Toniolo and Linder, Equation (13)

Percentage Accurate: 49.1% → 64.6%

Time: 9.0s

Alternatives: 15

Speedup: 1.9×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 49.1% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    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: 64.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ t_2 := \mathsf{fma}\left(l\_m \cdot \frac{l\_m}{Om}, -2, t\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(U* - U, \left(l\_m \cdot \frac{l\_m}{Om \cdot Om}\right) \cdot n, t\_2\right) \cdot U}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{\left(t\_2 - \left(\left(U - U*\right) \cdot \left(\frac{l\_m}{Om} \cdot n\right)\right) \cdot \frac{l\_m}{Om}\right) \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
        (t_2 (fma (* l_m (/ l_m Om)) -2.0 t)))
   (if (<= t_1 0.0)
     (*
      (sqrt (+ n n))
      (sqrt (* (fma (- U* U) (* (* l_m (/ l_m (* Om Om))) n) t_2) U)))
     (if (<= t_1 INFINITY)
       (sqrt
        (* (- t_2 (* (* (- U U*) (* (/ l_m Om) n)) (/ l_m Om))) (* (+ n n) U)))
       (*
        l_m
        (sqrt
         (*
          -2.0
          (*
           U
           (* n (fma 2.0 (/ 1.0 Om) (/ (* n (- U U*)) (pow Om 2.0))))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double t_2 = fma((l_m * (l_m / Om)), -2.0, t);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt((n + n)) * sqrt((fma((U_42_ - U), ((l_m * (l_m / (Om * Om))) * n), t_2) * U));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt(((t_2 - (((U - U_42_) * ((l_m / Om) * n)) * (l_m / Om))) * ((n + n) * U)));
	} else {
		tmp = l_m * sqrt((-2.0 * (U * (n * fma(2.0, (1.0 / Om), ((n * (U - U_42_)) / pow(Om, 2.0)))))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_)))))
	t_2 = fma(Float64(l_m * Float64(l_m / Om)), -2.0, t)
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(Float64(fma(Float64(U_42_ - U), Float64(Float64(l_m * Float64(l_m / Float64(Om * Om))) * n), t_2) * U)));
	elseif (t_1 <= Inf)
		tmp = sqrt(Float64(Float64(t_2 - Float64(Float64(Float64(U - U_42_) * Float64(Float64(l_m / Om) * n)) * Float64(l_m / Om))) * Float64(Float64(n + n) * U)));
	else
		tmp = Float64(l_m * sqrt(Float64(-2.0 * Float64(U * Float64(n * fma(2.0, Float64(1.0 / Om), Float64(Float64(n * Float64(U - U_42_)) / (Om ^ 2.0))))))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[(l$95$m * N[(l$95$m / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] + t$95$2), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[Sqrt[N[(N[(t$95$2 - N[(N[(N[(U - U$42$), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-2.0 * N[(U * N[(n * N[(2.0 * N[(1.0 / Om), $MachinePrecision] + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 49.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. *-commutative N/A

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

   3. Applied rewrites 25.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 26.5

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 26.5

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lift-/.f64 N/A

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

      13. lower-*.f64 28.7

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

   5. Applied rewrites 28.7%

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

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

   1. Initial program 49.1%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-out N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 53.0

