Toniolo and Linder, Equation (13)

Percentage Accurate: 49.8% → 65.0%

Time: 23.9s

Alternatives: 16

Speedup: 0.3×

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: 49.8% 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: 65.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;{\left(2 \cdot U\right)}^{0.5} \cdot \sqrt{n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\\ \mathbf{elif}\;t\_1 \leq 10^{+153}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left|n \cdot U\right|} \cdot \sqrt{\left|t + -2 \cdot \left(\frac{\ell}{\sqrt[3]{Om}} \cdot \left(\ell \cdot {\left(\sqrt[3]{Om}\right)}^{-2}\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(U \cdot {\ell}^{2}\right) \cdot \frac{2 \cdot n + \frac{\left(U - U*\right) \cdot {n}^{2}}{Om}}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U* U)))))))
   (if (<= t_1 0.0)
     (* (pow (* 2.0 U) 0.5) (sqrt (* n (+ t (* -2.0 (/ (pow l 2.0) Om))))))
     (if (<= t_1 1e+153)
       t_1
       (if (<= t_1 INFINITY)
         (*
          (sqrt (* 2.0 (fabs (* n U))))
          (sqrt
           (fabs
            (+ t (* -2.0 (* (/ l (cbrt Om)) (* l (pow (cbrt Om) -2.0))))))))
         (sqrt
          (*
           -2.0
           (*
            (* U (pow l 2.0))
            (/ (+ (* 2.0 n) (/ (* (- U U*) (pow n 2.0)) Om)) Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * pow((l / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = pow((2.0 * U), 0.5) * sqrt((n * (t + (-2.0 * (pow(l, 2.0) / Om)))));
	} else if (t_1 <= 1e+153) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((2.0 * fabs((n * U)))) * sqrt(fabs((t + (-2.0 * ((l / cbrt(Om)) * (l * pow(cbrt(Om), -2.0)))))));
	} else {
		tmp = sqrt((-2.0 * ((U * pow(l, 2.0)) * (((2.0 * n) + (((U - U_42_) * pow(n, 2.0)) / Om)) / Om))));
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + ((n * Math.pow((l / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.pow((2.0 * U), 0.5) * Math.sqrt((n * (t + (-2.0 * (Math.pow(l, 2.0) / Om)))));
	} else if (t_1 <= 1e+153) {
		tmp = t_1;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((2.0 * Math.abs((n * U)))) * Math.sqrt(Math.abs((t + (-2.0 * ((l / Math.cbrt(Om)) * (l * Math.pow(Math.cbrt(Om), -2.0)))))));
	} else {
		tmp = Math.sqrt((-2.0 * ((U * Math.pow(l, 2.0)) * (((2.0 * n) + (((U - U_42_) * Math.pow(n, 2.0)) / Om)) / Om))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = 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_42_ - U)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64((Float64(2.0 * U) ^ 0.5) * sqrt(Float64(n * Float64(t + Float64(-2.0 * Float64((l ^ 2.0) / Om))))));
	elseif (t_1 <= 1e+153)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = Float64(sqrt(Float64(2.0 * abs(Float64(n * U)))) * sqrt(abs(Float64(t + Float64(-2.0 * Float64(Float64(l / cbrt(Om)) * Float64(l * (cbrt(Om) ^ -2.0))))))));
	else
		tmp = sqrt(Float64(-2.0 * Float64(Float64(U * (l ^ 2.0)) * Float64(Float64(Float64(2.0 * n) + Float64(Float64(Float64(U - U_42_) * (n ^ 2.0)) / Om)) / Om))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = 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$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[Power[N[(2.0 * U), $MachinePrecision], 0.5], $MachinePrecision] * N[Sqrt[N[(n * N[(t + N[(-2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+153], t$95$1, If[LessEqual[t$95$1, Infinity], N[(N[Sqrt[N[(2.0 * N[Abs[N[(n * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[N[(t + N[(-2.0 * N[(N[(l / N[Power[Om, 1/3], $MachinePrecision]), $MachinePrecision] * N[(l * N[Power[N[Power[Om, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(U * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(2.0 * n), $MachinePrecision] + N[(N[(N[(U - U$42$), $MachinePrecision] * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 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 11.5%

