sqrt[3]{\sqrt{2 \cdot \frac{\left(-2 \cdot U\right) \cdot \color{blue}{\left(n \cdot {\ell}^{2}\right)}}{Om}}}\right)}^{3} \]

   10. Simplified 2.2%

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

       1. *-un-lft-identity 2.2%

          \[\leadsto {\color{blue}{\left(1 \cdot \sqrt[3]{\sqrt{2 \cdot \frac{\left(-2 \cdot U\right) \cdot \left(n \cdot {\ell}^{2}\right)}{Om}}}\right)}}^{3} \]

       2. pow1/3 2.2%

          \[\leadsto {\left(1 \cdot \color{blue}{{\left(\sqrt{2 \cdot \frac{\left(-2 \cdot U\right) \cdot \left(n \cdot {\ell}^{2}\right)}{Om}}\right)}^{0.3333333333333333}}\right)}^{3} \]

       3. pow1/2 29.6%

          \[\leadsto {\left(1 \cdot {\color{blue}{\left({\left(2 \cdot \frac{\left(-2 \cdot U\right) \cdot \left(n \cdot {\ell}^{2}\right)}{Om}\right)}^{0.5}\right)}}^{0.3333333333333333}\right)}^{3} \]

       4. metadata-eval 29.6%

          \[\leadsto {\left(1 \cdot {\left({\left(2 \cdot \frac{\left(-2 \cdot U\right) \cdot \left(n \cdot {\ell}^{2}\right)}{Om}\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}\right)}^{0.3333333333333333}\right)}^{3} \]

       5. pow-pow 29.6%

          \[\leadsto {\left(1 \cdot \color{blue}{{\left(2 \cdot \frac{\left(-2 \cdot U\right) \cdot \left(n \cdot {\ell}^{2}\right)}{Om}\right)}^{\left(\left(1.5 \cdot 0.3333333333333333\right) \cdot 0.3333333333333333\right)}}\right)}^{3} \]

       6. associate-/l* 32.3%

          \[\leadsto {\left(1 \cdot {\left(2 \cdot \color{blue}{\left(\left(-2 \cdot U\right) \cdot \frac{n \cdot {\ell}^{2}}{Om}\right)}\right)}^{\left(\left(1.5 \cdot 0.3333333333333333\right) \cdot 0.3333333333333333\right)}\right)}^{3} \]

       7. *-commutative 32.3%

          \[\leadsto {\left(1 \cdot {\left(2 \cdot \left(\color{blue}{\left(U \cdot -2\right)} \cdot \frac{n \cdot {\ell}^{2}}{Om}\right)\right)}^{\left(\left(1.5 \cdot 0.3333333333333333\right) \cdot 0.3333333333333333\right)}\right)}^{3} \]

       8. metadata-eval 32.3%

          \[\leadsto {\left(1 \cdot {\left(2 \cdot \left(\left(U \cdot -2\right) \cdot \frac{n \cdot {\ell}^{2}}{Om}\right)\right)}^{\left(\color{blue}{0.5} \cdot 0.3333333333333333\right)}\right)}^{3} \]

       9. metadata-eval 32.3%

          \[\leadsto {\left(1 \cdot {\left(2 \cdot \left(\left(U \cdot -2\right) \cdot \frac{n \cdot {\ell}^{2}}{Om}\right)\right)}^{\color{blue}{0.16666666666666666}}\right)}^{3} \]

   12. Applied egg-rr 32.3%

       \[\leadsto {\color{blue}{\left(1 \cdot {\left(2 \cdot \left(\left(U \cdot -2\right) \cdot \frac{n \cdot {\ell}^{2}}{Om}\right)\right)}^{0.16666666666666666}\right)}}^{3} \]
   13. Step-by-step derivation

       1. *-lft-identity 32.3%

          \[\leadsto {\color{blue}{\left({\left(2 \cdot \left(\left(U \cdot -2\right) \cdot \frac{n \cdot {\ell}^{2}}{Om}\right)\right)}^{0.16666666666666666}\right)}}^{3} \]

       2. associate-*r* 32.3%

          \[\leadsto {\left({\color{blue}{\left(\left(2 \cdot \left(U \cdot -2\right)\right) \cdot \frac{n \cdot {\ell}^{2}}{Om}\right)}}^{0.16666666666666666}\right)}^{3} \]

       3. associate-/l* 32.4%

          \[\leadsto {\left({\left(\left(2 \cdot \left(U \cdot -2\right)\right) \cdot \color{blue}{\left(n \cdot \frac{{\ell}^{2}}{Om}\right)}\right)}^{0.16666666666666666}\right)}^{3} \]