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

   3. Applied rewrites 53.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 54.6

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

   5. Applied rewrites 54.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 54.6

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

   7. Applied rewrites 54.6%

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

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

   1. Initial program 49.1%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 27.3

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

   4. Applied rewrites 27.3%

      \[\leadsto \color{blue}{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 64.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m}{Om} \cdot n\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 10^{-297}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(\left(\mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right) - \left(t\_1 \cdot \left(U - U*\right)\right) \cdot \frac{l\_m}{Om}\right) \cdot U\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(l\_m \cdot \frac{l\_m}{Om}, -2, t\right) - \left(\left(U - U*\right) \cdot t\_1\right) \cdot \frac{l\_m}{Om}\right) \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(l\_m \cdot \left(l\_m \cdot n\right)\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right)\right) \cdot U\right) \cdot -2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (/ l_m Om) n))
        (t_2
         (*
          (* (* 2.0 n) U)
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_2 1e-297)
     (sqrt
      (*
       (+ n n)
       (*
        (- (fma (* (/ l_m Om) l_m) -2.0 t) (* (* t_1 (- U U*)) (/ l_m Om)))
        U)))
     (if (<= t_2 INFINITY)
       (sqrt
        (*
         (- (fma (* l_m (/ l_m Om)) -2.0 t) (* (* (- U U*) t_1) (/ l_m Om)))
         (* (+ n n) U)))
       (sqrt
        (*
         (* (* (* l_m (* l_m n)) (fma n (/ (- U U*) (* Om Om)) (/ 2.0 Om))) U)
         -2.0))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (l_m / Om) * n;
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 1e-297) {
		tmp = sqrt(((n + n) * ((fma(((l_m / Om) * l_m), -2.0, t) - ((t_1 * (U - U_42_)) * (l_m / Om))) * U)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((fma((l_m * (l_m / Om)), -2.0, t) - (((U - U_42_) * t_1) * (l_m / Om))) * ((n + n) * U)));
	} else {
		tmp = sqrt(((((l_m * (l_m * n)) * fma(n, ((U - U_42_) / (Om * Om)), (2.0 / Om))) * U) * -2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m / Om) * n)
	t_2 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_2 <= 1e-297)
		tmp = sqrt(Float64(Float64(n + n) * Float64(Float64(fma(Float64(Float64(l_m / Om) * l_m), -2.0, t) - Float64(Float64(t_1 * Float64(U - U_42_)) * Float64(l_m / Om))) * U)));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(Float64(fma(Float64(l_m * Float64(l_m / Om)), -2.0, t) - Float64(Float64(Float64(U - U_42_) * t_1) * Float64(l_m / Om))) * Float64(Float64(n + n) * U)));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(l_m * Float64(l_m * n)) * fma(n, Float64(Float64(U - U_42_) / Float64(Om * Om)), Float64(2.0 / Om))) * U) * -2.0));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 1e-297], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision] - N[(N[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(N[(N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision] - N[(N[(N[(U - U$42$), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(l$95$m * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1.00000000000000004e-297

   1. Initial program 49.1%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-out N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 53.0

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

   3. Applied rewrites 53.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 54.6

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

   5. Applied rewrites 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-+.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 55.5

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

   7. Applied rewrites 55.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 55.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 55.5

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.5

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

   9. Applied rewrites 55.5%

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

   if 1.00000000000000004e-297 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 49.1%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-out N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 53.0

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

   3. Applied rewrites 53.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 54.6

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

   5. Applied rewrites 54.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 54.6

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

   7. Applied rewrites 54.6%

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

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

   1. Initial program 49.1%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 20.7

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

   4. Applied rewrites 20.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 20.7

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

   6. Applied rewrites 21.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 25.2

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

   8. Applied rewrites 25.2%

      \[\leadsto \sqrt{\left(\left(\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right)\right) \cdot U\right) \cdot -2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 60.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ t_3 := \sqrt{\left(n + n\right) \cdot \left(\left(\mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right) - \left(\left(\frac{l\_m}{Om} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \frac{l\_m}{Om}\right) \cdot U\right)}\\ \mathbf{if}\;t\_2 \leq 10^{+14}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 10^{+280}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(\mathsf{fma}\left(-2 \cdot l\_m, \frac{l\_m}{Om}, t\right) - \frac{l\_m}{Om} \cdot \frac{U \cdot \left(l\_m \cdot n\right)}{Om}\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(l\_m \cdot \left(l\_m \cdot n\right)\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right)\right) \cdot U\right) \cdot -2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))
        (t_3
         (sqrt
          (*
           (+ n n)
           (*
            (-
             (fma (* (/ l_m Om) l_m) -2.0 t)
             (* (* (* (/ l_m Om) n) (- U U*)) (/ l_m Om)))
            U)))))
   (if (<= t_2 1e+14)
     t_3
     (if (<= t_2 1e+280)
       (sqrt
        (*
         t_1
         (-
          (fma (* -2.0 l_m) (/ l_m Om) t)
          (* (/ l_m Om) (/ (* U (* l_m n)) Om)))))
       (if (<= t_2 INFINITY)
         t_3
         (sqrt
          (*
           (*
            (* (* l_m (* l_m n)) (fma n (/ (- U U*) (* Om Om)) (/ 2.0 Om)))
            U)
           -2.0)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double t_3 = sqrt(((n + n) * ((fma(((l_m / Om) * l_m), -2.0, t) - ((((l_m / Om) * n) * (U - U_42_)) * (l_m / Om))) * U)));
	double tmp;
	if (t_2 <= 1e+14) {
		tmp = t_3;
	} else if (t_2 <= 1e+280) {
		tmp = sqrt((t_1 * (fma((-2.0 * l_m), (l_m / Om), t) - ((l_m / Om) * ((U * (l_m * n)) / Om)))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = sqrt(((((l_m * (l_m * n)) * fma(n, ((U - U_42_) / (Om * Om)), (2.0 / Om))) * U) * -2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	t_3 = sqrt(Float64(Float64(n + n) * Float64(Float64(fma(Float64(Float64(l_m / Om) * l_m), -2.0, t) - Float64(Float64(Float64(Float64(l_m / Om) * n) * Float64(U - U_42_)) * Float64(l_m / Om))) * U)))
	tmp = 0.0
	if (t_2 <= 1e+14)
		tmp = t_3;
	elseif (t_2 <= 1e+280)
		tmp = sqrt(Float64(t_1 * Float64(fma(Float64(-2.0 * l_m), Float64(l_m / Om), t) - Float64(Float64(l_m / Om) * Float64(Float64(U * Float64(l_m * n)) / Om)))));
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(l_m * Float64(l_m * n)) * fma(n, Float64(Float64(U - U_42_) / Float64(Om * Om)), Float64(2.0 / Om))) * U) * -2.0));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision] - N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 1e+14], t$95$3, If[LessEqual[t$95$2, 1e+280], N[Sqrt[N[(t$95$1 * N[(N[(N[(-2.0 * l$95$m), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + t), $MachinePrecision] - N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(U * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$3, N[Sqrt[N[(N[(N[(N[(l$95$m * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1e14 or 1e280 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 49.1%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-out N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 53.0