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

   3. Simplified 28.1%

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

   5. Taylor expanded in n around 0 31.3%

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

      1. pow1/2 31.3%

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

      2. associate-*r* 31.3%

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

      3. unpow-prod-down 50.6%

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

      4. pow1/2 50.6%

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

      5. unpow2 50.6%

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

      6. add-cube-cbrt 50.6%

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

      7. unpow2 50.6%

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

      8. frac-times 50.6%

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

      9. cancel-sign-sub-inv 50.6%

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

      10. metadata-eval 50.6%

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

      11. frac-times 50.6%

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

      12. unpow2 50.6%

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

      13. unpow2 50.6%

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

      14. add-cube-cbrt 50.6%

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

   7. Applied egg-rr 50.6%

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

   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*))))) < 1e153

   1. Initial program 97.4%

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

   if 1e153 < (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 30.4%

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

   3. Simplified 42.4%

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

   5. Taylor expanded in n around 0 26.5%

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

      1. add-sqr-sqrt 26.5%

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

      2. pow1/2 26.5%

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

      3. pow1/2 26.7%

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

      4. pow-prod-down 30.3%

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

   7. Applied egg-rr 30.3%

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

      1. unpow1/2 30.3%

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

      2. unpow2 30.3%

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

      3. rem-sqrt-square 27.0%

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

      4. associate-*r* 27.0%

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

      5. metadata-eval 27.0%

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

      6. cancel-sign-sub-inv 27.0%

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

      7. associate-*r/ 27.0%

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

   9. Simplified 27.0%

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

       1. pow1/2 27.0%

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

       2. fabs-mul 27.0%

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

       3. unpow-prod-down 41.7%

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

       4. associate-*r* 41.7%

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

       5. associate-/l* 41.7%

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

   11. Applied egg-rr 41.7%

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

       1. unpow1/2 41.7%

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

       2. associate-*r* 41.7%

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

       3. fabs-mul 41.7%

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

       4. metadata-eval 41.7%

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

       5. *-commutative 41.7%

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

       6. unpow1/2 41.7%

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

       7. sub-neg 41.7%

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

       8. *-commutative 41.7%

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

       9. distribute-rgt-neg-in 41.7%

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

       10. metadata-eval 41.7%

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

   13. Simplified 41.7%

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

       1. unpow2 41.7%

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

       2. add-cube-cbrt 41.7%

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

       3. unpow2 41.7%

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

       4. frac-times 54.7%

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

       5. *-commutative 54.7%

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

       6. div-inv 54.7%

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

       7. pow-flip 54.7%

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

       8. metadata-eval 54.7%

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

   15. Applied egg-rr 54.7%

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

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

   1. Initial program 0.0%

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

   3. Simplified 3.9%

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

   5. Taylor expanded in l around inf 40.0%

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

      1. associate-*r* 45.6%