   14. Simplified 32.4%

       \[\leadsto {\color{blue}{\left({\left(\left(2 \cdot \left(U \cdot -2\right)\right) \cdot \left(n \cdot \frac{{\ell}^{2}}{Om}\right)\right)}^{0.16666666666666666}\right)}}^{3} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\left(2 \cdot \left(U \cdot -2\right)\right) \cdot \left(n \cdot \frac{{\ell}^{2}}{Om}\right)\right)}^{0.16666666666666666}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(n \cdot U\right)\\ t_2 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := t\_2 \cdot \left(U* - U\right)\\ t_4 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t\_3\right)}\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{\left(t + \left(t\_3 - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t + t\_2 \cdot U*\right) \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = 2.0 * (n * U);
	t_2 = n * ((l / Om) ^ 2.0);
	t_3 = t_2 * (U_42_ - U);
	t_4 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + t_3)));
	tmp = 0.0;
	if (t_4 <= 0.0)
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * ((l ^ 2.0) / Om)))))));
	elseif (t_4 <= Inf)
		tmp = sqrt(((t + (t_3 - (2.0 * (l * (l / Om))))) * t_1));
	else
		tmp = sqrt(((t + (t_2 * U_42_)) * t_1));
	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]}, Block[{t$95$2 = N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * 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]}, If[LessEqual[t$95$4, 0.0], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(N[(t + N[(t$95$3 - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(t + N[(t$95$2 * U$42$), $MachinePrecision]), $MachinePrecision] * t$95$1), $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*))))) < 0.0

   1. Initial program 15.9%

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

   3. Simplified 46.6%

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

   5. Taylor expanded in n around 0 46.7%

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

   if 0.0 < (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*))))) < +inf.0

   1. Initial program 68.3%

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

   3. Simplified 72.8%

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

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

   1. Initial program 0.0%

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

   3. Simplified 3.0%

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

   5. Taylor expanded in U* around inf 31.0%

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

      1. mul-1-neg 31.0%

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

      2. associate-/l* 31.0%

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

      3. distribute-rgt-neg-in 31.0%

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

      4. distribute-neg-frac2 31.0%

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

      5. *-commutative 31.0%

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

   7. Simplified 31.0%

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

      1. *-un-lft-identity 31.0%

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

      2. associate-*r* 30.4%

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

      3. associate-*l* 30.4%

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

      4. associate-/l* 30.4%

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

   9. Applied egg-rr 30.4%

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

       1. *-lft-identity 30.4%

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

       2. distribute-frac-neg2 30.4%

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

       3. unpow2 30.4%

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

       4. unpow2 30.4%

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

       5. times-frac 31.1%

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

       6. unpow2 31.1%

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

   11. Simplified 31.1%

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

4. Final simplification 63.8%

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

Alternative 4: 50.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U* \leq -3.4 \cdot 10^{-243} \lor \neg \left(U* \leq 9.5 \cdot 10^{+127}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= U* -3.4e-243) (not (<= U* 9.5e+127)))
   (sqrt (* (* 2.0 (* n U)) (+ t (* n (* (pow (/ l Om) 2.0) U*)))))
   (sqrt (* 2.0 (* U (* n (- t (* 2.0 (/ (pow l 2.0) Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((U_42_ <= -3.4e-243) || !(U_42_ <= 9.5e+127)) {
		tmp = sqrt(((2.0 * (n * U)) * (t + (n * (pow((l / Om), 2.0) * U_42_)))));
	} else {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (pow(l, 2.0) / Om)))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((u_42 <= (-3.4d-243)) .or. (.not. (u_42 <= 9.5d+127))) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + (n * (((l / om) ** 2.0d0) * u_42)))))
    else
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * ((l ** 2.0d0) / om)))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((U_42_ <= -3.4e-243) || !(U_42_ <= 9.5e+127)) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + (n * (Math.pow((l / Om), 2.0) * U_42_)))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (Math.pow(l, 2.0) / Om)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (U_42_ <= -3.4e-243) or not (U_42_ <= 9.5e+127):
		tmp = math.sqrt(((2.0 * (n * U)) * (t + (n * (math.pow((l / Om), 2.0) * U_42_)))))
	else:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (math.pow(l, 2.0) / Om)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((U_42_ <= -3.4e-243) || !(U_42_ <= 9.5e+127))
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(n * Float64((Float64(l / Om) ^ 2.0) * U_42_)))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64((l ^ 2.0) / Om)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((U_42_ <= -3.4e-243) || ~((U_42_ <= 9.5e+127)))
		tmp = sqrt(((2.0 * (n * U)) * (t + (n * (((l / Om) ^ 2.0) * U_42_)))));
	else
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * ((l ^ 2.0) / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[U$42$, -3.4e-243], N[Not[LessEqual[U$42$, 9.5e+127]], $MachinePrecision]], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(n * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < -3.39999999999999996e-243 or 9.49999999999999975e127 < U*