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

   3. Applied rewrites 53.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 54.6

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

   5. Applied rewrites 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-+.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 55.5

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

   7. Applied rewrites 55.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 55.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 55.5

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.5

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

   9. Applied rewrites 55.5%

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

   if 1e14 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1e280

   1. Initial program 49.1%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-out N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 53.0

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

   3. Applied rewrites 53.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 54.6

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

   5. Applied rewrites 54.6%

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

   6. Taylor expanded in U around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 39.2

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

   8. Applied rewrites 39.2%

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

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

   1. Initial program 49.1%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 20.7

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

   4. Applied rewrites 20.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 20.7

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

   6. Applied rewrites 21.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 25.2

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

   8. Applied rewrites 25.2%

      \[\leadsto \sqrt{\left(\left(\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right)\right) \cdot U\right) \cdot -2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 58.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right) \cdot U}\\ \mathbf{elif}\;t\_2 \leq 10^{+280}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(\mathsf{fma}\left(-2 \cdot l\_m, \frac{l\_m}{Om}, t\right) - \frac{l\_m}{Om} \cdot \frac{U \cdot \left(l\_m \cdot n\right)}{Om}\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(\mathsf{fma}\left(\left(l\_m \cdot \frac{l\_m}{Om \cdot Om}\right) \cdot n, U* - U, \mathsf{fma}\left(l\_m \cdot \frac{l\_m}{Om}, -2, t\right)\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(l\_m \cdot \left(l\_m \cdot n\right)\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right)\right) \cdot U\right) \cdot -2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_2 0.0)
     (* (sqrt (+ n n)) (sqrt (* (fma (* (/ l_m Om) l_m) -2.0 t) U)))
     (if (<= t_2 1e+280)
       (sqrt
        (*
         t_1
         (-
          (fma (* -2.0 l_m) (/ l_m Om) t)
          (* (/ l_m Om) (/ (* U (* l_m n)) Om)))))
       (if (<= t_2 INFINITY)
         (sqrt
          (*
           (+ n n)
           (*
            (fma
             (* (* l_m (/ l_m (* Om Om))) n)
             (- U* U)
             (fma (* l_m (/ l_m Om)) -2.0 t))
            U)))
         (sqrt
          (*
           (*
            (* (* l_m (* l_m n)) (fma n (/ (- U U*) (* Om Om)) (/ 2.0 Om)))
            U)
           -2.0)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((n + n)) * sqrt((fma(((l_m / Om) * l_m), -2.0, t) * U));
	} else if (t_2 <= 1e+280) {
		tmp = sqrt((t_1 * (fma((-2.0 * l_m), (l_m / Om), t) - ((l_m / Om) * ((U * (l_m * n)) / Om)))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((n + n) * (fma(((l_m * (l_m / (Om * Om))) * n), (U_42_ - U), fma((l_m * (l_m / Om)), -2.0, t)) * U)));
	} else {
		tmp = sqrt(((((l_m * (l_m * n)) * fma(n, ((U - U_42_) / (Om * Om)), (2.0 / Om))) * U) * -2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(Float64(fma(Float64(Float64(l_m / Om) * l_m), -2.0, t) * U)));
	elseif (t_2 <= 1e+280)
		tmp = sqrt(Float64(t_1 * Float64(fma(Float64(-2.0 * l_m), Float64(l_m / Om), t) - Float64(Float64(l_m / Om) * Float64(Float64(U * Float64(l_m * n)) / Om)))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(Float64(n + n) * Float64(fma(Float64(Float64(l_m * Float64(l_m / Float64(Om * Om))) * n), Float64(U_42_ - U), fma(Float64(l_m * Float64(l_m / Om)), -2.0, t)) * U)));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(l_m * Float64(l_m * n)) * fma(n, Float64(Float64(U - U_42_) / Float64(Om * Om)), Float64(2.0 / Om))) * U) * -2.0));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+280], N[Sqrt[N[(t$95$1 * N[(N[(N[(-2.0 * l$95$m), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + t), $MachinePrecision] - N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(U * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(N[(N[(N[(l$95$m * N[(l$95$m / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision] + N[(N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(l$95$m * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 49.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. *-commutative N/A

