Toniolo and Linder, Equation (13)

Percentage Accurate: 50.1% → 62.3%

Time: 28.2s

Alternatives: 20

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 62.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := U \cdot \left(2 \cdot n\right)\\ t_3 := \sqrt{t_2 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot t_1\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_3 \leq 2 \cdot 10^{-148}:\\ \;\;\;\;{\left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left(t_1 \cdot \left(U - U*\right)\right)\right)\right)\right)}^{0.5} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + {\left({\left(\sqrt[3]{\frac{\ell}{Om}}\right)}^{2} \cdot \sqrt[3]{n}\right)}^{3} \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(-2 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om}\right)\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (pow (/ l Om) 2.0))
        (t_2 (* U (* 2.0 n)))
        (t_3
         (sqrt
          (* t_2 (+ (- t (* 2.0 (/ (* l l) Om))) (* (* n t_1) (- U* U)))))))
   (if (<= t_3 2e-148)
     (*
      (pow (* U (- t (fma 2.0 (* l (/ l Om)) (* n (* t_1 (- U U*)))))) 0.5)
      (sqrt (* 2.0 n)))
     (if (<= t_3 INFINITY)
       (sqrt
        (*
         t_2
         (+
          (- t (* 2.0 (/ l (/ Om l))))
          (* (pow (* (pow (cbrt (/ l Om)) 2.0) (cbrt n)) 3.0) (- U* U)))))
       (cbrt (pow (* 2.0 (* -2.0 (/ (* U (* n (pow l 2.0))) Om))) 1.5))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = pow((l / Om), 2.0);
	double t_2 = U * (2.0 * n);
	double t_3 = sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + ((n * t_1) * (U_42_ - U)))));
	double tmp;
	if (t_3 <= 2e-148) {
		tmp = pow((U * (t - fma(2.0, (l * (l / Om)), (n * (t_1 * (U - U_42_)))))), 0.5) * sqrt((2.0 * n));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * ((t - (2.0 * (l / (Om / l)))) + (pow((pow(cbrt((l / Om)), 2.0) * cbrt(n)), 3.0) * (U_42_ - U)))));
	} else {
		tmp = cbrt(pow((2.0 * (-2.0 * ((U * (n * pow(l, 2.0))) / Om))), 1.5));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l / Om) ^ 2.0
	t_2 = Float64(U * Float64(2.0 * n))
	t_3 = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(Float64(n * t_1) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_3 <= 2e-148)
		tmp = Float64((Float64(U * Float64(t - fma(2.0, Float64(l * Float64(l / Om)), Float64(n * Float64(t_1 * Float64(U - U_42_)))))) ^ 0.5) * sqrt(Float64(2.0 * n)));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(l / Float64(Om / l)))) + Float64((Float64((cbrt(Float64(l / Om)) ^ 2.0) * cbrt(n)) ^ 3.0) * Float64(U_42_ - U)))));
	else
		tmp = cbrt((Float64(2.0 * Float64(-2.0 * Float64(Float64(U * Float64(n * (l ^ 2.0))) / Om))) ^ 1.5));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * t$95$1), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 2e-148], N[(N[Power[N[(U * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + N[(n * N[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(N[Power[N[Power[N[(l / Om), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[n, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[Power[N[(2.0 * N[(-2.0 * N[(N[(U * N[(n * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 1.99999999999999987e-148

   1. Initial program 14.7%

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

      1. associate-/l* 14.7%

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

      2. add-sqr-sqrt 6.7%

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

      3. *-un-lft-identity 6.7%

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

      4. times-frac 6.7%

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

   3. Applied egg-rr 6.7%

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

      1. /-rgt-identity 6.7%

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

      2. associate-*r/ 6.7%

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

      3. rem-square-sqrt 14.7%

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

   5. Simplified 14.7%

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

      1. pow1/2 14.7%

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

      2. associate-*l* 35.4%

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

      3. unpow-prod-down 46.8%

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

      4. pow1/2 46.8%

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

      5. associate--l- 46.8%

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

      6. fma-def 46.8%

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

      7. div-inv 46.8%

         \[\leadsto \sqrt{2 \cdot n} \cdot {\left(U \cdot \left(t - \mathsf{fma}\left(2, \color{blue}{\ell \cdot \frac{1}{\frac{Om}{\ell}}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}^{0.5} \]

      8. clear-num 46.8%

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

      9. associate-*l* 46.8%

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

   7. Applied egg-rr 46.8%

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

   if 1.99999999999999987e-148 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 63.5%

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

      1. associate-/l* 70.5%

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

      2. add-sqr-sqrt 31.5%

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

      3. *-un-lft-identity 31.5%

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

      4. times-frac 31.5%

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

   3. Applied egg-rr 31.5%

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

      1. /-rgt-identity 31.5%

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

      2. associate-*r/ 31.5%

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

      3. rem-square-sqrt 70.5%

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

   5. Simplified 70.5%

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

      1. add-cube-cbrt 70.4%

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

      2. pow3 70.4%

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

   7. Applied egg-rr 70.4%

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

      1. *-commutative 70.4%

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

      2. cbrt-prod 70.4%

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

      3. unpow2 70.4%

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

      4. cbrt-prod 71.2%

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

      5. pow2 71.2%

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

   9. Applied egg-rr 71.2%

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

   if +inf.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.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 2.9%

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

   4. Taylor expanded in Om around inf 6.2%

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

      1. *-commutative 6.2%

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

      2. associate-*l/ 6.2%

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

      3. associate-*r/ 6.2%

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

   6. Simplified 6.2%

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

      1. add-cbrt-cube 6.1%

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

      2. pow1/3 6.0%

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

      3. add-sqr-sqrt 6.0%

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

      4. pow1 6.0%

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

      5. pow1/2 27.2%

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

      6. pow-prod-up 27.2%

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

      7. associate-*r* 27.2%

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

      8. metadata-eval 27.2%

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

   8. Applied egg-rr 27.2%

      \[\leadsto \color{blue}{{\left({\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - {\ell}^{2} \cdot \frac{2}{Om}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   9. Step-by-step derivation

      1. unpow1/3 27.4%

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

      2. associate-*l* 27.4%

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

      3. associate-*l* 28.3%

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

   10. Simplified 28.3%

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

   11. Taylor expanded in t around 0 30.6%

       \[\leadsto \sqrt[3]{{\left(2 \cdot \color{blue}{\left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}\right)}^{1.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{-148}:\\ \;\;\;\;{\left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}^{0.5} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + {\left({\left(\sqrt[3]{\frac{\ell}{Om}}\right)}^{2} \cdot \sqrt[3]{n}\right)}^{3} \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(-2 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om}\right)\right)}^{1.5}}\\ \end{array} \]

Alternative 2: 62.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := U \cdot \left(2 \cdot n\right)\\ t_3 := \sqrt{t_2 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_1\right)}\\ \mathbf{if}\;t_3 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(-2 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om}\right)\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l Om) 2.0)) (- U* U)))
        (t_2 (* U (* 2.0 n)))
        (t_3 (sqrt (* t_2 (+ (- t (* 2.0 (/ (* l l) Om))) t_1)))))
   (if (<= t_3 0.0)
     (sqrt (* 2.0 (* U (* n (- t (* 2.0 (/ (pow l 2.0) Om)))))))
     (if (<= t_3 INFINITY)
       (sqrt (* t_2 (+ (- t (* 2.0 (/ l (/ Om l)))) t_1)))
       (cbrt (pow (* 2.0 (* -2.0 (/ (* U (* n (pow l 2.0))) Om))) 1.5))))))

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)) * (U_42_ - U);
	double t_2 = U * (2.0 * n);
	double t_3 = sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + t_1)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (pow(l, 2.0) / Om)))))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * ((t - (2.0 * (l / (Om / l)))) + t_1)));
	} else {
		tmp = cbrt(pow((2.0 * (-2.0 * ((U * (n * pow(l, 2.0))) / Om))), 1.5));
	}
	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)) * (U_42_ - U);
	double t_2 = U * (2.0 * n);
	double t_3 = Math.sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + t_1)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (Math.pow(l, 2.0) / Om)))))));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_2 * ((t - (2.0 * (l / (Om / l)))) + t_1)));
	} else {
		tmp = Math.cbrt(Math.pow((2.0 * (-2.0 * ((U * (n * Math.pow(l, 2.0))) / Om))), 1.5));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_2 = Float64(U * Float64(2.0 * n))
	t_3 = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_1)))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64((l ^ 2.0) / Om)))))));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(l / Float64(Om / l)))) + t_1)));
	else
		tmp = cbrt((Float64(2.0 * Float64(-2.0 * Float64(Float64(U * Float64(n * (l ^ 2.0))) / Om))) ^ 1.5));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[Power[N[(2.0 * N[(-2.0 * N[(N[(U * N[(n * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 12.4%