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

      2. associate-*r/ 45.6%

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

      3. metadata-eval 45.6%

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

      4. associate-/l* 42.8%

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

   7. Simplified 42.8%

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

   8. Taylor expanded in Om around inf 45.0%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;{\left(2 \cdot U\right)}^{0.5} \cdot \sqrt{n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 10^{+153}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left|n \cdot U\right|} \cdot \sqrt{\left|t + -2 \cdot \left(\frac{\ell}{\sqrt[3]{Om}} \cdot \left(\ell \cdot {\left(\sqrt[3]{Om}\right)}^{-2}\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(U \cdot {\ell}^{2}\right) \cdot \frac{2 \cdot n + \frac{\left(U - U*\right) \cdot {n}^{2}}{Om}}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t\_1 \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;{\left(2 \cdot U\right)}^{0.5} \cdot \sqrt{n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) + t\_1 \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(U \cdot {\ell}^{2}\right) \cdot \frac{2 \cdot n + \frac{\left(U - U*\right) \cdot {n}^{2}}{Om}}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (pow (/ l Om) 2.0)))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+ (- t (* 2.0 (/ (* l l) Om))) (* t_1 (- U* U)))))))
   (if (<= t_2 0.0)
     (* (pow (* 2.0 U) 0.5) (sqrt (* n (+ t (* -2.0 (/ (pow l 2.0) Om))))))
     (if (<= t_2 INFINITY)
       (sqrt
        (* (* 2.0 (* n U)) (- t (+ (* 2.0 (* l (/ l Om))) (* t_1 (- U U*))))))
       (sqrt
        (*
         -2.0
         (*
          (* U (pow l 2.0))
          (/ (+ (* 2.0 n) (/ (* (- U U*) (pow n 2.0)) Om)) Om))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * pow((l / Om), 2.0);
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + (t_1 * (U_42_ - U)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = pow((2.0 * U), 0.5) * sqrt((n * (t + (-2.0 * (pow(l, 2.0) / Om)))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (n * U)) * (t - ((2.0 * (l * (l / Om))) + (t_1 * (U - U_42_))))));
	} else {
		tmp = sqrt((-2.0 * ((U * pow(l, 2.0)) * (((2.0 * n) + (((U - U_42_) * pow(n, 2.0)) / Om)) / Om))));
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * Math.pow((l / Om), 2.0);
	double t_2 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + (t_1 * (U_42_ - U)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = Math.pow((2.0 * U), 0.5) * Math.sqrt((n * (t + (-2.0 * (Math.pow(l, 2.0) / Om)))));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t - ((2.0 * (l * (l / Om))) + (t_1 * (U - U_42_))))));
	} else {
		tmp = Math.sqrt((-2.0 * ((U * Math.pow(l, 2.0)) * (((2.0 * n) + (((U - U_42_) * Math.pow(n, 2.0)) / Om)) / Om))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = n * math.pow((l / Om), 2.0)
	t_2 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + (t_1 * (U_42_ - U)))))
	tmp = 0
	if t_2 <= 0.0:
		tmp = math.pow((2.0 * U), 0.5) * math.sqrt((n * (t + (-2.0 * (math.pow(l, 2.0) / Om)))))
	elif t_2 <= math.inf:
		tmp = math.sqrt(((2.0 * (n * U)) * (t - ((2.0 * (l * (l / Om))) + (t_1 * (U - U_42_))))))
	else:
		tmp = math.sqrt((-2.0 * ((U * math.pow(l, 2.0)) * (((2.0 * n) + (((U - U_42_) * math.pow(n, 2.0)) / Om)) / Om))))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Power[N[(2.0 * U), $MachinePrecision], 0.5], $MachinePrecision] * N[Sqrt[N[(n * N[(t + N[(-2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(U * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(2.0 * n), $MachinePrecision] + N[(N[(N[(U - U$42$), $MachinePrecision] * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 11.5%

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

   3. Simplified 28.1%

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

   5. Taylor expanded in n around 0 31.3%

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

      1. pow1/2 31.3%

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

      2. associate-*r* 31.3%

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

      3. unpow-prod-down 50.6%

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

      4. pow1/2 50.6%

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

      5. unpow2 50.6%

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

      6. add-cube-cbrt 50.6%

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

      7. unpow2 50.6%

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

      8. frac-times 50.6%

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

      9. cancel-sign-sub-inv 50.6%

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

      10. metadata-eval 50.6%

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

      11. frac-times 50.6%

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

      12. unpow2 50.6%

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

      13. unpow2 50.6%

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

      14. add-cube-cbrt 50.6%

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

   7. Applied egg-rr 50.6%

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

   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 66.0%

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

   3. Simplified 71.6%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
   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 3.9%