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

   3. Applied rewrites 25.9%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 25.3

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

   6. Applied rewrites 25.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-*.f64 N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. associate-/l* N/A

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

      17. lift-/.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 27.4

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

   8. Applied rewrites 27.4%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1e280

   1. Initial program 49.1%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-out N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 53.0

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

   3. Applied rewrites 53.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 54.6

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

   5. Applied rewrites 54.6%

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

   6. Taylor expanded in U around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 39.2

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

   8. Applied rewrites 39.2%

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

   if 1e280 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 49.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. *-commutative N/A

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

   3. Applied rewrites 25.9%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 44.3

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

   5. Applied rewrites 49.3%

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

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

   1. Initial program 49.1%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 20.7

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

   4. Applied rewrites 20.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 20.7

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

   6. Applied rewrites 21.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 25.2

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

   8. Applied rewrites 25.2%

      \[\leadsto \sqrt{\left(\left(\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right)\right) \cdot U\right) \cdot -2} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 58.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right) \cdot U}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+283}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(\mathsf{fma}\left(-2 \cdot l\_m, \frac{l\_m}{Om}, t\right) - \frac{l\_m}{Om} \cdot \frac{U \cdot \left(l\_m \cdot n\right)}{Om}\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(U* - U, \left(l\_m \cdot \frac{l\_m}{Om \cdot Om}\right) \cdot n, \mathsf{fma}\left(l\_m \cdot \frac{l\_m}{Om}, -2, t\right)\right) \cdot \left(n + n\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(l\_m \cdot \left(l\_m \cdot n\right)\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right)\right) \cdot U\right) \cdot -2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_2 0.0)
     (* (sqrt (+ n n)) (sqrt (* (fma (* (/ l_m Om) l_m) -2.0 t) U)))
     (if (<= t_2 5e+283)
       (sqrt
        (*
         t_1
         (-
          (fma (* -2.0 l_m) (/ l_m Om) t)
          (* (/ l_m Om) (/ (* U (* l_m n)) Om)))))
       (if (<= t_2 INFINITY)
         (sqrt
          (*
           (*
            (fma
             (- U* U)
             (* (* l_m (/ l_m (* Om Om))) n)
             (fma (* l_m (/ l_m Om)) -2.0 t))
            (+ n n))
           U))
         (sqrt
          (*
           (*
            (* (* l_m (* l_m n)) (fma n (/ (- U U*) (* Om Om)) (/ 2.0 Om)))
            U)
           -2.0)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((n + n)) * sqrt((fma(((l_m / Om) * l_m), -2.0, t) * U));
	} else if (t_2 <= 5e+283) {
		tmp = sqrt((t_1 * (fma((-2.0 * l_m), (l_m / Om), t) - ((l_m / Om) * ((U * (l_m * n)) / Om)))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((fma((U_42_ - U), ((l_m * (l_m / (Om * Om))) * n), fma((l_m * (l_m / Om)), -2.0, t)) * (n + n)) * U));
	} else {
		tmp = sqrt(((((l_m * (l_m * n)) * fma(n, ((U - U_42_) / (Om * Om)), (2.0 / Om))) * U) * -2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(Float64(fma(Float64(Float64(l_m / Om) * l_m), -2.0, t) * U)));
	elseif (t_2 <= 5e+283)
		tmp = sqrt(Float64(t_1 * Float64(fma(Float64(-2.0 * l_m), Float64(l_m / Om), t) - Float64(Float64(l_m / Om) * Float64(Float64(U * Float64(l_m * n)) / Om)))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(Float64(fma(Float64(U_42_ - U), Float64(Float64(l_m * Float64(l_m / Float64(Om * Om))) * n), fma(Float64(l_m * Float64(l_m / Om)), -2.0, t)) * Float64(n + n)) * U));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(l_m * Float64(l_m * n)) * fma(n, Float64(Float64(U - U_42_) / Float64(Om * Om)), Float64(2.0 / Om))) * U) * -2.0));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+283], N[Sqrt[N[(t$95$1 * N[(N[(N[(-2.0 * l$95$m), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + t), $MachinePrecision] - N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(U * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[(l$95$m * N[(l$95$m / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] + N[(N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(l$95$m * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 49.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. *-commutative N/A