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

   3. Simplified 12.4%

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

   4. Taylor expanded in n around 0 33.8%

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

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

   1. Initial program 63.7%

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

      1. associate-/l* 70.6%

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

      2. add-sqr-sqrt 31.9%

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

      3. *-un-lft-identity 31.9%

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

      4. times-frac 31.8%

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

   3. Applied egg-rr 31.8%

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

      1. /-rgt-identity 31.8%

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

      2. associate-*r/ 31.9%

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

      3. rem-square-sqrt 70.6%

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

   5. Simplified 70.6%

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

   if +inf.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.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 2.9%

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

   4. Taylor expanded in Om around inf 6.2%

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

      1. *-commutative 6.2%

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

      2. associate-*l/ 6.2%

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

      3. associate-*r/ 6.2%

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

   6. Simplified 6.2%

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

      1. add-cbrt-cube 6.1%

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

      2. pow1/3 6.0%

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

      3. add-sqr-sqrt 6.0%

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

      4. pow1 6.0%

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

      5. pow1/2 27.2%

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

      6. pow-prod-up 27.2%

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

      7. associate-*r* 27.2%

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

      8. metadata-eval 27.2%

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

   8. Applied egg-rr 27.2%

      \[\leadsto \color{blue}{{\left({\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - {\ell}^{2} \cdot \frac{2}{Om}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   9. Step-by-step derivation

      1. unpow1/3 27.4%

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

      2. associate-*l* 27.4%

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

      3. associate-*l* 28.3%

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

   10. Simplified 28.3%

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

   11. Taylor expanded in t around 0 30.6%

       \[\leadsto \sqrt[3]{{\left(2 \cdot \color{blue}{\left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}\right)}^{1.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(-2 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om}\right)\right)}^{1.5}}\\ \end{array} \]

Alternative 3: 61.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := U \cdot \left(2 \cdot n\right)\\ t_3 := \sqrt{t_2 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_1\right)}\\ \mathbf{if}\;t_3 \leq 2 \cdot 10^{-148}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - {\ell}^{2} \cdot \frac{2}{Om}\right)}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(-2 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om}\right)\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l Om) 2.0)) (- U* U)))
        (t_2 (* U (* 2.0 n)))
        (t_3 (sqrt (* t_2 (+ (- t (* 2.0 (/ (* l l) Om))) t_1)))))
   (if (<= t_3 2e-148)
     (* (sqrt (* 2.0 n)) (sqrt (* U (- t (* (pow l 2.0) (/ 2.0 Om))))))
     (if (<= t_3 INFINITY)
       (sqrt (* t_2 (+ (- t (* 2.0 (/ l (/ Om l)))) t_1)))
       (cbrt (pow (* 2.0 (* -2.0 (/ (* U (* n (pow l 2.0))) Om))) 1.5))))))

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)) * (U_42_ - U);
	double t_2 = U * (2.0 * n);
	double t_3 = sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + t_1)));
	double tmp;
	if (t_3 <= 2e-148) {
		tmp = sqrt((2.0 * n)) * sqrt((U * (t - (pow(l, 2.0) * (2.0 / Om)))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * ((t - (2.0 * (l / (Om / l)))) + t_1)));
	} else {
		tmp = cbrt(pow((2.0 * (-2.0 * ((U * (n * pow(l, 2.0))) / Om))), 1.5));
	}
	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)) * (U_42_ - U);
	double t_2 = U * (2.0 * n);
	double t_3 = Math.sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + t_1)));
	double tmp;
	if (t_3 <= 2e-148) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((U * (t - (Math.pow(l, 2.0) * (2.0 / Om)))));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_2 * ((t - (2.0 * (l / (Om / l)))) + t_1)));
	} else {
		tmp = Math.cbrt(Math.pow((2.0 * (-2.0 * ((U * (n * Math.pow(l, 2.0))) / Om))), 1.5));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_2 = Float64(U * Float64(2.0 * n))
	t_3 = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_1)))
	tmp = 0.0
	if (t_3 <= 2e-148)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * Float64(t - Float64((l ^ 2.0) * Float64(2.0 / Om))))));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(l / Float64(Om / l)))) + t_1)));
	else
		tmp = cbrt((Float64(2.0 * Float64(-2.0 * Float64(Float64(U * Float64(n * (l ^ 2.0))) / Om))) ^ 1.5));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 2e-148], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t - N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[Power[N[(2.0 * N[(-2.0 * N[(N[(U * N[(n * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 1.99999999999999987e-148

   1. Initial program 14.7%

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

   3. Simplified 14.7%

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

   5. Applied egg-rr 46.8%

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

      1. *-commutative 46.8%

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

      2. fma-def 46.8%

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

      3. +-commutative 46.8%

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

      4. fma-def 46.8%

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

      5. *-commutative 46.8%

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

      6. *-commutative 46.8%

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

      7. associate-*l/ 46.8%

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

      8. associate-*r/ 46.8%

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

   7. Simplified 46.8%

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

   8. Taylor expanded in n around 0 44.5%

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

      1. associate-*r/ 44.5%

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

      2. *-commutative 44.5%

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

      3. associate-*r/ 44.5%

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

   10. Simplified 44.5%

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

   if 1.99999999999999987e-148 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 63.5%

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

      1. associate-/l* 70.5%

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

      2. add-sqr-sqrt 31.5%

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

      3. *-un-lft-identity 31.5%

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

      4. times-frac 31.5%

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

   3. Applied egg-rr 31.5%

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

      1. /-rgt-identity 31.5%

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

      2. associate-*r/ 31.5%

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

      3. rem-square-sqrt 70.5%

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

   5. Simplified 70.5%

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

   if +inf.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.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 2.9%

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

   4. Taylor expanded in Om around inf 6.2%

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

      1. *-commutative 6.2%

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

      2. associate-*l/ 6.2%

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

      3. associate-*r/ 6.2%

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

   6. Simplified 6.2%

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

      1. add-cbrt-cube 6.1%

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

      2. pow1/3 6.0%

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

      3. add-sqr-sqrt 6.0%

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

      4. pow1 6.0%

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

      5. pow1/2 27.2%

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

      6. pow-prod-up 27.2%

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

      7. associate-*r* 27.2%

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

      8. metadata-eval 27.2%

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

   8. Applied egg-rr 27.2%

      \[\leadsto \color{blue}{{\left({\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - {\ell}^{2} \cdot \frac{2}{Om}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   9. Step-by-step derivation

      1. unpow1/3 27.4%

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

      2. associate-*l* 27.4%

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

      3. associate-*l* 28.3%

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

   10. Simplified 28.3%

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

   11. Taylor expanded in t around 0 30.6%

       \[\leadsto \sqrt[3]{{\left(2 \cdot \color{blue}{\left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}\right)}^{1.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{-148}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - {\ell}^{2} \cdot \frac{2}{Om}\right)}\\ \mathbf{elif}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(-2 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om}\right)\right)}^{1.5}}\\ \end{array} \]

Alternative 4: 61.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := U \cdot \left(2 \cdot n\right)\\ t_3 := \sqrt{t_2 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_1\right)}\\ \mathbf{if}\;t_3 \leq 2 \cdot 10^{-148}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\mathsf{fma}\left(-2, \frac{U}{\frac{Om}{{\ell}^{2}}}, U \cdot t\right)}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(-2 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om}\right)\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l Om) 2.0)) (- U* U)))
        (t_2 (* U (* 2.0 n)))
        (t_3 (sqrt (* t_2 (+ (- t (* 2.0 (/ (* l l) Om))) t_1)))))
   (if (<= t_3 2e-148)
     (* (sqrt (* 2.0 n)) (sqrt (fma -2.0 (/ U (/ Om (pow l 2.0))) (* U t))))
     (if (<= t_3 INFINITY)
       (sqrt (* t_2 (+ (- t (* 2.0 (/ l (/ Om l)))) t_1)))
       (cbrt (pow (* 2.0 (* -2.0 (/ (* U (* n (pow l 2.0))) Om))) 1.5))))))