   1. Initial program 54.5%

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

   3. Simplified 53.2%

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

   5. Taylor expanded in U* around inf 47.2%

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

      1. mul-1-neg 47.2%

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

      2. associate-/l* 49.9%

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

      3. distribute-rgt-neg-in 49.9%

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

      4. distribute-neg-frac2 49.9%

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

      5. *-commutative 49.9%

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

   7. Simplified 49.9%

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

      1. *-un-lft-identity 49.9%

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

      2. associate-*r* 51.8%

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

      3. associate-*l* 51.8%

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

      4. associate-/l* 51.9%

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

   9. Applied egg-rr 51.9%

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

       1. *-lft-identity 51.9%

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

       2. distribute-frac-neg2 51.9%

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

       3. unpow2 51.9%

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

       4. unpow2 51.9%

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

       5. times-frac 60.6%

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

       6. unpow2 60.6%

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

   11. Simplified 60.6%

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

       1. *-un-lft-identity 60.6%

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

       2. associate-*r* 56.5%

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

   13. Applied egg-rr 56.5%

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

       1. *-lft-identity 56.5%

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

       2. *-commutative 56.5%

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

       3. sub-neg 56.5%

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

       4. distribute-rgt-neg-out 56.5%

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

       5. remove-double-neg 56.5%

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

       6. *-commutative 56.5%

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

   15. Simplified 56.5%

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

       1. pow1 56.5%

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

       2. associate-*l* 60.3%

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

   17. Applied egg-rr 60.3%

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

       1. unpow1 60.3%

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

   19. Simplified 60.3%

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

   if -3.39999999999999996e-243 < U* < 9.49999999999999975e127

   1. Initial program 49.4%

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

   3. Simplified 53.0%

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

   5. Taylor expanded in n around 0 53.4%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq -3.4 \cdot 10^{-243} \lor \neg \left(U* \leq 9.5 \cdot 10^{+127}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U* \leq -1.96 \cdot 10^{-230} \lor \neg \left(U* \leq 2.8 \cdot 10^{+111}\right):\\ \;\;\;\;\sqrt{\left(t + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot U*\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= U* -1.96e-230) (not (<= U* 2.8e+111)))
   (sqrt (* (+ t (* (* n (pow (/ l Om) 2.0)) U*)) (* 2.0 (* n U))))
   (sqrt (* 2.0 (* U (* n (- t (* 2.0 (/ (pow l 2.0) Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((U_42_ <= -1.96e-230) || !(U_42_ <= 2.8e+111)) {
		tmp = sqrt(((t + ((n * pow((l / Om), 2.0)) * U_42_)) * (2.0 * (n * U))));
	} else {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (pow(l, 2.0) / Om)))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((u_42 <= (-1.96d-230)) .or. (.not. (u_42 <= 2.8d+111))) then
        tmp = sqrt(((t + ((n * ((l / om) ** 2.0d0)) * u_42)) * (2.0d0 * (n * u))))
    else
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * ((l ** 2.0d0) / om)))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((U_42_ <= -1.96e-230) || !(U_42_ <= 2.8e+111)) {
		tmp = Math.sqrt(((t + ((n * Math.pow((l / Om), 2.0)) * U_42_)) * (2.0 * (n * U))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (Math.pow(l, 2.0) / Om)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (U_42_ <= -1.96e-230) or not (U_42_ <= 2.8e+111):
		tmp = math.sqrt(((t + ((n * math.pow((l / Om), 2.0)) * U_42_)) * (2.0 * (n * U))))
	else:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (math.pow(l, 2.0) / Om)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((U_42_ <= -1.96e-230) || !(U_42_ <= 2.8e+111))
		tmp = sqrt(Float64(Float64(t + Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * U_42_)) * Float64(2.0 * Float64(n * U))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64((l ^ 2.0) / Om)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((U_42_ <= -1.96e-230) || ~((U_42_ <= 2.8e+111)))
		tmp = sqrt(((t + ((n * ((l / Om) ^ 2.0)) * U_42_)) * (2.0 * (n * U))));
	else
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * ((l ^ 2.0) / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[U$42$, -1.96e-230], N[Not[LessEqual[U$42$, 2.8e+111]], $MachinePrecision]], N[Sqrt[N[(N[(t + N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * U$42$), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < -1.96000000000000001e-230 or 2.7999999999999999e111 < U*

   1. Initial program 53.3%

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

   3. Simplified 52.0%

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

   5. Taylor expanded in U* around inf 46.7%

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

      1. mul-1-neg 46.7%

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

      2. associate-/l* 49.3%

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

      3. distribute-rgt-neg-in 49.3%

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

      4. distribute-neg-frac2 49.3%

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

      5. *-commutative 49.3%

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

   7. Simplified 49.3%

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

      1. *-un-lft-identity 49.3%

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

      2. associate-*r* 50.6%

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

      3. associate-*l* 50.6%

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

      4. associate-/l* 50.7%

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

   9. Applied egg-rr 50.7%

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

       1. *-lft-identity 50.7%

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

       2. distribute-frac-neg2 50.7%

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

       3. unpow2 50.7%

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

       4. unpow2 50.7%

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

       5. times-frac 59.2%

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

       6. unpow2 59.2%

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

   11. Simplified 59.2%

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

   if -1.96000000000000001e-230 < U* < 2.7999999999999999e111

   1. Initial program 51.1%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in n around 0 55.3%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq -1.96 \cdot 10^{-230} \lor \neg \left(U* \leq 2.8 \cdot 10^{+111}\right):\\ \;\;\;\;\sqrt{\left(t + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot U*\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U* \leq -5 \cdot 10^{-62} \lor \neg \left(U* \leq 2.5 \cdot 10^{+111}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot U*\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= U* -5e-62) (not (<= U* 2.5e+111)))
   (sqrt (* (* 2.0 (* n U)) (+ t (* (* n U*) (* (/ l Om) (/ l Om))))))
   (sqrt (* 2.0 (* U (* n (- t (* 2.0 (/ (pow l 2.0) Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((U_42_ <= -5e-62) || !(U_42_ <= 2.5e+111)) {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * U_42_) * ((l / Om) * (l / Om))))));
	} else {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (pow(l, 2.0) / Om)))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((u_42 <= (-5d-62)) .or. (.not. (u_42 <= 2.5d+111))) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + ((n * u_42) * ((l / om) * (l / om))))))
    else
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * ((l ** 2.0d0) / om)))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((U_42_ <= -5e-62) || !(U_42_ <= 2.5e+111)) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((n * U_42_) * ((l / Om) * (l / Om))))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (Math.pow(l, 2.0) / Om)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (U_42_ <= -5e-62) or not (U_42_ <= 2.5e+111):
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((n * U_42_) * ((l / Om) * (l / Om))))))
	else:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (math.pow(l, 2.0) / Om)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((U_42_ <= -5e-62) || !(U_42_ <= 2.5e+111))
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(n * U_42_) * Float64(Float64(l / Om) * Float64(l / Om))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64((l ^ 2.0) / Om)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((U_42_ <= -5e-62) || ~((U_42_ <= 2.5e+111)))
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * U_42_) * ((l / Om) * (l / Om))))));
	else
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * ((l ^ 2.0) / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[U$42$, -5e-62], N[Not[LessEqual[U$42$, 2.5e+111]], $MachinePrecision]], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(n * U$42$), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < -5.0000000000000002e-62 or 2.4999999999999998e111 < U*