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

   3. Applied rewrites 25.9%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 25.3

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

   6. Applied rewrites 25.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-*.f64 N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. associate-/l* N/A

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

      17. lift-/.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 27.4

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

   8. Applied rewrites 27.4%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.0000000000000004e283

   1. Initial program 49.1%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-out N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 53.0

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

   3. Applied rewrites 53.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 54.6

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

   5. Applied rewrites 54.6%

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

   6. Taylor expanded in U around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 39.2

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

   8. Applied rewrites 39.2%

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

   if 5.0000000000000004e283 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 49.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 44.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 45.3

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 45.3

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lift-/.f64 N/A

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

      13. lower-*.f64 49.2

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

   5. Applied rewrites 49.2%

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

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

   1. Initial program 49.1%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 20.7

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

   4. Applied rewrites 20.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 20.7

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

   6. Applied rewrites 21.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 25.2

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

   8. Applied rewrites 25.2%

      \[\leadsto \sqrt{\left(\left(\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right)\right) \cdot U\right) \cdot -2} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 55.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right) \cdot U}\\ \mathbf{elif}\;t\_2 \leq 10^{+286}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(\mathsf{fma}\left(-2 \cdot l\_m, \frac{l\_m}{Om}, t\right) - \frac{l\_m}{Om} \cdot \frac{U \cdot \left(l\_m \cdot n\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(l\_m \cdot \left(l\_m \cdot n\right)\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right)\right) \cdot U\right) \cdot -2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_2 0.0)
     (* (sqrt (+ n n)) (sqrt (* (fma (* (/ l_m Om) l_m) -2.0 t) U)))
     (if (<= t_2 1e+286)
       (sqrt
        (*
         t_1
         (-
          (fma (* -2.0 l_m) (/ l_m Om) t)
          (* (/ l_m Om) (/ (* U (* l_m n)) Om)))))
       (sqrt
        (*
         (* (* (* l_m (* l_m n)) (fma n (/ (- U U*) (* Om Om)) (/ 2.0 Om))) U)
         -2.0))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((n + n)) * sqrt((fma(((l_m / Om) * l_m), -2.0, t) * U));
	} else if (t_2 <= 1e+286) {
		tmp = sqrt((t_1 * (fma((-2.0 * l_m), (l_m / Om), t) - ((l_m / Om) * ((U * (l_m * n)) / Om)))));
	} else {
		tmp = sqrt(((((l_m * (l_m * n)) * fma(n, ((U - U_42_) / (Om * Om)), (2.0 / Om))) * U) * -2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(Float64(fma(Float64(Float64(l_m / Om) * l_m), -2.0, t) * U)));
	elseif (t_2 <= 1e+286)
		tmp = sqrt(Float64(t_1 * Float64(fma(Float64(-2.0 * l_m), Float64(l_m / Om), t) - Float64(Float64(l_m / Om) * Float64(Float64(U * Float64(l_m * n)) / Om)))));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(l_m * Float64(l_m * n)) * fma(n, Float64(Float64(U - U_42_) / Float64(Om * Om)), Float64(2.0 / Om))) * U) * -2.0));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+286], N[Sqrt[N[(t$95$1 * N[(N[(N[(-2.0 * l$95$m), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + t), $MachinePrecision] - N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(U * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(l$95$m * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 49.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. *-commutative N/A