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

   5. Taylor expanded in l around inf 40.0%

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

      1. associate-*r* 45.6%

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

      2. associate-*r/ 45.6%

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

      3. metadata-eval 45.6%

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

      4. associate-/l* 42.8%

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

   7. Simplified 42.8%

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

   8. Taylor expanded in Om around inf 45.0%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;{\left(2 \cdot U\right)}^{0.5} \cdot \sqrt{n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(U \cdot {\ell}^{2}\right) \cdot \frac{2 \cdot n + \frac{\left(U - U*\right) \cdot {n}^{2}}{Om}}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t\_1 \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;{\left(2 \cdot U\right)}^{0.5} \cdot \sqrt{n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) + t\_1 \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(n \cdot \left(\frac{2}{Om} + n \cdot \frac{U - U*}{{Om}^{2}}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (pow (/ l Om) 2.0)))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (+ (- t (* 2.0 (/ (* l l) Om))) (* t_1 (- U* U)))))))
   (if (<= t_2 0.0)
     (* (pow (* 2.0 U) 0.5) (sqrt (* n (+ t (* -2.0 (/ (pow l 2.0) Om))))))
     (if (<= t_2 INFINITY)
       (sqrt
        (* (* 2.0 (* n U)) (- t (+ (* 2.0 (* l (/ l Om))) (* t_1 (- U U*))))))
       (sqrt
        (*
         -2.0
         (*
          (* U (* l l))
          (* n (+ (/ 2.0 Om) (* n (/ (- U U*) (pow Om 2.0))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * pow((l / Om), 2.0);
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + (t_1 * (U_42_ - U)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = pow((2.0 * U), 0.5) * sqrt((n * (t + (-2.0 * (pow(l, 2.0) / Om)))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (n * U)) * (t - ((2.0 * (l * (l / Om))) + (t_1 * (U - U_42_))))));
	} else {
		tmp = sqrt((-2.0 * ((U * (l * l)) * (n * ((2.0 / Om) + (n * ((U - U_42_) / pow(Om, 2.0))))))));
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * Math.pow((l / Om), 2.0);
	double t_2 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + (t_1 * (U_42_ - U)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = Math.pow((2.0 * U), 0.5) * Math.sqrt((n * (t + (-2.0 * (Math.pow(l, 2.0) / Om)))));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t - ((2.0 * (l * (l / Om))) + (t_1 * (U - U_42_))))));
	} else {
		tmp = Math.sqrt((-2.0 * ((U * (l * l)) * (n * ((2.0 / Om) + (n * ((U - U_42_) / Math.pow(Om, 2.0))))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = n * math.pow((l / Om), 2.0)
	t_2 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) + (t_1 * (U_42_ - U)))))
	tmp = 0
	if t_2 <= 0.0:
		tmp = math.pow((2.0 * U), 0.5) * math.sqrt((n * (t + (-2.0 * (math.pow(l, 2.0) / Om)))))
	elif t_2 <= math.inf:
		tmp = math.sqrt(((2.0 * (n * U)) * (t - ((2.0 * (l * (l / Om))) + (t_1 * (U - U_42_))))))
	else:
		tmp = math.sqrt((-2.0 * ((U * (l * l)) * (n * ((2.0 / Om) + (n * ((U - U_42_) / math.pow(Om, 2.0))))))))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Power[N[(2.0 * U), $MachinePrecision], 0.5], $MachinePrecision] * N[Sqrt[N[(n * N[(t + N[(-2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-2.0 * N[(N[(U * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(n * N[(N[(2.0 / Om), $MachinePrecision] + N[(n * N[(N[(U - U$42$), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (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 11.5%

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

   3. Simplified 28.1%

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

   5. Taylor expanded in n around 0 31.3%

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

      1. pow1/2 31.3%

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

      2. associate-*r* 31.3%

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

      3. unpow-prod-down 50.6%

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

      4. pow1/2 50.6%

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

      5. unpow2 50.6%

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

      6. add-cube-cbrt 50.6%

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

      7. unpow2 50.6%

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

      8. frac-times 50.6%

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

      9. cancel-sign-sub-inv 50.6%

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

      10. metadata-eval 50.6%

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

      11. frac-times 50.6%

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

      12. unpow2 50.6%

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

      13. unpow2 50.6%

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

      14. add-cube-cbrt 50.6%

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

   7. Applied egg-rr 50.6%

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

   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 66.0%

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

   3. Simplified 71.6%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
   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 3.9%

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

   5. Taylor expanded in l around inf 40.0%

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

      1. associate-*r* 45.6%

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

      2. associate-*r/ 45.6%

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

      3. metadata-eval 45.6%

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

      4. associate-/l* 42.8%

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

   7. Simplified 42.8%

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

      1. unpow2 42.8%

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

   9. Applied egg-rr 42.8%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;{\left(2 \cdot U\right)}^{0.5} \cdot \sqrt{n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(n \cdot \left(\frac{2}{Om} + n \cdot \frac{U - U*}{{Om}^{2}}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (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 11.5%

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

   3. Simplified 28.1%

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

   5. Taylor expanded in n around 0 31.3%

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

      1. pow1/2 31.3%

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

      2. associate-*r* 31.3%

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

      3. unpow-prod-down 50.6%

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

      4. pow1/2 50.6%

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

      5. unpow2 50.6%

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

      6. add-cube-cbrt 50.6%

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

      7. unpow2 50.6%

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

      8. frac-times 50.6%

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

      9. cancel-sign-sub-inv 50.6%

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

      10. metadata-eval 50.6%

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

      11. frac-times 50.6%

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

      12. unpow2 50.6%

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

      13. unpow2 50.6%

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

      14. add-cube-cbrt 50.6%

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

   7. Applied egg-rr 50.6%

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

      1. unpow1/2 50.5%

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

      2. metadata-eval 50.5%

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

      3. cancel-sign-sub-inv 50.5%

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

      4. associate-*r/ 50.5%

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

   9. Simplified 50.5%

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

   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 66.0%

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

   3. Simplified 71.6%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
   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 3.9%