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)) * (U_42_ - U);
	double t_2 = U * (2.0 * n);
	double t_3 = sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + t_1)));
	double tmp;
	if (t_3 <= 2e-148) {
		tmp = sqrt((2.0 * n)) * sqrt(fma(-2.0, (U / (Om / pow(l, 2.0))), (U * t)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * ((t - (2.0 * (l / (Om / l)))) + t_1)));
	} else {
		tmp = cbrt(pow((2.0 * (-2.0 * ((U * (n * pow(l, 2.0))) / Om))), 1.5));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U_42_ - U))
	t_2 = Float64(U * Float64(2.0 * n))
	t_3 = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_1)))
	tmp = 0.0
	if (t_3 <= 2e-148)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(fma(-2.0, Float64(U / Float64(Om / (l ^ 2.0))), Float64(U * t))));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(l / Float64(Om / l)))) + t_1)));
	else
		tmp = cbrt((Float64(2.0 * Float64(-2.0 * Float64(Float64(U * Float64(n * (l ^ 2.0))) / Om))) ^ 1.5));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 2e-148], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(-2.0 * N[(U / N[(Om / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[Power[N[(2.0 * N[(-2.0 * N[(N[(U * N[(n * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 1.99999999999999987e-148

   1. Initial program 14.7%

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

   3. Simplified 14.7%

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

   5. Applied egg-rr 46.8%

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

      1. *-commutative 46.8%

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

      2. fma-def 46.8%

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

      3. +-commutative 46.8%

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

      4. fma-def 46.8%

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

      5. *-commutative 46.8%

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

      6. *-commutative 46.8%

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

      7. associate-*l/ 46.8%

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

      8. associate-*r/ 46.8%

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

   7. Simplified 46.8%

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

   8. Taylor expanded in n around 0 44.5%

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

      1. associate-*r/ 44.5%

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

      2. *-commutative 44.5%

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

      3. associate-*r/ 44.5%

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

   10. Simplified 44.5%

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

   11. Taylor expanded in t around 0 42.2%

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

       1. fma-def 42.2%

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

       2. associate-/l* 44.5%

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

   13. Simplified 44.5%

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

   if 1.99999999999999987e-148 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 63.5%

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

      1. associate-/l* 70.5%

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

      2. add-sqr-sqrt 31.5%

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

      3. *-un-lft-identity 31.5%

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

      4. times-frac 31.5%

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

   3. Applied egg-rr 31.5%

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

      1. /-rgt-identity 31.5%

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

      2. associate-*r/ 31.5%

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

      3. rem-square-sqrt 70.5%

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

   5. Simplified 70.5%

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

   if +inf.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.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 2.9%

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

   4. Taylor expanded in Om around inf 6.2%

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

      1. *-commutative 6.2%

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

      2. associate-*l/ 6.2%

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

      3. associate-*r/ 6.2%

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

   6. Simplified 6.2%

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

      1. add-cbrt-cube 6.1%

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

      2. pow1/3 6.0%

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

      3. add-sqr-sqrt 6.0%

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

      4. pow1 6.0%

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

      5. pow1/2 27.2%

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

      6. pow-prod-up 27.2%

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

      7. associate-*r* 27.2%

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

      8. metadata-eval 27.2%

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

   8. Applied egg-rr 27.2%

      \[\leadsto \color{blue}{{\left({\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - {\ell}^{2} \cdot \frac{2}{Om}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   9. Step-by-step derivation

      1. unpow1/3 27.4%

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

      2. associate-*l* 27.4%

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

      3. associate-*l* 28.3%

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

   10. Simplified 28.3%

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

   11. Taylor expanded in t around 0 30.6%

       \[\leadsto \sqrt[3]{{\left(2 \cdot \color{blue}{\left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}\right)}^{1.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{-148}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\mathsf{fma}\left(-2, \frac{U}{\frac{Om}{{\ell}^{2}}}, U \cdot t\right)}\\ \mathbf{elif}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(-2 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om}\right)\right)}^{1.5}}\\ \end{array} \]

Alternative 5: 61.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \left(n \cdot t_1\right) \cdot \left(U* - U\right)\\ t_3 := U \cdot \left(2 \cdot n\right)\\ t_4 := \sqrt{t_3 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_2\right)}\\ \mathbf{if}\;t_4 \leq 2 \cdot 10^{-148}:\\ \;\;\;\;{\left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left(t_1 \cdot \left(U - U*\right)\right)\right)\right)\right)}^{0.5} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\sqrt{t_3 \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(-2 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om}\right)\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (pow (/ l Om) 2.0))
        (t_2 (* (* n t_1) (- U* U)))
        (t_3 (* U (* 2.0 n)))
        (t_4 (sqrt (* t_3 (+ (- t (* 2.0 (/ (* l l) Om))) t_2)))))
   (if (<= t_4 2e-148)
     (*
      (pow (* U (- t (fma 2.0 (* l (/ l Om)) (* n (* t_1 (- U U*)))))) 0.5)
      (sqrt (* 2.0 n)))
     (if (<= t_4 INFINITY)
       (sqrt (* t_3 (+ (- t (* 2.0 (/ l (/ Om l)))) t_2)))
       (cbrt (pow (* 2.0 (* -2.0 (/ (* U (* n (pow l 2.0))) Om))) 1.5))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = pow((l / Om), 2.0);
	double t_2 = (n * t_1) * (U_42_ - U);
	double t_3 = U * (2.0 * n);
	double t_4 = sqrt((t_3 * ((t - (2.0 * ((l * l) / Om))) + t_2)));
	double tmp;
	if (t_4 <= 2e-148) {
		tmp = pow((U * (t - fma(2.0, (l * (l / Om)), (n * (t_1 * (U - U_42_)))))), 0.5) * sqrt((2.0 * n));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_3 * ((t - (2.0 * (l / (Om / l)))) + t_2)));
	} else {
		tmp = cbrt(pow((2.0 * (-2.0 * ((U * (n * pow(l, 2.0))) / Om))), 1.5));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l / Om) ^ 2.0
	t_2 = Float64(Float64(n * t_1) * Float64(U_42_ - U))
	t_3 = Float64(U * Float64(2.0 * n))
	t_4 = sqrt(Float64(t_3 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + t_2)))
	tmp = 0.0
	if (t_4 <= 2e-148)
		tmp = Float64((Float64(U * Float64(t - fma(2.0, Float64(l * Float64(l / Om)), Float64(n * Float64(t_1 * Float64(U - U_42_)))))) ^ 0.5) * sqrt(Float64(2.0 * n)));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(t_3 * Float64(Float64(t - Float64(2.0 * Float64(l / Float64(Om / l)))) + t_2)));
	else
		tmp = cbrt((Float64(2.0 * Float64(-2.0 * Float64(Float64(U * Float64(n * (l ^ 2.0))) / Om))) ^ 1.5));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(n * t$95$1), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t$95$3 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 2e-148], N[(N[Power[N[(U * N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + N[(n * N[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(t$95$3 * N[(N[(t - N[(2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[Power[N[(2.0 * N[(-2.0 * N[(N[(U * N[(n * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 1.99999999999999987e-148

   1. Initial program 14.7%

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

      1. associate-/l* 14.7%

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

      2. add-sqr-sqrt 6.7%

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

      3. *-un-lft-identity 6.7%

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

      4. times-frac 6.7%

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

   3. Applied egg-rr 6.7%

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

      1. /-rgt-identity 6.7%

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

      2. associate-*r/ 6.7%

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

      3. rem-square-sqrt 14.7%

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

   5. Simplified 14.7%

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

      1. pow1/2 14.7%

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

      2. associate-*l* 35.4%

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

      3. unpow-prod-down 46.8%

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

      4. pow1/2 46.8%

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

      5. associate--l- 46.8%

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

      6. fma-def 46.8%

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

      7. div-inv 46.8%

         \[\leadsto \sqrt{2 \cdot n} \cdot {\left(U \cdot \left(t - \mathsf{fma}\left(2, \color{blue}{\ell \cdot \frac{1}{\frac{Om}{\ell}}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}^{0.5} \]