   1. Initial program 53.1%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in U* around inf 50.3%

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

      1. mul-1-neg 50.3%

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

      2. associate-/l* 53.5%

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

      3. distribute-rgt-neg-in 53.5%

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

      4. distribute-neg-frac2 53.5%

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

      5. *-commutative 53.5%

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

   7. Simplified 53.5%

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

      1. *-un-lft-identity 53.5%

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

      2. associate-*r* 52.8%

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

      3. associate-*l* 52.8%

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

      4. associate-/l* 52.9%

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

   9. Applied egg-rr 52.9%

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

       1. *-lft-identity 52.9%

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

       2. distribute-frac-neg2 52.9%

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

       3. unpow2 52.9%

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

       4. unpow2 52.9%

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

       5. times-frac 61.9%

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

       6. unpow2 61.9%

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

   11. Simplified 61.9%

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

       1. *-un-lft-identity 61.9%

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

       2. associate-*r* 56.9%

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

   13. Applied egg-rr 56.9%

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

       1. *-lft-identity 56.9%

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

       2. *-commutative 56.9%

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

       3. sub-neg 56.9%

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

       4. distribute-rgt-neg-out 56.9%

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

       5. remove-double-neg 56.9%

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

       6. *-commutative 56.9%

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

   15. Simplified 56.9%

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

       1. unpow2 56.9%

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

   17. Applied egg-rr 56.9%

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

   if -5.0000000000000002e-62 < U* < 2.4999999999999998e111

   1. Initial program 51.8%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in n around 0 53.7%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq -5 \cdot 10^{-62} \lor \neg \left(U* \leq 2.5 \cdot 10^{+111}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot U*\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq -1.85 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 8.2 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot U*\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om -1.85e+15)
   (sqrt (* (* 2.0 n) (* U (- t (/ (* 2.0 (pow l 2.0)) Om)))))
   (if (<= Om 8.2e+83)
     (sqrt (* (* 2.0 (* n U)) (+ t (* (* n U*) (* (/ l Om) (/ l Om))))))
     (sqrt (fabs (* (* 2.0 U) (* n t)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= -1.85e+15) {
		tmp = sqrt(((2.0 * n) * (U * (t - ((2.0 * pow(l, 2.0)) / Om)))));
	} else if (Om <= 8.2e+83) {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * U_42_) * ((l / Om) * (l / Om))))));
	} else {
		tmp = sqrt(fabs(((2.0 * U) * (n * t))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (om <= (-1.85d+15)) then
        tmp = sqrt(((2.0d0 * n) * (u * (t - ((2.0d0 * (l ** 2.0d0)) / om)))))
    else if (om <= 8.2d+83) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + ((n * u_42) * ((l / om) * (l / om))))))
    else
        tmp = sqrt(abs(((2.0d0 * u) * (n * t))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= -1.85e+15) {
		tmp = Math.sqrt(((2.0 * n) * (U * (t - ((2.0 * Math.pow(l, 2.0)) / Om)))));
	} else if (Om <= 8.2e+83) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((n * U_42_) * ((l / Om) * (l / Om))))));
	} else {
		tmp = Math.sqrt(Math.abs(((2.0 * U) * (n * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if Om <= -1.85e+15:
		tmp = math.sqrt(((2.0 * n) * (U * (t - ((2.0 * math.pow(l, 2.0)) / Om)))))
	elif Om <= 8.2e+83:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((n * U_42_) * ((l / Om) * (l / Om))))))
	else:
		tmp = math.sqrt(math.fabs(((2.0 * U) * (n * t))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (Om <= -1.85e+15)
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(Float64(2.0 * (l ^ 2.0)) / Om)))));
	elseif (Om <= 8.2e+83)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(n * U_42_) * Float64(Float64(l / Om) * Float64(l / Om))))));
	else
		tmp = sqrt(abs(Float64(Float64(2.0 * U) * Float64(n * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (Om <= -1.85e+15)
		tmp = sqrt(((2.0 * n) * (U * (t - ((2.0 * (l ^ 2.0)) / Om)))));
	elseif (Om <= 8.2e+83)
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * U_42_) * ((l / Om) * (l / Om))))));
	else
		tmp = sqrt(abs(((2.0 * U) * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[Om, -1.85e+15], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 8.2e+83], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(n * U$42$), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if Om < -1.85e15