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

   5. Taylor expanded in l around inf 40.0%

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

      1. associate-*r* 45.6%

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

      2. associate-*r/ 45.6%

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

      3. metadata-eval 45.6%

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

      4. associate-/l* 42.8%

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

   7. Simplified 42.8%

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

      1. unpow2 42.8%

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

   9. Applied egg-rr 42.8%

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

4. Final simplification 65.0%

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

Alternative 5: 60.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (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 11.5%

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

   3. Simplified 28.1%

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

   5. Taylor expanded in t around inf 31.3%

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

      1. pow1/2 31.3%

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

      2. associate-*r* 31.3%

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

      3. unpow-prod-down 39.7%

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

      4. pow1/2 39.7%

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

   7. Applied egg-rr 39.7%

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

      1. unpow1/2 39.7%

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

   9. Simplified 39.7%

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

   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 66.0%

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

   3. Simplified 71.6%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]
   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 3.9%

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

   5. Taylor expanded in l around inf 40.0%

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

      1. associate-*r* 45.6%

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

      2. associate-*r/ 45.6%

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

      3. metadata-eval 45.6%

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

      4. associate-/l* 42.8%

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

   7. Simplified 42.8%

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

      1. unpow2 42.8%

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

   9. Applied egg-rr 42.8%

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

4. Final simplification 63.6%

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

Alternative 6: 45.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.7 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \left(U \cdot \frac{n \cdot {\ell}^{2}}{Om}\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.7e+95)
   (sqrt (* (* (* 2.0 n) U) (- t (/ (* 2.0 (pow l 2.0)) Om))))
   (pow (* -4.0 (* U (/ (* n (pow l 2.0)) Om))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.7e+95) {
		tmp = sqrt((((2.0 * n) * U) * (t - ((2.0 * pow(l, 2.0)) / Om))));
	} else {
		tmp = pow((-4.0 * (U * ((n * pow(l, 2.0)) / Om))), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 2.7d+95) then
        tmp = sqrt((((2.0d0 * n) * u) * (t - ((2.0d0 * (l ** 2.0d0)) / om))))
    else
        tmp = ((-4.0d0) * (u * ((n * (l ** 2.0d0)) / om))) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.7e+95) {
		tmp = Math.sqrt((((2.0 * n) * U) * (t - ((2.0 * Math.pow(l, 2.0)) / Om))));
	} else {
		tmp = Math.pow((-4.0 * (U * ((n * Math.pow(l, 2.0)) / Om))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.7e+95:
		tmp = math.sqrt((((2.0 * n) * U) * (t - ((2.0 * math.pow(l, 2.0)) / Om))))
	else:
		tmp = math.pow((-4.0 * (U * ((n * math.pow(l, 2.0)) / Om))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.7e+95)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t - Float64(Float64(2.0 * (l ^ 2.0)) / Om))));
	else
		tmp = Float64(-4.0 * Float64(U * Float64(Float64(n * (l ^ 2.0)) / Om))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2.7e+95)
		tmp = sqrt((((2.0 * n) * U) * (t - ((2.0 * (l ^ 2.0)) / Om))));
	else
		tmp = (-4.0 * (U * ((n * (l ^ 2.0)) / Om))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.7e+95], 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], N[Power[N[(-4.0 * N[(U * N[(N[(n * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.7e95

   1. Initial program 56.6%

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

      1. add-cube-cbrt 56.6%

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

      2. times-frac 57.4%

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

      3. pow2 57.4%

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

   4. Applied egg-rr 57.4%

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

   5. Taylor expanded in Om around inf 51.9%

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

      1. metadata-eval 51.9%

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

      2. cancel-sign-sub-inv 51.9%

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

      3. associate-*r/ 51.9%

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

   7. Simplified 51.9%

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

   if 2.7e95 < l

   1. Initial program 18.4%

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

   3. Simplified 40.6%

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

   5. Taylor expanded in n around 0 18.8%

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

   6. Taylor expanded in t around 0 23.2%

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

      1. pow1/2 32.5%

         \[\leadsto \color{blue}{{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}^{0.5}} \]

      2. associate-/l* 32.6%

         \[\leadsto {\left(-4 \cdot \color{blue}{\left(U \cdot \frac{{\ell}^{2} \cdot n}{Om}\right)}\right)}^{0.5} \]