      8. clear-num 46.8%

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

      9. associate-*l* 46.8%

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

   7. Applied egg-rr 46.8%

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

   if 1.99999999999999987e-148 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 63.5%

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

      1. associate-/l* 70.5%

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

      2. add-sqr-sqrt 31.5%

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

      3. *-un-lft-identity 31.5%

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

      4. times-frac 31.5%

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

   3. Applied egg-rr 31.5%

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

      1. /-rgt-identity 31.5%

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

      2. associate-*r/ 31.5%

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

      3. rem-square-sqrt 70.5%

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

   5. Simplified 70.5%

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

   if +inf.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.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 2.9%

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

   4. Taylor expanded in Om around inf 6.2%

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

      1. *-commutative 6.2%

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

      2. associate-*l/ 6.2%

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

      3. associate-*r/ 6.2%

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

   6. Simplified 6.2%

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

      1. add-cbrt-cube 6.1%

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

      2. pow1/3 6.0%

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

      3. add-sqr-sqrt 6.0%

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

      4. pow1 6.0%

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

      5. pow1/2 27.2%

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

      6. pow-prod-up 27.2%

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

      7. associate-*r* 27.2%

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

      8. metadata-eval 27.2%

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

   8. Applied egg-rr 27.2%

      \[\leadsto \color{blue}{{\left({\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - {\ell}^{2} \cdot \frac{2}{Om}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   9. Step-by-step derivation

      1. unpow1/3 27.4%

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

      2. associate-*l* 27.4%

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

      3. associate-*l* 28.3%

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

   10. Simplified 28.3%

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

   11. Taylor expanded in t around 0 30.6%

       \[\leadsto \sqrt[3]{{\left(2 \cdot \color{blue}{\left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}\right)}^{1.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{-148}:\\ \;\;\;\;{\left(U \cdot \left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)}^{0.5} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(-2 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om}\right)\right)}^{1.5}}\\ \end{array} \]

Alternative 6: 61.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\\ t_2 := U \cdot \left(2 \cdot n\right)\\ t_3 := \sqrt{t_2 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_1\right)}\\ \mathbf{if}\;t_3 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* n (pow (/ l Om) 2.0)) (- U* U)))
        (t_2 (* U (* 2.0 n)))
        (t_3 (sqrt (* t_2 (+ (- t (* 2.0 (/ (* l l) Om))) t_1)))))
   (if (<= t_3 0.0)
     (sqrt (* 2.0 (* U (* n (- t (* 2.0 (/ (pow l 2.0) Om)))))))
     (if (<= t_3 INFINITY)
       (sqrt (* t_2 (+ (- t (* 2.0 (/ l (/ Om l)))) t_1)))
       (pow (* (* 2.0 n) (* U (- t (/ (* 2.0 (pow l 2.0)) Om)))) 0.5)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n * pow((l / Om), 2.0)) * (U_42_ - U);
	double t_2 = U * (2.0 * n);
	double t_3 = sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + t_1)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (pow(l, 2.0) / Om)))))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * ((t - (2.0 * (l / (Om / l)))) + t_1)));
	} else {
		tmp = pow(((2.0 * n) * (U * (t - ((2.0 * pow(l, 2.0)) / Om)))), 0.5);
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n * Math.pow((l / Om), 2.0)) * (U_42_ - U);
	double t_2 = U * (2.0 * n);
	double t_3 = Math.sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + t_1)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (Math.pow(l, 2.0) / Om)))))));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_2 * ((t - (2.0 * (l / (Om / l)))) + t_1)));
	} else {
		tmp = Math.pow(((2.0 * n) * (U * (t - ((2.0 * Math.pow(l, 2.0)) / Om)))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (n * math.pow((l / Om), 2.0)) * (U_42_ - U)
	t_2 = U * (2.0 * n)
	t_3 = math.sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + t_1)))
	tmp = 0
	if t_3 <= 0.0:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (math.pow(l, 2.0) / Om)))))))
	elif t_3 <= math.inf:
		tmp = math.sqrt((t_2 * ((t - (2.0 * (l / (Om / l)))) + t_1)))
	else:
		tmp = math.pow(((2.0 * n) * (U * (t - ((2.0 * math.pow(l, 2.0)) / Om)))), 0.5)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 12.4%

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

   3. Simplified 12.4%

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

   4. Taylor expanded in n around 0 33.8%

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

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

   1. Initial program 63.7%

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

      1. associate-/l* 70.6%

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

      2. add-sqr-sqrt 31.9%

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

      3. *-un-lft-identity 31.9%

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

      4. times-frac 31.8%

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

   3. Applied egg-rr 31.8%

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

      1. /-rgt-identity 31.8%

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

      2. associate-*r/ 31.9%

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

      3. rem-square-sqrt 70.6%

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

   5. Simplified 70.6%

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

   if +inf.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.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 2.9%

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

   5. Applied egg-rr 0.5%

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

      1. *-commutative 0.5%

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

      2. fma-def 0.5%

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

      3. +-commutative 0.5%

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

      4. fma-def 0.5%

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

      5. *-commutative 0.5%

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

      6. *-commutative 0.5%

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

      7. associate-*l/ 0.5%

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

      8. associate-*r/ 0.5%

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

   7. Simplified 0.5%

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

   8. Taylor expanded in n around 0 3.5%

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

      1. associate-*r/ 3.5%

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

      2. *-commutative 3.5%

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

      3. associate-*r/ 3.5%

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

   10. Simplified 3.5%

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

       1. sqrt-unprod 7.1%

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

       2. pow1/2 28.4%

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

       3. associate-*r/ 28.4%

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

   12. Applied egg-rr 28.4%

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

4. Final simplification 58.8%

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

Alternative 7: 55.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq -0.95 \lor \neg \left(U \leq 5.3 \cdot 10^{-5}\right):\\ \;\;\;\;{\left(\left(t - {\ell}^{2} \cdot \frac{2}{Om}\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t + \frac{\ell}{\frac{Om}{\ell}} \cdot -2\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= U -0.95) (not (<= U 5.3e-5)))
   (pow (* (- t (* (pow l 2.0) (/ 2.0 Om))) (* 2.0 (* n U))) 0.5)
   (sqrt
    (*
     (* 2.0 n)
     (*
      U
      (+
       (+ t (* (/ l (/ Om l)) -2.0))
       (* n (* (pow (/ l Om) 2.0) (- U* U)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((U <= -0.95) || !(U <= 5.3e-5)) {
		tmp = pow(((t - (pow(l, 2.0) * (2.0 / Om))) * (2.0 * (n * U))), 0.5);
	} else {
		tmp = sqrt(((2.0 * n) * (U * ((t + ((l / (Om / l)) * -2.0)) + (n * (pow((l / Om), 2.0) * (U_42_ - 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 ((u <= (-0.95d0)) .or. (.not. (u <= 5.3d-5))) then
        tmp = ((t - ((l ** 2.0d0) * (2.0d0 / om))) * (2.0d0 * (n * u))) ** 0.5d0
    else
        tmp = sqrt(((2.0d0 * n) * (u * ((t + ((l / (om / l)) * (-2.0d0))) + (n * (((l / om) ** 2.0d0) * (u_42 - 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 ((U <= -0.95) || !(U <= 5.3e-5)) {
		tmp = Math.pow(((t - (Math.pow(l, 2.0) * (2.0 / Om))) * (2.0 * (n * U))), 0.5);
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * ((t + ((l / (Om / l)) * -2.0)) + (n * (Math.pow((l / Om), 2.0) * (U_42_ - U)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (U <= -0.95) or not (U <= 5.3e-5):
		tmp = math.pow(((t - (math.pow(l, 2.0) * (2.0 / Om))) * (2.0 * (n * U))), 0.5)
	else:
		tmp = math.sqrt(((2.0 * n) * (U * ((t + ((l / (Om / l)) * -2.0)) + (n * (math.pow((l / Om), 2.0) * (U_42_ - U)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((U <= -0.95) || !(U <= 5.3e-5))
		tmp = Float64(Float64(t - Float64((l ^ 2.0) * Float64(2.0 / Om))) * Float64(2.0 * Float64(n * U))) ^ 0.5;
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(Float64(t + Float64(Float64(l / Float64(Om / l)) * -2.0)) + Float64(n * Float64((Float64(l / Om) ^ 2.0) * Float64(U_42_ - U)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((U <= -0.95) || ~((U <= 5.3e-5)))
		tmp = ((t - ((l ^ 2.0) * (2.0 / Om))) * (2.0 * (n * U))) ^ 0.5;
	else
		tmp = sqrt(((2.0 * n) * (U * ((t + ((l / (Om / l)) * -2.0)) + (n * (((l / Om) ^ 2.0) * (U_42_ - U)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[U, -0.95], N[Not[LessEqual[U, 5.3e-5]], $MachinePrecision]], N[Power[N[(N[(t - N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(N[(t + N[(N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] + N[(n * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -0.94999999999999996 or 5.3000000000000001e-5 < U