   1. Initial program 63.3%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in Om around inf 58.4%

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

      1. associate-*r/ 58.4%

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

   7. Simplified 58.4%

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

   if -1.85e15 < Om < 8.2000000000000002e83

   1. Initial program 49.7%

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

   3. Simplified 46.6%

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

   5. Taylor expanded in U* around inf 43.5%

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

      1. mul-1-neg 43.5%

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

      2. associate-/l* 43.5%

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

      3. distribute-rgt-neg-in 43.5%

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

      4. distribute-neg-frac2 43.5%

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

      5. *-commutative 43.5%

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

   7. Simplified 43.5%

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

      1. *-un-lft-identity 43.5%

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

      2. associate-*r* 44.3%

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

      3. associate-*l* 44.3%

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

      4. associate-/l* 44.3%

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

   9. Applied egg-rr 44.3%

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

       1. *-lft-identity 44.3%

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

       2. distribute-frac-neg2 44.3%

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

       3. unpow2 44.3%

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

       4. unpow2 44.3%

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

       5. times-frac 57.0%

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

       6. unpow2 57.0%

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

   11. Simplified 57.0%

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

       1. *-un-lft-identity 57.0%

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

       2. associate-*r* 56.9%

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

   13. Applied egg-rr 56.9%

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

       1. *-lft-identity 56.9%

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

       2. *-commutative 56.9%

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

       3. sub-neg 56.9%

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

       4. distribute-rgt-neg-out 56.9%

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

       5. remove-double-neg 56.9%

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

       6. *-commutative 56.9%

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

   15. Simplified 56.9%

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

       1. unpow2 56.9%

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

   17. Applied egg-rr 56.9%

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

   if 8.2000000000000002e83 < Om

   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 48.3%

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

   5. Taylor expanded in t around inf 48.9%

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

      1. add-sqr-sqrt 48.9%

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

      2. pow1/2 48.9%

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

      3. pow1/2 48.9%

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

      4. pow-prod-down 37.4%

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

      5. pow2 37.4%

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

      6. associate-*r* 37.4%

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

   7. Applied egg-rr 37.4%

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

      1. unpow1/2 37.4%

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

      2. unpow2 37.4%

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

      3. rem-sqrt-square 49.2%

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

   9. Simplified 49.2%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1.85 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;Om \leq 8.2 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot U*\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq -1 \cdot 10^{+16}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)}\\ \mathbf{elif}\;Om \leq 9.5 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot U*\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om -1e+16)
   (sqrt (* (* (* 2.0 n) U) (- t (/ (* 2.0 (pow l 2.0)) Om))))
   (if (<= Om 9.5e+83)
     (sqrt (* (* 2.0 (* n U)) (+ t (* (* n U*) (* (/ l Om) (/ l Om))))))
     (sqrt (fabs (* (* 2.0 U) (* n t)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= -1e+16) {
		tmp = sqrt((((2.0 * n) * U) * (t - ((2.0 * pow(l, 2.0)) / Om))));
	} else if (Om <= 9.5e+83) {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * U_42_) * ((l / Om) * (l / Om))))));
	} else {
		tmp = sqrt(fabs(((2.0 * U) * (n * t))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (om <= (-1d+16)) then
        tmp = sqrt((((2.0d0 * n) * u) * (t - ((2.0d0 * (l ** 2.0d0)) / om))))
    else if (om <= 9.5d+83) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + ((n * u_42) * ((l / om) * (l / om))))))
    else
        tmp = sqrt(abs(((2.0d0 * u) * (n * t))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= -1e+16) {
		tmp = Math.sqrt((((2.0 * n) * U) * (t - ((2.0 * Math.pow(l, 2.0)) / Om))));
	} else if (Om <= 9.5e+83) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((n * U_42_) * ((l / Om) * (l / Om))))));
	} else {
		tmp = Math.sqrt(Math.abs(((2.0 * U) * (n * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if Om <= -1e+16:
		tmp = math.sqrt((((2.0 * n) * U) * (t - ((2.0 * math.pow(l, 2.0)) / Om))))
	elif Om <= 9.5e+83:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((n * U_42_) * ((l / Om) * (l / Om))))))
	else:
		tmp = math.sqrt(math.fabs(((2.0 * U) * (n * t))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (Om <= -1e+16)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t - Float64(Float64(2.0 * (l ^ 2.0)) / Om))));
	elseif (Om <= 9.5e+83)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(n * U_42_) * Float64(Float64(l / Om) * Float64(l / Om))))));
	else
		tmp = sqrt(abs(Float64(Float64(2.0 * U) * Float64(n * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (Om <= -1e+16)
		tmp = sqrt((((2.0 * n) * U) * (t - ((2.0 * (l ^ 2.0)) / Om))));
	elseif (Om <= 9.5e+83)
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * U_42_) * ((l / Om) * (l / Om))))));
	else
		tmp = sqrt(abs(((2.0 * U) * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[Om, -1e+16], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t - N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 9.5e+83], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(n * U$42$), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if Om < -1e16