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

   3. Applied rewrites 25.9%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 25.3

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

   6. Applied rewrites 25.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-*.f64 N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. associate-/l* N/A

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

      17. lift-/.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 27.4

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

   8. Applied rewrites 27.4%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1.00000000000000003e286

   1. Initial program 49.1%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-out N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 53.0

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

   3. Applied rewrites 53.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 54.6

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

   5. Applied rewrites 54.6%

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

   6. Taylor expanded in U around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 39.2

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

   8. Applied rewrites 39.2%

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

   if 1.00000000000000003e286 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 49.1%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 20.7

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

   4. Applied rewrites 20.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 20.7

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

   6. Applied rewrites 21.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 25.2

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

   8. Applied rewrites 25.2%

      \[\leadsto \sqrt{\left(\left(\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right)\right) \cdot U\right) \cdot -2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 55.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right) \cdot U}\\ \mathbf{elif}\;t\_2 \leq 10^{+286}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t - 2 \cdot \frac{{l\_m}^{2}}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(l\_m \cdot \left(l\_m \cdot n\right)\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right)\right) \cdot U\right) \cdot -2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_2 0.0)
     (* (sqrt (+ n n)) (sqrt (* (fma (* (/ l_m Om) l_m) -2.0 t) U)))
     (if (<= t_2 1e+286)
       (sqrt (* t_1 (- t (* 2.0 (/ (pow l_m 2.0) Om)))))
       (sqrt
        (*
         (* (* (* l_m (* l_m n)) (fma n (/ (- U U*) (* Om Om)) (/ 2.0 Om))) U)
         -2.0))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((n + n)) * sqrt((fma(((l_m / Om) * l_m), -2.0, t) * U));
	} else if (t_2 <= 1e+286) {
		tmp = sqrt((t_1 * (t - (2.0 * (pow(l_m, 2.0) / Om)))));
	} else {
		tmp = sqrt(((((l_m * (l_m * n)) * fma(n, ((U - U_42_) / (Om * Om)), (2.0 / Om))) * U) * -2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(Float64(fma(Float64(Float64(l_m / Om) * l_m), -2.0, t) * U)));
	elseif (t_2 <= 1e+286)
		tmp = sqrt(Float64(t_1 * Float64(t - Float64(2.0 * Float64((l_m ^ 2.0) / Om)))));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(l_m * Float64(l_m * n)) * fma(n, Float64(Float64(U - U_42_) / Float64(Om * Om)), Float64(2.0 / Om))) * U) * -2.0));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+286], N[Sqrt[N[(t$95$1 * N[(t - N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(l$95$m * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 49.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. *-commutative N/A

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

   3. Applied rewrites 25.9%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 25.3

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

   6. Applied rewrites 25.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-*.f64 N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. associate-/l* N/A

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

      17. lift-/.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 27.4

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

   8. Applied rewrites 27.4%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1.00000000000000003e286

   1. Initial program 49.1%

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

   2. Taylor expanded in n around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 43.1

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

   4. Applied rewrites 43.1%

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

   if 1.00000000000000003e286 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 49.1%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 20.7