      3. *-commutative 32.6%

         \[\leadsto {\left(-4 \cdot \left(U \cdot \frac{\color{blue}{n \cdot {\ell}^{2}}}{Om}\right)\right)}^{0.5} \]

   8. Applied egg-rr 32.6%

      \[\leadsto \color{blue}{{\left(-4 \cdot \left(U \cdot \frac{n \cdot {\ell}^{2}}{Om}\right)\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 45.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.3 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \left(U \cdot \frac{n \cdot {\ell}^{2}}{Om}\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2.3e+94)
   (sqrt (* (* 2.0 (* n U)) (- t (/ (* 2.0 (pow l 2.0)) Om))))
   (pow (* -4.0 (* U (/ (* n (pow l 2.0)) Om))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.3e+94) {
		tmp = sqrt(((2.0 * (n * U)) * (t - ((2.0 * pow(l, 2.0)) / Om))));
	} else {
		tmp = pow((-4.0 * (U * ((n * pow(l, 2.0)) / Om))), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 2.3d+94) then
        tmp = sqrt(((2.0d0 * (n * u)) * (t - ((2.0d0 * (l ** 2.0d0)) / om))))
    else
        tmp = ((-4.0d0) * (u * ((n * (l ** 2.0d0)) / om))) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2.3e+94) {
		tmp = Math.sqrt(((2.0 * (n * U)) * (t - ((2.0 * Math.pow(l, 2.0)) / Om))));
	} else {
		tmp = Math.pow((-4.0 * (U * ((n * Math.pow(l, 2.0)) / Om))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 2.3e+94:
		tmp = math.sqrt(((2.0 * (n * U)) * (t - ((2.0 * math.pow(l, 2.0)) / Om))))
	else:
		tmp = math.pow((-4.0 * (U * ((n * math.pow(l, 2.0)) / Om))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2.3e+94)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * U)) * Float64(t - Float64(Float64(2.0 * (l ^ 2.0)) / Om))));
	else
		tmp = Float64(-4.0 * Float64(U * Float64(Float64(n * (l ^ 2.0)) / Om))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 2.3e+94)
		tmp = sqrt(((2.0 * (n * U)) * (t - ((2.0 * (l ^ 2.0)) / Om))));
	else
		tmp = (-4.0 * (U * ((n * (l ^ 2.0)) / Om))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2.3e+94], N[Sqrt[N[(N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(-4.0 * N[(U * N[(N[(n * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.3e94

   1. Initial program 56.6%

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

   3. Simplified 57.5%

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

   5. Taylor expanded in Om around inf 51.9%

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

      1. associate-*r/ 51.9%

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

   7. Simplified 51.9%

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

   if 2.3e94 < l

   1. Initial program 18.4%

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

   3. Simplified 40.6%

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

   5. Taylor expanded in n around 0 18.8%

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

   6. Taylor expanded in t around 0 23.2%

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

      1. pow1/2 32.5%

         \[\leadsto \color{blue}{{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}^{0.5}} \]

      2. associate-/l* 32.6%

         \[\leadsto {\left(-4 \cdot \color{blue}{\left(U \cdot \frac{{\ell}^{2} \cdot n}{Om}\right)}\right)}^{0.5} \]

      3. *-commutative 32.6%

         \[\leadsto {\left(-4 \cdot \left(U \cdot \frac{\color{blue}{n \cdot {\ell}^{2}}}{Om}\right)\right)}^{0.5} \]

   8. Applied egg-rr 32.6%

      \[\leadsto \color{blue}{{\left(-4 \cdot \left(U \cdot \frac{n \cdot {\ell}^{2}}{Om}\right)\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 46.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 8 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \left(U \cdot \frac{n \cdot {\ell}^{2}}{Om}\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 8e+97)
   (sqrt (* 2.0 (* U (* n (- t (* 2.0 (/ (* l l) Om)))))))
   (pow (* -4.0 (* U (/ (* n (pow l 2.0)) Om))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 8e+97) {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * ((l * l) / Om)))))));
	} else {
		tmp = pow((-4.0 * (U * ((n * pow(l, 2.0)) / Om))), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 8d+97) then
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * ((l * l) / om)))))))
    else
        tmp = ((-4.0d0) * (u * ((n * (l ** 2.0d0)) / om))) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 8e+97) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * ((l * l) / Om)))))));
	} else {
		tmp = Math.pow((-4.0 * (U * ((n * Math.pow(l, 2.0)) / Om))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 8e+97:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * ((l * l) / Om)))))))
	else:
		tmp = math.pow((-4.0 * (U * ((n * math.pow(l, 2.0)) / Om))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 8e+97)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om)))))));
	else
		tmp = Float64(-4.0 * Float64(U * Float64(Float64(n * (l ^ 2.0)) / Om))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 8e+97)
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * ((l * l) / Om)))))));
	else
		tmp = (-4.0 * (U * ((n * (l ^ 2.0)) / Om))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 8e+97], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(-4.0 * N[(U * N[(N[(n * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 8.0000000000000006e97