   1. Initial program 64.2%

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

   3. Taylor expanded in Om around inf 62.8%

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

      1. metadata-eval 62.8%

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

      2. cancel-sign-sub-inv 62.8%

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

      3. associate-*r/ 62.8%

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

   5. Simplified 62.8%

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

   if -1e16 < Om < 9.5000000000000002e83

   1. Initial program 49.3%

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

   3. Simplified 46.2%

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

   5. Taylor expanded in U* around inf 43.2%

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

      1. mul-1-neg 43.2%

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

      2. associate-/l* 43.2%

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

      3. distribute-rgt-neg-in 43.2%

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

      4. distribute-neg-frac2 43.2%

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

      5. *-commutative 43.2%

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

   7. Simplified 43.2%

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

      1. *-un-lft-identity 43.2%

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

      2. associate-*r* 43.9%

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

      3. associate-*l* 43.9%

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

      4. associate-/l* 44.0%

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

   9. Applied egg-rr 44.0%

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

       1. *-lft-identity 44.0%

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

       2. distribute-frac-neg2 44.0%

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

       3. unpow2 44.0%

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

       4. unpow2 44.0%

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

       5. times-frac 56.5%

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

       6. unpow2 56.5%

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

   11. Simplified 56.5%

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

       1. *-un-lft-identity 56.5%

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

       2. associate-*r* 56.4%

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

   13. Applied egg-rr 56.4%

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

       1. *-lft-identity 56.4%

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

       2. *-commutative 56.4%

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

       3. sub-neg 56.4%

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

       4. distribute-rgt-neg-out 56.4%

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

       5. remove-double-neg 56.4%

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

       6. *-commutative 56.4%

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

   15. Simplified 56.4%

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

       1. unpow2 56.4%

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

   17. Applied egg-rr 56.4%

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

   if 9.5000000000000002e83 < Om

   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 48.3%

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

   5. Taylor expanded in t around inf 48.9%

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

      1. add-sqr-sqrt 48.9%

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

      2. pow1/2 48.9%

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

      3. pow1/2 48.9%

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

      4. pow-prod-down 37.4%

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

      5. pow2 37.4%

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

      6. associate-*r* 37.4%

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

   7. Applied egg-rr 37.4%

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

      1. unpow1/2 37.4%

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

      2. unpow2 37.4%

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

      3. rem-sqrt-square 49.2%

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

   9. Simplified 49.2%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1 \cdot 10^{+16}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)}\\ \mathbf{elif}\;Om \leq 9.5 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot U*\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 4.3 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot U*\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om 4.3e+83)
   (sqrt (* (* 2.0 (* n U)) (+ t (* (* n U*) (* (/ l Om) (/ l Om))))))
   (sqrt (fabs (* (* 2.0 U) (* n t))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= 4.3e+83) {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * U_42_) * ((l / Om) * (l / Om))))));
	} else {
		tmp = sqrt(fabs(((2.0 * U) * (n * t))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (om <= 4.3d+83) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + ((n * u_42) * ((l / om) * (l / om))))))
    else
        tmp = sqrt(abs(((2.0d0 * u) * (n * t))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= 4.3e+83) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((n * U_42_) * ((l / Om) * (l / Om))))));
	} else {
		tmp = Math.sqrt(Math.abs(((2.0 * U) * (n * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if Om <= 4.3e+83:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((n * U_42_) * ((l / Om) * (l / Om))))))
	else:
		tmp = math.sqrt(math.fabs(((2.0 * U) * (n * t))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (Om <= 4.3e+83)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(n * U_42_) * Float64(Float64(l / Om) * Float64(l / Om))))));
	else
		tmp = sqrt(abs(Float64(Float64(2.0 * U) * Float64(n * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (Om <= 4.3e+83)
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * U_42_) * ((l / Om) * (l / Om))))));
	else
		tmp = sqrt(abs(((2.0 * U) * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[Om, 4.3e+83], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(n * U$42$), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < 4.3e83