   1. Initial program 54.4%

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

   3. Simplified 56.5%

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

   4. Taylor expanded in Om around inf 49.1%

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

      1. *-commutative 49.1%

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

      2. associate-*l/ 49.1%

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

      3. associate-*r/ 49.1%

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

   6. Simplified 49.1%

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

      1. pow1/2 55.2%

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

      2. associate-*r* 55.2%

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

   8. Applied egg-rr 55.2%

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

   if -0.94999999999999996 < U < 5.3000000000000001e-5

   1. Initial program 40.9%

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

   3. Simplified 50.5%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -0.95 \lor \neg \left(U \leq 5.3 \cdot 10^{-5}\right):\\ \;\;\;\;{\left(\left(t - {\ell}^{2} \cdot \frac{2}{Om}\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t + \frac{\ell}{\frac{Om}{\ell}} \cdot -2\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)}\\ \end{array} \]

Alternative 8: 54.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt
  (*
   (* U (* 2.0 n))
   (- (- 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(((U * (2.0 * n)) * ((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(((u * (2.0d0 * n)) * ((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(((U * (2.0 * n)) * ((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(((U * (2.0 * n)) * ((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(U * Float64(2.0 * n)) * Float64(Float64(t - Float64(2.0 * Float64(l * Float64(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(((U * (2.0 * n)) * ((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[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $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]

Derivation

1. Initial program 46.5%

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

   1. associate-*l/ 51.8%

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

3. Applied egg-rr 51.8%

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

4. Final simplification 51.8%

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

Alternative 9: 48.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - {\ell}^{2} \cdot \frac{2}{Om}\\ \mathbf{if}\;Om \leq -2.75 \cdot 10^{+159}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(t_1 \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (- t (* (pow l 2.0) (/ 2.0 Om)))))
   (if (<= Om -2.75e+159)
     (sqrt (* 2.0 (* n (* U t_1))))
     (pow (* t_1 (* 2.0 (* n U))) 0.5))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t - (pow(l, 2.0) * (2.0 / Om));
	double tmp;
	if (Om <= -2.75e+159) {
		tmp = sqrt((2.0 * (n * (U * t_1))));
	} else {
		tmp = pow((t_1 * (2.0 * (n * U))), 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) :: t_1
    real(8) :: tmp
    t_1 = t - ((l ** 2.0d0) * (2.0d0 / om))
    if (om <= (-2.75d+159)) then
        tmp = sqrt((2.0d0 * (n * (u * t_1))))
    else
        tmp = (t_1 * (2.0d0 * (n * u))) ** 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 t_1 = t - (Math.pow(l, 2.0) * (2.0 / Om));
	double tmp;
	if (Om <= -2.75e+159) {
		tmp = Math.sqrt((2.0 * (n * (U * t_1))));
	} else {
		tmp = Math.pow((t_1 * (2.0 * (n * U))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = t - (math.pow(l, 2.0) * (2.0 / Om))
	tmp = 0
	if Om <= -2.75e+159:
		tmp = math.sqrt((2.0 * (n * (U * t_1))))
	else:
		tmp = math.pow((t_1 * (2.0 * (n * U))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(t - Float64((l ^ 2.0) * Float64(2.0 / Om)))
	tmp = 0.0
	if (Om <= -2.75e+159)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * t_1))));
	else
		tmp = Float64(t_1 * Float64(2.0 * Float64(n * U))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = t - ((l ^ 2.0) * (2.0 / Om));
	tmp = 0.0;
	if (Om <= -2.75e+159)
		tmp = sqrt((2.0 * (n * (U * t_1))));
	else
		tmp = (t_1 * (2.0 * (n * U))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(t - N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Om, -2.75e+159], N[Sqrt[N[(2.0 * N[(n * N[(U * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(t$95$1 * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if Om < -2.7499999999999999e159

   1. Initial program 45.5%

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

      1. associate-/l* 55.9%

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

      2. add-sqr-sqrt 27.6%

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

      3. *-un-lft-identity 27.6%

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

      4. times-frac 27.5%

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

   3. Applied egg-rr 27.5%

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

      1. /-rgt-identity 27.5%

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

      2. associate-*r/ 27.6%

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

      3. rem-square-sqrt 55.9%

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

   5. Simplified 55.9%

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

   6. Taylor expanded in n around 0 48.1%

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

      1. associate-*r* 42.8%

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

      2. associate-*r/ 42.8%

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

      3. *-commutative 42.8%

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

      4. associate-*r/ 42.8%

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

      5. *-commutative 42.8%

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

      6. associate-*l* 55.8%

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

   8. Simplified 55.8%

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

   if -2.7499999999999999e159 < Om

   1. Initial program 46.7%

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

   3. Simplified 47.3%

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

   4. Taylor expanded in Om around inf 36.8%

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

      1. *-commutative 36.8%

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

      2. associate-*l/ 36.8%

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

      3. associate-*r/ 36.8%

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

   6. Simplified 36.8%

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

      1. pow1/2 42.5%

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

      2. associate-*r* 42.5%

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

   8. Applied egg-rr 42.5%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -2.75 \cdot 10^{+159}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - {\ell}^{2} \cdot \frac{2}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(t - {\ell}^{2} \cdot \frac{2}{Om}\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 10: 48.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq -7 \cdot 10^{+158}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(t - {\ell}^{2} \cdot \frac{2}{Om}\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om -7e+158)
   (pow (* (* 2.0 n) (* U (- t (/ (* 2.0 (pow l 2.0)) Om)))) 0.5)
   (pow (* (- t (* (pow l 2.0) (/ 2.0 Om))) (* 2.0 (* n U))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= -7e+158) {
		tmp = pow(((2.0 * n) * (U * (t - ((2.0 * pow(l, 2.0)) / Om)))), 0.5);
	} else {
		tmp = pow(((t - (pow(l, 2.0) * (2.0 / Om))) * (2.0 * (n * U))), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (om <= (-7d+158)) then
        tmp = ((2.0d0 * n) * (u * (t - ((2.0d0 * (l ** 2.0d0)) / om)))) ** 0.5d0
    else
        tmp = ((t - ((l ** 2.0d0) * (2.0d0 / om))) * (2.0d0 * (n * u))) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= -7e+158) {
		tmp = Math.pow(((2.0 * n) * (U * (t - ((2.0 * Math.pow(l, 2.0)) / Om)))), 0.5);
	} else {
		tmp = Math.pow(((t - (Math.pow(l, 2.0) * (2.0 / Om))) * (2.0 * (n * U))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if Om <= -7e+158:
		tmp = math.pow(((2.0 * n) * (U * (t - ((2.0 * math.pow(l, 2.0)) / Om)))), 0.5)
	else:
		tmp = math.pow(((t - (math.pow(l, 2.0) * (2.0 / Om))) * (2.0 * (n * U))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (Om <= -7e+158)
		tmp = Float64(Float64(2.0 * n) * Float64(U * Float64(t - Float64(Float64(2.0 * (l ^ 2.0)) / Om)))) ^ 0.5;
	else
		tmp = Float64(Float64(t - Float64((l ^ 2.0) * Float64(2.0 / Om))) * Float64(2.0 * Float64(n * U))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (Om <= -7e+158)
		tmp = ((2.0 * n) * (U * (t - ((2.0 * (l ^ 2.0)) / Om)))) ^ 0.5;
	else
		tmp = ((t - ((l ^ 2.0) * (2.0 / Om))) * (2.0 * (n * U))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[Om, -7e+158], N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(t - N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Power[N[(N[(t - N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -7.0000000000000003e158