   1. Initial program 56.6%

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

   3. Simplified 54.3%

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

   5. Taylor expanded in n around 0 50.7%

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

      1. unpow2 16.8%

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

   7. Applied egg-rr 50.7%

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

   if 8.0000000000000006e97 < l

   1. Initial program 18.4%

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

   3. Simplified 40.6%

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

   5. Taylor expanded in n around 0 18.8%

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

   6. Taylor expanded in t around 0 23.2%

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

      1. pow1/2 32.5%

         \[\leadsto \color{blue}{{\left(-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}^{0.5}} \]

      2. associate-/l* 32.6%

         \[\leadsto {\left(-4 \cdot \color{blue}{\left(U \cdot \frac{{\ell}^{2} \cdot n}{Om}\right)}\right)}^{0.5} \]

      3. *-commutative 32.6%

         \[\leadsto {\left(-4 \cdot \left(U \cdot \frac{\color{blue}{n \cdot {\ell}^{2}}}{Om}\right)\right)}^{0.5} \]

   8. Applied egg-rr 32.6%

      \[\leadsto \color{blue}{{\left(-4 \cdot \left(U \cdot \frac{n \cdot {\ell}^{2}}{Om}\right)\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 49.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.0%

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

3. Simplified 51.9%

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

5. Taylor expanded in n around 0 45.2%

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

   1. pow1/2 48.5%

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

7. Applied egg-rr 49.3%

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

8. Final simplification 49.3%

   \[\leadsto {\left(2 \cdot \left(\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot \left(n \cdot U\right)\right)\right)}^{0.5} \]
9. Add Preprocessing

Alternative 10: 46.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5.3 \cdot 10^{+201}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 5.3e+201)
   (sqrt (* 2.0 (* U (* n (- t (* 2.0 (/ (* l l) Om)))))))
   (* (sqrt (* n (* 2.0 U))) (sqrt t))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 5.3e+201) {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * ((l * l) / Om)))))));
	} else {
		tmp = sqrt((n * (2.0 * U))) * sqrt(t);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= 5.3d+201) then
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * ((l * l) / om)))))))
    else
        tmp = sqrt((n * (2.0d0 * u))) * sqrt(t)
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 5.3e+201) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * ((l * l) / Om)))))));
	} else {
		tmp = Math.sqrt((n * (2.0 * U))) * Math.sqrt(t);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= 5.3e+201:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * ((l * l) / Om)))))))
	else:
		tmp = math.sqrt((n * (2.0 * U))) * math.sqrt(t)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 5.3e+201)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om)))))));
	else
		tmp = Float64(sqrt(Float64(n * Float64(2.0 * U))) * sqrt(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= 5.3e+201)
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * ((l * l) / Om)))))));
	else
		tmp = sqrt((n * (2.0 * U))) * sqrt(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 5.3e+201], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.30000000000000035e201