   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 54.4%

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

   5. Taylor expanded in U* around inf 43.5%

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

      1. mul-1-neg 43.5%

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

      2. associate-/l* 44.4%

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

      3. distribute-rgt-neg-in 44.4%

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

      4. distribute-neg-frac2 44.4%

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

      5. *-commutative 44.4%

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

   7. Simplified 44.4%

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

      1. *-un-lft-identity 44.4%

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

      2. associate-*r* 46.3%

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

      3. associate-*l* 46.3%

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

      4. associate-/l* 46.4%

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

   9. Applied egg-rr 46.4%

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

       1. *-lft-identity 46.4%

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

       2. distribute-frac-neg2 46.4%

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

       3. unpow2 46.4%

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

       4. unpow2 46.4%

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

       5. times-frac 55.1%

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

       6. unpow2 55.1%

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

   11. Simplified 55.1%

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

       1. *-un-lft-identity 55.1%

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

       2. associate-*r* 53.5%

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

   13. Applied egg-rr 53.5%

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

       1. *-lft-identity 53.5%

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

       2. *-commutative 53.5%

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

       3. sub-neg 53.5%

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

       4. distribute-rgt-neg-out 53.5%

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

       5. remove-double-neg 53.5%

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

       6. *-commutative 53.5%

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

   15. Simplified 53.5%

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

       1. unpow2 53.5%

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

   17. Applied egg-rr 53.5%

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

   if 4.3e83 < Om

   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 48.3%

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

   5. Taylor expanded in t around inf 48.9%

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

      1. add-sqr-sqrt 48.9%

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

      2. pow1/2 48.9%

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

      3. pow1/2 48.9%

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

      4. pow-prod-down 37.4%

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

      5. pow2 37.4%

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

      6. associate-*r* 37.4%

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

   7. Applied egg-rr 37.4%

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

      1. unpow1/2 37.4%

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

      2. unpow2 37.4%

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

      3. rem-sqrt-square 49.2%

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

   9. Simplified 49.2%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 4.3 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot U*\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 1.9 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot U*\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om 1.9e+83)
   (sqrt (* (* 2.0 (* n U)) (+ t (* (* n U*) (* (/ l Om) (/ l Om))))))
   (sqrt (* 2.0 (* U (* n t))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= 1.9e+83) {
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * U_42_) * ((l / Om) * (l / Om))))));
	} else {
		tmp = sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (om <= 1.9d+83) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t + ((n * u_42) * ((l / om) * (l / om))))))
    else
        tmp = sqrt((2.0d0 * (u * (n * t))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= 1.9e+83) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t + ((n * U_42_) * ((l / Om) * (l / Om))))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if Om <= 1.9e+83:
		tmp = math.sqrt(((2.0 * (n * U)) * (t + ((n * U_42_) * ((l / Om) * (l / Om))))))
	else:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (Om <= 1.9e+83)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t + Float64(Float64(n * U_42_) * Float64(Float64(l / Om) * Float64(l / Om))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (Om <= 1.9e+83)
		tmp = sqrt(((2.0 * (n * U)) * (t + ((n * U_42_) * ((l / Om) * (l / Om))))));
	else
		tmp = sqrt((2.0 * (U * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[Om, 1.9e+83], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(n * U$42$), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < 1.9000000000000001e83

   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 54.4%

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

   5. Taylor expanded in U* around inf 43.5%

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

      1. mul-1-neg 43.5%

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

      2. associate-/l* 44.4%

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

      3. distribute-rgt-neg-in 44.4%

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

      4. distribute-neg-frac2 44.4%

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

      5. *-commutative 44.4%

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

   7. Simplified 44.4%

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

      1. *-un-lft-identity 44.4%

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

      2. associate-*r* 46.3%

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

      3. associate-*l* 46.3%

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

      4. associate-/l* 46.4%

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

   9. Applied egg-rr 46.4%

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

       1. *-lft-identity 46.4%

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

       2. distribute-frac-neg2 46.4%

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

       3. unpow2 46.4%

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

       4. unpow2 46.4%

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

       5. times-frac 55.1%

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

       6. unpow2 55.1%

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

   11. Simplified 55.1%

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

       1. *-un-lft-identity 55.1%

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

       2. associate-*r* 53.5%

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

   13. Applied egg-rr 53.5%

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

       1. *-lft-identity 53.5%

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

       2. *-commutative 53.5%

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

       3. sub-neg 53.5%

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

       4. distribute-rgt-neg-out 53.5%

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

       5. remove-double-neg 53.5%

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

       6. *-commutative 53.5%

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

   15. Simplified 53.5%

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

       1. unpow2 53.5%

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

   17. Applied egg-rr 53.5%

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

   if 1.9000000000000001e83 < Om

   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 48.3%

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

   5. Taylor expanded in t around inf 48.9%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 1.9 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \left(n \cdot U*\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 37.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq -8.6 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om -8.6e+15)
   (sqrt (* t (* 2.0 (* n U))))
   (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 (Om <= -8.6e+15) {
		tmp = sqrt((t * (2.0 * (n * U))));
	} 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 (om <= (-8.6d+15)) then
        tmp = sqrt((t * (2.0d0 * (n * u))))
    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 (Om <= -8.6e+15) {
		tmp = Math.sqrt((t * (2.0 * (n * U))));
	} 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 Om <= -8.6e+15:
		tmp = math.sqrt((t * (2.0 * (n * U))))
	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 (Om <= -8.6e+15)
		tmp = sqrt(Float64(t * Float64(2.0 * Float64(n * U))));
	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 (Om <= -8.6e+15)
		tmp = sqrt((t * (2.0 * (n * U))));
	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[Om, -8.6e+15], N[Sqrt[N[(t * N[(2.0 * N[(n * U), $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 Om < -8.6e15