   1. Initial program 45.5%

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

   3. Simplified 46.9%

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

   5. Applied egg-rr 52.8%

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

      1. *-commutative 52.8%

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

      2. fma-def 52.8%

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

      3. +-commutative 52.8%

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

      4. fma-def 52.8%

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

      5. *-commutative 52.8%

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

      6. *-commutative 52.8%

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

      7. associate-*l/ 52.8%

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

      8. associate-*r/ 52.8%

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

   7. Simplified 52.8%

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

   8. Taylor expanded in n around 0 50.1%

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

      1. associate-*r/ 50.1%

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

      2. *-commutative 50.1%

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

      3. associate-*r/ 50.1%

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

   10. Simplified 50.1%

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

       1. sqrt-unprod 55.8%

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

       2. pow1/2 55.8%

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

       3. associate-*r/ 55.9%

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

   12. Applied egg-rr 55.9%

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

   if -7.0000000000000003e158 < Om

   1. Initial program 46.7%

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

   3. Simplified 47.3%

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

   4. Taylor expanded in Om around inf 36.8%

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

      1. *-commutative 36.8%

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

      2. associate-*l/ 36.8%

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

      3. associate-*r/ 36.8%

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

   6. Simplified 36.8%

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

      1. pow1/2 42.5%

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

      2. associate-*r* 42.5%

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

   8. Applied egg-rr 42.5%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -7 \cdot 10^{+158}:\\ \;\;\;\;{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \frac{2 \cdot {\ell}^{2}}{Om}\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(t - {\ell}^{2} \cdot \frac{2}{Om}\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 11: 44.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U* \leq 4.6 \cdot 10^{-177}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - {\ell}^{2} \cdot \frac{2}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U* 4.6e-177)
   (sqrt (* 2.0 (* U (* n (- t (* 2.0 (/ (pow l 2.0) Om)))))))
   (sqrt (* 2.0 (* n (* U (- t (* (pow l 2.0) (/ 2.0 Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= 4.6e-177) {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (pow(l, 2.0) / Om)))))));
	} else {
		tmp = sqrt((2.0 * (n * (U * (t - (pow(l, 2.0) * (2.0 / Om)))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u_42 <= 4.6d-177) then
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * ((l ** 2.0d0) / om)))))))
    else
        tmp = sqrt((2.0d0 * (n * (u * (t - ((l ** 2.0d0) * (2.0d0 / om)))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= 4.6e-177) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (Math.pow(l, 2.0) / Om)))))));
	} else {
		tmp = Math.sqrt((2.0 * (n * (U * (t - (Math.pow(l, 2.0) * (2.0 / Om)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U_42_ <= 4.6e-177:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (math.pow(l, 2.0) / Om)))))))
	else:
		tmp = math.sqrt((2.0 * (n * (U * (t - (math.pow(l, 2.0) * (2.0 / Om)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= 4.6e-177)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64((l ^ 2.0) / Om)))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * Float64(t - Float64((l ^ 2.0) * Float64(2.0 / Om)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U_42_ <= 4.6e-177)
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * ((l ^ 2.0) / Om)))))));
	else
		tmp = sqrt((2.0 * (n * (U * (t - ((l ^ 2.0) * (2.0 / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U$42$, 4.6e-177], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(n * N[(U * N[(t - N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < 4.60000000000000044e-177

   1. Initial program 47.2%

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

   3. Simplified 49.5%

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

   4. Taylor expanded in n around 0 42.7%

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

   if 4.60000000000000044e-177 < U*

   1. Initial program 45.6%

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

      1. associate-/l* 49.2%

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

      2. add-sqr-sqrt 20.2%

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

      3. *-un-lft-identity 20.2%

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

      4. times-frac 20.2%

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

   3. Applied egg-rr 20.2%

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

      1. /-rgt-identity 20.2%

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

      2. associate-*r/ 20.2%

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

      3. rem-square-sqrt 49.2%

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

   5. Simplified 49.2%

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

   6. Taylor expanded in n around 0 28.5%

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

      1. associate-*r* 34.5%

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

      2. associate-*r/ 34.5%

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

      3. *-commutative 34.5%

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

      4. associate-*r/ 34.5%

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

      5. *-commutative 34.5%

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

      6. associate-*l* 37.3%

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

   8. Simplified 37.3%

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

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq 4.6 \cdot 10^{-177}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t - {\ell}^{2} \cdot \frac{2}{Om}\right)\right)\right)}\\ \end{array} \]

Alternative 12: 44.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{-70}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 4.8e-70)
   (sqrt (* 2.0 (* (- t (* 2.0 (/ (pow l 2.0) Om))) (* n U))))
   (sqrt (fabs (* 2.0 (* U (* n t)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 4.8e-70) {
		tmp = sqrt((2.0 * ((t - (2.0 * (pow(l, 2.0) / Om))) * (n * U))));
	} else {
		tmp = sqrt(fabs((2.0 * (U * (n * t)))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= 4.8d-70) then
        tmp = sqrt((2.0d0 * ((t - (2.0d0 * ((l ** 2.0d0) / om))) * (n * u))))
    else
        tmp = sqrt(abs((2.0d0 * (u * (n * t)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 4.8e-70) {
		tmp = Math.sqrt((2.0 * ((t - (2.0 * (Math.pow(l, 2.0) / Om))) * (n * U))));
	} else {
		tmp = Math.sqrt(Math.abs((2.0 * (U * (n * t)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= 4.8e-70:
		tmp = math.sqrt((2.0 * ((t - (2.0 * (math.pow(l, 2.0) / Om))) * (n * U))))
	else:
		tmp = math.sqrt(math.fabs((2.0 * (U * (n * t)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 4.8e-70)
		tmp = sqrt(Float64(2.0 * Float64(Float64(t - Float64(2.0 * Float64((l ^ 2.0) / Om))) * Float64(n * U))));
	else
		tmp = sqrt(abs(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 (t <= 4.8e-70)
		tmp = sqrt((2.0 * ((t - (2.0 * ((l ^ 2.0) / Om))) * (n * U))));
	else
		tmp = sqrt(abs((2.0 * (U * (n * t)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 4.8e-70], N[Sqrt[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]], $MachinePrecision], N[Sqrt[N[Abs[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.8000000000000002e-70

   1. Initial program 46.3%

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

   3. Simplified 48.2%

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

   4. Taylor expanded in Om around inf 37.4%

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

   if 4.8000000000000002e-70 < t

   1. Initial program 46.9%

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

   3. Simplified 49.5%

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

   4. Taylor expanded in t around inf 40.7%

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

      1. associate-*r* 40.7%

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

   6. Simplified 40.7%

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

      1. add-sqr-sqrt 40.7%

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

      2. pow1/2 40.7%

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

      3. pow1/2 47.2%

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

      4. pow-prod-down 35.6%

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

      5. pow2 35.6%

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

      6. associate-*l* 35.6%

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

   8. Applied egg-rr 35.6%

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

      1. unpow1/2 35.6%

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

      2. unpow2 35.6%

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

      3. rem-sqrt-square 47.9%

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

   10. Simplified 47.9%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{-70}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right|}\\ \end{array} \]

Alternative 13: 41.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 9.4 \cdot 10^{+30}:\\ \;\;\;\;{\left(t \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \left(\frac{U}{Om} \cdot \left(n \cdot {\ell}^{2}\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 9.4e+30)
   (pow (* t (* 2.0 (* n U))) 0.5)
   (pow (* -4.0 (* (/ U Om) (* n (pow l 2.0)))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 9.4e+30) {
		tmp = pow((t * (2.0 * (n * U))), 0.5);
	} else {
		tmp = pow((-4.0 * ((U / Om) * (n * pow(l, 2.0)))), 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 <= 9.4d+30) then
        tmp = (t * (2.0d0 * (n * u))) ** 0.5d0
    else
        tmp = ((-4.0d0) * ((u / om) * (n * (l ** 2.0d0)))) ** 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 <= 9.4e+30) {
		tmp = Math.pow((t * (2.0 * (n * U))), 0.5);
	} else {
		tmp = Math.pow((-4.0 * ((U / Om) * (n * Math.pow(l, 2.0)))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 9.4e+30:
		tmp = math.pow((t * (2.0 * (n * U))), 0.5)
	else:
		tmp = math.pow((-4.0 * ((U / Om) * (n * math.pow(l, 2.0)))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 9.4e+30)
		tmp = Float64(t * Float64(2.0 * Float64(n * U))) ^ 0.5;
	else
		tmp = Float64(-4.0 * Float64(Float64(U / Om) * Float64(n * (l ^ 2.0)))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 9.4e+30)
		tmp = (t * (2.0 * (n * U))) ^ 0.5;
	else
		tmp = (-4.0 * ((U / Om) * (n * (l ^ 2.0)))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 9.4e+30], N[Power[N[(t * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Power[N[(-4.0 * N[(N[(U / Om), $MachinePrecision] * N[(n * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 9.39999999999999979e30