   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 51.5%

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

   5. Taylor expanded in n around 0 45.3%

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

      1. unpow2 20.6%

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

   7. Applied egg-rr 45.3%

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

   if 5.30000000000000035e201 < t

   1. Initial program 55.7%

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

   3. Simplified 55.8%

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

   5. Taylor expanded in t around inf 36.8%

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

      1. associate-*r* 47.9%

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

   7. Simplified 47.9%

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

      1. pow1/2 63.2%

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

      2. associate-*r* 63.2%

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

      3. unpow-prod-down 76.9%

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

      4. pow1/2 73.1%

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

      5. associate-*r* 73.1%

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

      6. pow1/2 73.1%

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

   9. Applied egg-rr 73.1%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.3 \cdot 10^{+201}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(2 \cdot U\right)} \cdot \sqrt{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 45.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.2 \cdot 10^{+163}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 3.2e+163)
   (sqrt (* 2.0 (* U (* n (- t (* 2.0 (/ (* l l) Om)))))))
   (sqrt (fabs (* 2.0 (* t (* n U)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 3.2e+163) {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * ((l * l) / Om)))))));
	} else {
		tmp = sqrt(fabs((2.0 * (t * (n * U)))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= 3.2d+163) then
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * ((l * l) / om)))))))
    else
        tmp = sqrt(abs((2.0d0 * (t * (n * u)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 3.2e+163) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * ((l * l) / Om)))))));
	} else {
		tmp = Math.sqrt(Math.abs((2.0 * (t * (n * U)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= 3.2e+163:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * ((l * l) / Om)))))))
	else:
		tmp = math.sqrt(math.fabs((2.0 * (t * (n * U)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 3.2e+163)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om)))))));
	else
		tmp = sqrt(abs(Float64(2.0 * Float64(t * Float64(n * U)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= 3.2e+163)
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * ((l * l) / Om)))))));
	else
		tmp = sqrt(abs((2.0 * (t * (n * U)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 3.2e+163], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.1999999999999998e163

   1. Initial program 49.8%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in n around 0 45.5%

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

      1. unpow2 20.3%

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

   7. Applied egg-rr 45.5%

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

   if 3.1999999999999998e163 < t

   1. Initial program 51.9%

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

   3. Simplified 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 t around inf 37.1%

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

      1. associate-*r* 45.6%

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

   7. Simplified 45.6%

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

      1. add-sqr-sqrt 45.6%

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

      2. pow1/2 45.6%

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

      3. pow1/2 57.4%

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

      4. pow-prod-down 49.4%

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

      5. pow2 49.4%

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

      6. associate-*r* 49.4%

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

      7. associate-*r* 49.4%

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

   9. Applied egg-rr 49.4%

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

       1. unpow1/2 49.4%

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

       2. unpow2 49.4%

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

       3. rem-sqrt-square 57.9%

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

       4. associate-*l* 49.6%

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

       5. associate-*r* 49.6%

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

       6. associate-*r* 57.9%

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

       7. *-commutative 57.9%

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

   11. Simplified 57.9%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.2 \cdot 10^{+163}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 45.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.0%

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

3. Simplified 51.9%

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

5. Taylor expanded in n around 0 45.2%

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

   1. unpow2 20.2%

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

7. Applied egg-rr 45.2%

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

Alternative 13: 37.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.2 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -3.2e+75)
   (sqrt (* 2.0 (* t (* n U))))
   (sqrt (* 2.0 (* U (* n t))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -3.2e+75) {
		tmp = sqrt((2.0 * (t * (n * U))));
	} else {
		tmp = sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= (-3.2d+75)) then
        tmp = sqrt((2.0d0 * (t * (n * u))))
    else
        tmp = sqrt((2.0d0 * (u * (n * t))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -3.2e+75) {
		tmp = Math.sqrt((2.0 * (t * (n * U))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= -3.2e+75:
		tmp = math.sqrt((2.0 * (t * (n * U))))
	else:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -3.2e+75)
		tmp = sqrt(Float64(2.0 * Float64(t * Float64(n * U))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= -3.2e+75)
		tmp = sqrt((2.0 * (t * (n * U))));
	else
		tmp = sqrt((2.0 * (U * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -3.2e+75], N[Sqrt[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -3.19999999999999985e75

   1. Initial program 59.0%

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

   3. Simplified 55.8%

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

   5. Taylor expanded in t around inf 34.2%

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

      1. associate-*r* 45.1%

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

   7. Simplified 45.1%

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

   if -3.19999999999999985e75 < n

   1. Initial program 48.2%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in t around inf 36.9%

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

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.2 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 38.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.0%

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

3. Simplified 51.9%

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

5. Taylor expanded in t around inf 36.4%

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

   1. pow1/2 38.9%

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

   2. associate-*r* 38.9%

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

7. Applied egg-rr 38.9%

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

Alternative 15: 37.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.0%

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

3. Simplified 51.9%

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

5. Taylor expanded in t around inf 36.0%

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

   1. pow1/2 38.0%

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

   2. associate-*l* 38.0%

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

7. Applied egg-rr 38.0%

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

Alternative 16: 36.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.0%

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

3. Simplified 51.9%

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

5. Taylor expanded in t around inf 36.4%

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

Reproduce

herbie shell --seed 2024160 
(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*))))))