   1. Initial program 64.2%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in U* around inf 43.9%

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

      1. mul-1-neg 43.9%

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

      2. associate-/l* 46.5%

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

      3. distribute-rgt-neg-in 46.5%

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

      4. distribute-neg-frac2 46.5%

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

      5. *-commutative 46.5%

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

   7. Simplified 46.5%

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

      1. *-un-lft-identity 46.5%

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

      2. associate-*r* 50.4%

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

      3. associate-*l* 50.4%

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

      4. associate-/l* 50.4%

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

   9. Applied egg-rr 50.4%

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

       1. *-lft-identity 50.4%

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

       2. distribute-frac-neg2 50.4%

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

       3. unpow2 50.4%

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

       4. unpow2 50.4%

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

       5. times-frac 52.7%

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

       6. unpow2 52.7%

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

   11. Simplified 52.7%

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

   12. Taylor expanded in n around 0 40.5%

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

       1. associate-*r* 40.5%

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

       2. associate-*r* 48.6%

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

       3. associate-*r* 48.6%

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

       4. *-commutative 48.6%

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

   14. Simplified 48.6%

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

   if -8.6e15 < Om

   1. Initial program 47.5%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in t around inf 35.1%

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

      1. pow1/2 37.4%

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

      2. associate-*r* 37.4%

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

   7. Applied egg-rr 37.4%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -8.6 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 37.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq -7 \cdot 10^{+15}:\\ \;\;\;\;{\left(\left(n \cdot U\right) \cdot \left(2 \cdot t\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om -7e+15)
   (pow (* (* n U) (* 2.0 t)) 0.5)
   (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 (Om <= -7e+15) {
		tmp = pow(((n * U) * (2.0 * t)), 0.5);
	} 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 (om <= (-7d+15)) then
        tmp = ((n * u) * (2.0d0 * t)) ** 0.5d0
    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 (Om <= -7e+15) {
		tmp = Math.pow(((n * U) * (2.0 * t)), 0.5);
	} 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 Om <= -7e+15:
		tmp = math.pow(((n * U) * (2.0 * t)), 0.5)
	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 (Om <= -7e+15)
		tmp = Float64(Float64(n * U) * Float64(2.0 * t)) ^ 0.5;
	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 (Om <= -7e+15)
		tmp = ((n * U) * (2.0 * t)) ^ 0.5;
	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[Om, -7e+15], N[Power[N[(N[(n * U), $MachinePrecision] * N[(2.0 * t), $MachinePrecision]), $MachinePrecision], 0.5], $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 Om < -7e15

   1. Initial program 64.2%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in U* around inf 43.9%

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

      1. mul-1-neg 43.9%

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

      2. associate-/l* 46.5%

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

      3. distribute-rgt-neg-in 46.5%

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

      4. distribute-neg-frac2 46.5%

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

      5. *-commutative 46.5%

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

   7. Simplified 46.5%

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

      1. *-un-lft-identity 46.5%

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

      2. associate-*r* 50.4%

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

      3. associate-*l* 50.4%

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

      4. associate-/l* 50.4%

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

   9. Applied egg-rr 50.4%

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

       1. *-lft-identity 50.4%

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

       2. distribute-frac-neg2 50.4%

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

       3. unpow2 50.4%

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

       4. unpow2 50.4%

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

       5. times-frac 52.7%

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

       6. unpow2 52.7%

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

   11. Simplified 52.7%

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

   12. Taylor expanded in n around 0 40.5%

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

       1. associate-*r* 40.5%

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

       2. associate-*r* 48.6%

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

       3. associate-*r* 48.6%

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

       4. *-commutative 48.6%

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

   14. Simplified 48.6%

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

       1. pow1/2 48.7%

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

       2. associate-*r* 48.7%

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

   16. Applied egg-rr 48.7%

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

   if -7e15 < Om

   1. Initial program 47.5%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in t around inf 35.1%

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

      1. pow1/2 37.4%

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

      2. associate-*r* 37.4%

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

   7. Applied egg-rr 37.4%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -7 \cdot 10^{+15}:\\ \;\;\;\;{\left(\left(n \cdot U\right) \cdot \left(2 \cdot t\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 35.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq -9 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om -9e+15) (sqrt (* t (* 2.0 (* n U)))) (sqrt (* 2.0 (* U (* n t))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= -9e+15) {
		tmp = sqrt((t * (2.0 * (n * U))));
	} else {
		tmp = sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (om <= (-9d+15)) then
        tmp = sqrt((t * (2.0d0 * (n * u))))
    else
        tmp = sqrt((2.0d0 * (u * (n * t))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= -9e+15) {
		tmp = Math.sqrt((t * (2.0 * (n * U))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if Om <= -9e+15:
		tmp = math.sqrt((t * (2.0 * (n * U))))
	else:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (Om <= -9e+15)
		tmp = sqrt(Float64(t * Float64(2.0 * Float64(n * U))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (Om <= -9e+15)
		tmp = sqrt((t * (2.0 * (n * U))));
	else
		tmp = sqrt((2.0 * (U * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[Om, -9e+15], N[Sqrt[N[(t * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -9e15

   1. Initial program 64.2%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in U* around inf 43.9%

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

      1. mul-1-neg 43.9%

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

      2. associate-/l* 46.5%

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

      3. distribute-rgt-neg-in 46.5%

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

      4. distribute-neg-frac2 46.5%

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

      5. *-commutative 46.5%

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

   7. Simplified 46.5%

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

      1. *-un-lft-identity 46.5%

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

      2. associate-*r* 50.4%

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

      3. associate-*l* 50.4%

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

      4. associate-/l* 50.4%

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

   9. Applied egg-rr 50.4%

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

       1. *-lft-identity 50.4%

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

       2. distribute-frac-neg2 50.4%

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

       3. unpow2 50.4%

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

       4. unpow2 50.4%

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

       5. times-frac 52.7%

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

       6. unpow2 52.7%

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

   11. Simplified 52.7%

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

   12. Taylor expanded in n around 0 40.5%

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

       1. associate-*r* 40.5%

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

       2. associate-*r* 48.6%

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

       3. associate-*r* 48.6%

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

       4. *-commutative 48.6%

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

   14. Simplified 48.6%

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

   if -9e15 < Om

   1. Initial program 47.5%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in t around inf 35.1%

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

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -9 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 35.1% 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 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 53.1%

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

5. Taylor expanded in t around inf 36.7%

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

6. Final simplification 36.7%

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

Reproduce

herbie shell --seed 2024131 
(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*))))))