   1. Initial program 52.4%

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

   3. Simplified 50.9%

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

   4. Taylor expanded in t around inf 37.9%

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

      1. associate-*r* 39.7%

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

      2. *-commutative 39.7%

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

   6. Simplified 39.7%

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

      1. pow1/2 41.2%

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

      2. associate-*r* 41.2%

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

   8. Applied egg-rr 41.2%

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

   if 9.39999999999999979e30 < l

   1. Initial program 21.9%

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

   3. Simplified 31.7%

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

   4. Taylor expanded in Om around inf 20.2%

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

      1. *-commutative 20.2%

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

      2. associate-*l/ 20.2%

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

      3. associate-*r/ 20.2%

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

   6. Simplified 20.2%

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

   7. Taylor expanded in t around 0 19.6%

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

      1. associate-/l* 19.6%

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

      2. *-commutative 19.6%

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

   9. Simplified 19.6%

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

       1. pow1/2 36.3%

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

       2. associate-*r* 36.3%

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

       3. metadata-eval 36.3%

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

       4. associate-/r/ 34.1%

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

   11. Applied egg-rr 34.1%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9.4 \cdot 10^{+30}:\\ \;\;\;\;{\left(t \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(-4 \cdot \left(\frac{U}{Om} \cdot \left(n \cdot {\ell}^{2}\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 14: 38.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{-80}:\\ \;\;\;\;{\left(t \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.5e-80)
   (pow (* t (* 2.0 (* n U))) 0.5)
   (sqrt (fabs (* 2.0 (* U (* n t)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.5e-80) {
		tmp = pow((t * (2.0 * (n * U))), 0.5);
	} else {
		tmp = sqrt(fabs((2.0 * (U * (n * t)))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 1.5d-80) then
        tmp = (t * (2.0d0 * (n * u))) ** 0.5d0
    else
        tmp = sqrt(abs((2.0d0 * (u * (n * t)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.5e-80) {
		tmp = Math.pow((t * (2.0 * (n * U))), 0.5);
	} else {
		tmp = Math.sqrt(Math.abs((2.0 * (U * (n * t)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 1.5e-80:
		tmp = math.pow((t * (2.0 * (n * U))), 0.5)
	else:
		tmp = math.sqrt(math.fabs((2.0 * (U * (n * t)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.5e-80)
		tmp = Float64(t * Float64(2.0 * Float64(n * U))) ^ 0.5;
	else
		tmp = sqrt(abs(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 (l <= 1.5e-80)
		tmp = (t * (2.0 * (n * U))) ^ 0.5;
	else
		tmp = sqrt(abs((2.0 * (U * (n * t)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.5e-80], N[Power[N[(t * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[Abs[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.50000000000000004e-80

   1. Initial program 51.7%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in t around inf 38.2%

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

      1. associate-*r* 40.2%

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

      2. *-commutative 40.2%

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

   6. Simplified 40.2%

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

      1. pow1/2 42.0%

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

      2. associate-*r* 42.0%

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

   8. Applied egg-rr 42.0%

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

   if 1.50000000000000004e-80 < l

   1. Initial program 33.9%

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

   3. Simplified 31.6%

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

   4. Taylor expanded in t around inf 18.8%

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

      1. associate-*r* 18.8%

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

   6. Simplified 18.8%

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

      1. add-sqr-sqrt 18.8%

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

      2. pow1/2 18.8%

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

      3. pow1/2 25.7%

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

      4. pow-prod-down 21.7%

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

      5. pow2 21.7%

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

      6. associate-*l* 21.7%

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

   8. Applied egg-rr 21.7%

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

      1. unpow1/2 21.7%

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

      2. unpow2 21.7%

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

      3. rem-sqrt-square 27.0%

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

   10. Simplified 27.0%

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

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{-80}:\\ \;\;\;\;{\left(t \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right|}\\ \end{array} \]

Alternative 15: 41.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq 1.5 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot t} \cdot \sqrt{2 \cdot U}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U 1.5e-308)
   (sqrt (fabs (* 2.0 (* U (* n t)))))
   (* (sqrt (* n t)) (sqrt (* 2.0 U)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 1.5e-308) {
		tmp = sqrt(fabs((2.0 * (U * (n * t)))));
	} else {
		tmp = sqrt((n * t)) * sqrt((2.0 * 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 (u <= 1.5d-308) then
        tmp = sqrt(abs((2.0d0 * (u * (n * t)))))
    else
        tmp = sqrt((n * t)) * sqrt((2.0d0 * 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 (U <= 1.5e-308) {
		tmp = Math.sqrt(Math.abs((2.0 * (U * (n * t)))));
	} else {
		tmp = Math.sqrt((n * t)) * Math.sqrt((2.0 * U));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= 1.5e-308:
		tmp = math.sqrt(math.fabs((2.0 * (U * (n * t)))))
	else:
		tmp = math.sqrt((n * t)) * math.sqrt((2.0 * U))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= 1.5e-308)
		tmp = sqrt(abs(Float64(2.0 * Float64(U * Float64(n * t)))));
	else
		tmp = Float64(sqrt(Float64(n * t)) * sqrt(Float64(2.0 * U)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= 1.5e-308)
		tmp = sqrt(abs((2.0 * (U * (n * t)))));
	else
		tmp = sqrt((n * t)) * sqrt((2.0 * U));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, 1.5e-308], N[Sqrt[N[Abs[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n * t), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 1.4999999999999999e-308

   1. Initial program 46.5%

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

   3. Simplified 49.9%

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

   4. Taylor expanded in t around inf 34.1%

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

      1. associate-*r* 34.1%

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

   6. Simplified 34.1%

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

      1. add-sqr-sqrt 34.1%

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

      2. pow1/2 34.1%

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

      3. pow1/2 38.0%

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

      4. pow-prod-down 27.3%

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

      5. pow2 27.3%

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

      6. associate-*l* 27.3%

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

   8. Applied egg-rr 27.3%

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

      1. unpow1/2 27.3%

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

      2. unpow2 27.3%

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

      3. rem-sqrt-square 38.7%

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

   10. Simplified 38.7%

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

   if 1.4999999999999999e-308 < U

   1. Initial program 46.6%

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

   3. Simplified 46.0%

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

   4. Taylor expanded in t around inf 31.0%

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

      1. associate-*r* 31.0%

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

   6. Simplified 31.0%

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

      1. pow1/2 32.8%

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

      2. *-commutative 32.8%

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

      3. unpow-prod-down 40.9%

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

      4. pow1/2 40.0%

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

      5. pow1/2 40.0%

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

   8. Applied egg-rr 40.0%

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

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.5 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot t} \cdot \sqrt{2 \cdot U}\\ \end{array} \]

Alternative 16: 37.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U* \leq 1.04 \cdot 10^{-171}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U* 1.04e-171)
   (pow (* 2.0 (* U (* n t))) 0.5)
   (sqrt (* 2.0 (* n (* U t))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= 1.04e-171) {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = sqrt((2.0 * (n * (U * t))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u_42 <= 1.04d-171) then
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    else
        tmp = sqrt((2.0d0 * (n * (u * t))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= 1.04e-171) {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	} else {
		tmp = Math.sqrt((2.0 * (n * (U * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U_42_ <= 1.04e-171:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	else:
		tmp = math.sqrt((2.0 * (n * (U * t))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= 1.04e-171)
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	else
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U_42_ <= 1.04e-171)
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	else
		tmp = sqrt((2.0 * (n * (U * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U$42$, 1.04e-171], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < 1.0399999999999999e-171

   1. Initial program 46.5%

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

   3. Simplified 47.0%

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

   4. Taylor expanded in t around inf 37.4%

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

      1. associate-*r* 37.4%

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

   6. Simplified 37.4%

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

      1. pow1/2 40.8%

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

      2. associate-*l* 40.8%

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

   8. Applied egg-rr 40.8%

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

   if 1.0399999999999999e-171 < U*

   1. Initial program 46.5%

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

      1. associate-/l* 50.1%

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

      2. add-sqr-sqrt 20.6%

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

      3. *-un-lft-identity 20.6%

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

      4. times-frac 20.6%

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

   3. Applied egg-rr 20.6%

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

      1. /-rgt-identity 20.6%

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

      2. associate-*r/ 20.6%

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

      3. rem-square-sqrt 50.1%

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

   5. Simplified 50.1%

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

   6. Taylor expanded in t around inf 25.8%

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

      1. associate-*r* 31.1%

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

      2. *-commutative 31.1%

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

      3. associate-*r* 33.0%

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

   8. Simplified 33.0%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq 1.04 \cdot 10^{-171}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 17: 38.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.5 \cdot 10^{-82}:\\ \;\;\;\;{\left(t \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 4.5e-82)
   (pow (* t (* 2.0 (* n U))) 0.5)
   (pow (* 2.0 (* U (* n t))) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 4.5e-82) {
		tmp = pow((t * (2.0 * (n * U))), 0.5);
	} else {
		tmp = pow((2.0 * (U * (n * t))), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (l <= 4.5d-82) then
        tmp = (t * (2.0d0 * (n * u))) ** 0.5d0
    else
        tmp = (2.0d0 * (u * (n * t))) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 4.5e-82) {
		tmp = Math.pow((t * (2.0 * (n * U))), 0.5);
	} else {
		tmp = Math.pow((2.0 * (U * (n * t))), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 4.5e-82:
		tmp = math.pow((t * (2.0 * (n * U))), 0.5)
	else:
		tmp = math.pow((2.0 * (U * (n * t))), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 4.5e-82)
		tmp = Float64(t * Float64(2.0 * Float64(n * U))) ^ 0.5;
	else
		tmp = Float64(2.0 * Float64(U * Float64(n * t))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 4.5e-82)
		tmp = (t * (2.0 * (n * U))) ^ 0.5;
	else
		tmp = (2.0 * (U * (n * t))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 4.5e-82], N[Power[N[(t * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Power[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.4999999999999998e-82

   1. Initial program 51.7%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in t around inf 38.2%

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

      1. associate-*r* 40.2%

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

      2. *-commutative 40.2%

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

   6. Simplified 40.2%

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

      1. pow1/2 42.0%

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

      2. associate-*r* 42.0%

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

   8. Applied egg-rr 42.0%

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

   if 4.4999999999999998e-82 < l

   1. Initial program 33.9%

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

   3. Simplified 31.6%

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

   4. Taylor expanded in t around inf 18.8%

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

      1. associate-*r* 18.8%

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

   6. Simplified 18.8%

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

      1. pow1/2 25.7%

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

      2. associate-*l* 25.7%

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

   8. Applied egg-rr 25.7%

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

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.5 \cdot 10^{-82}:\\ \;\;\;\;{\left(t \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 18: 36.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq -1.02 \cdot 10^{+158}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om -1.02e+158)
   (sqrt (* 2.0 (* n (* U t))))
   (sqrt (* 2.0 (* t (* n U))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= -1.02e+158) {
		tmp = sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = sqrt((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 (om <= (-1.02d+158)) then
        tmp = sqrt((2.0d0 * (n * (u * t))))
    else
        tmp = sqrt((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 (Om <= -1.02e+158) {
		tmp = Math.sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = Math.sqrt((2.0 * (t * (n * U))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if Om <= -1.02e+158:
		tmp = math.sqrt((2.0 * (n * (U * t))))
	else:
		tmp = math.sqrt((2.0 * (t * (n * U))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (Om <= -1.02e+158)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * t))));
	else
		tmp = sqrt(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 (Om <= -1.02e+158)
		tmp = sqrt((2.0 * (n * (U * t))));
	else
		tmp = sqrt((2.0 * (t * (n * U))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[Om, -1.02e+158], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -1.02e158

   1. Initial program 45.5%

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

      1. associate-/l* 55.9%

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

      2. add-sqr-sqrt 27.6%

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

      3. *-un-lft-identity 27.6%

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

      4. times-frac 27.5%

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

   3. Applied egg-rr 27.5%

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

      1. /-rgt-identity 27.5%

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

      2. associate-*r/ 27.6%

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

      3. rem-square-sqrt 55.9%

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

   5. Simplified 55.9%

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

   6. Taylor expanded in t around inf 45.4%

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

      1. associate-*r* 40.1%

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

      2. *-commutative 40.1%

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

      3. associate-*r* 53.2%

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

   8. Simplified 53.2%

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

   if -1.02e158 < Om

   1. Initial program 46.7%

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

   3. Simplified 47.3%

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

   4. Taylor expanded in t around inf 30.4%

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

      1. associate-*r* 32.1%

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

      2. *-commutative 32.1%

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

   6. Simplified 32.1%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1.02 \cdot 10^{+158}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \end{array} \]

Alternative 19: 36.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U* \leq 2.42 \cdot 10^{-169}:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U* 2.42e-169)
   (sqrt (* (* n t) (* 2.0 U)))
   (sqrt (* 2.0 (* n (* U t))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= 2.42e-169) {
		tmp = sqrt(((n * t) * (2.0 * U)));
	} else {
		tmp = sqrt((2.0 * (n * (U * t))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u_42 <= 2.42d-169) then
        tmp = sqrt(((n * t) * (2.0d0 * u)))
    else
        tmp = sqrt((2.0d0 * (n * (u * t))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U_42_ <= 2.42e-169) {
		tmp = Math.sqrt(((n * t) * (2.0 * U)));
	} else {
		tmp = Math.sqrt((2.0 * (n * (U * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U_42_ <= 2.42e-169:
		tmp = math.sqrt(((n * t) * (2.0 * U)))
	else:
		tmp = math.sqrt((2.0 * (n * (U * t))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U_42_ <= 2.42e-169)
		tmp = sqrt(Float64(Float64(n * t) * Float64(2.0 * U)));
	else
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U_42_ <= 2.42e-169)
		tmp = sqrt(((n * t) * (2.0 * U)));
	else
		tmp = sqrt((2.0 * (n * (U * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U$42$, 2.42e-169], N[Sqrt[N[(N[(n * t), $MachinePrecision] * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U* < 2.42000000000000005e-169

   1. Initial program 46.5%

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

   3. Simplified 47.0%

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

   4. Taylor expanded in t around inf 37.4%

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

      1. associate-*r* 37.4%

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

   6. Simplified 37.4%

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

   if 2.42000000000000005e-169 < U*

   1. Initial program 46.5%

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

      1. associate-/l* 50.1%

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

      2. add-sqr-sqrt 20.6%

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

      3. *-un-lft-identity 20.6%

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

      4. times-frac 20.6%

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

   3. Applied egg-rr 20.6%

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

      1. /-rgt-identity 20.6%

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

      2. associate-*r/ 20.6%

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

      3. rem-square-sqrt 50.1%

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

   5. Simplified 50.1%

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

   6. Taylor expanded in t around inf 25.8%

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

      1. associate-*r* 31.1%

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

      2. *-commutative 31.1%

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

      3. associate-*r* 33.0%

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

   8. Simplified 33.0%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq 2.42 \cdot 10^{-169}:\\ \;\;\;\;\sqrt{\left(n \cdot t\right) \cdot \left(2 \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \end{array} \]

Alternative 20: 36.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.5%

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

   1. associate-/l* 51.8%

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

   2. add-sqr-sqrt 23.0%

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

   3. *-un-lft-identity 23.0%

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

   4. times-frac 23.0%

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

3. Applied egg-rr 23.0%

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

   1. /-rgt-identity 23.0%

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

   2. associate-*r/ 23.0%

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

   3. rem-square-sqrt 51.8%

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

5. Simplified 51.8%

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

6. Taylor expanded in t around inf 32.6%

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

   1. associate-*r* 33.2%

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

   2. *-commutative 33.2%

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

   3. associate-*r* 31.7%

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

8. Simplified 31.7%

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

9. Final simplification 31.7%

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

Reproduce

herbie shell --seed 2023311 
(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*))))))