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

   4. Applied rewrites 20.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 20.7

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

   6. Applied rewrites 21.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 25.2

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

   8. Applied rewrites 25.2%

      \[\leadsto \sqrt{\left(\left(\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right)\right) \cdot U\right) \cdot -2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 53.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right) \cdot U\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{t\_1}\\ \mathbf{elif}\;t\_2 \leq 10^{+14}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot t\_1}\\ \mathbf{elif}\;t\_2 \leq 10^{+286}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(l\_m \cdot \left(l\_m \cdot n\right)\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right)\right) \cdot U\right) \cdot -2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (fma (* (/ l_m Om) l_m) -2.0 t) U))
        (t_2
         (*
          (* (* 2.0 n) U)
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_2 0.0)
     (* (sqrt (+ n n)) (sqrt t_1))
     (if (<= t_2 1e+14)
       (sqrt (* (+ n n) t_1))
       (if (<= t_2 1e+286)
         (sqrt (* (* (+ n n) U) t))
         (sqrt
          (*
           (*
            (* (* l_m (* l_m n)) (fma n (/ (- U U*) (* Om Om)) (/ 2.0 Om)))
            U)
           -2.0)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = fma(((l_m / Om) * l_m), -2.0, t) * U;
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((n + n)) * sqrt(t_1);
	} else if (t_2 <= 1e+14) {
		tmp = sqrt(((n + n) * t_1));
	} else if (t_2 <= 1e+286) {
		tmp = sqrt((((n + n) * U) * t));
	} else {
		tmp = sqrt(((((l_m * (l_m * n)) * fma(n, ((U - U_42_) / (Om * Om)), (2.0 / Om))) * U) * -2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(fma(Float64(Float64(l_m / Om) * l_m), -2.0, t) * U)
	t_2 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(t_1));
	elseif (t_2 <= 1e+14)
		tmp = sqrt(Float64(Float64(n + n) * t_1));
	elseif (t_2 <= 1e+286)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * t));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(l_m * Float64(l_m * n)) * fma(n, Float64(Float64(U - U_42_) / Float64(Om * Om)), Float64(2.0 / Om))) * U) * -2.0));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+14], N[Sqrt[N[(N[(n + n), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 1e+286], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(l$95$m * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 49.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. *-commutative N/A

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

   3. Applied rewrites 25.9%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 25.3

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

   6. Applied rewrites 25.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-*.f64 N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. associate-/l* N/A

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

      17. lift-/.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 27.4

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

   8. Applied rewrites 27.4%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1e14

   1. Initial program 49.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. *-commutative N/A

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

   3. Applied rewrites 25.9%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 25.3

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

   6. Applied rewrites 25.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 43.9

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

   8. Applied rewrites 47.5%

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

   if 1e14 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1.00000000000000003e286

   1. Initial program 49.1%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 36.2

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

   4. Applied rewrites 36.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 35.9

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

   6. Applied rewrites 35.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 35.9

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

      10. lift-*.f64 N/A

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

      11. count-2-rev N/A

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

      12. lift-+.f64 35.9

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

   8. Applied rewrites 35.9%

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

   if 1.00000000000000003e286 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 49.1%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 20.7

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

   4. Applied rewrites 20.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 20.7

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

   6. Applied rewrites 21.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 25.2

         \[\leadsto \sqrt{\left(\left(\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right)\right) \cdot U\right) \cdot -2} \]

   8. Applied rewrites 25.2%

      \[\leadsto \sqrt{\left(\left(\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right)\right) \cdot U\right) \cdot -2} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 53.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right) \cdot U\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ t_3 := \sqrt{\left(n + n\right) \cdot t\_1}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{t\_1}\\ \mathbf{elif}\;t\_2 \leq 10^{+14}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 10^{+280}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(\left(l\_m \cdot l\_m\right) \cdot n\right) \cdot \frac{2 + \frac{n \cdot \left(U - U*\right)}{Om}}{Om}\right) \cdot U\right) \cdot -2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (fma (* (/ l_m Om) l_m) -2.0 t) U))
        (t_2
         (*
          (* (* 2.0 n) U)
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))
        (t_3 (sqrt (* (+ n n) t_1))))
   (if (<= t_2 0.0)
     (* (sqrt (+ n n)) (sqrt t_1))
     (if (<= t_2 1e+14)
       t_3
       (if (<= t_2 1e+280)
         (sqrt (* (* (+ n n) U) t))
         (if (<= t_2 INFINITY)
           t_3
           (sqrt
            (*
             (* (* (* (* l_m l_m) n) (/ (+ 2.0 (/ (* n (- U U*)) Om)) Om)) U)
             -2.0))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = fma(((l_m / Om) * l_m), -2.0, t) * U;
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double t_3 = sqrt(((n + n) * t_1));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((n + n)) * sqrt(t_1);
	} else if (t_2 <= 1e+14) {
		tmp = t_3;
	} else if (t_2 <= 1e+280) {
		tmp = sqrt((((n + n) * U) * t));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = sqrt((((((l_m * l_m) * n) * ((2.0 + ((n * (U - U_42_)) / Om)) / Om)) * U) * -2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(fma(Float64(Float64(l_m / Om) * l_m), -2.0, t) * U)
	t_2 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	t_3 = sqrt(Float64(Float64(n + n) * t_1))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(t_1));
	elseif (t_2 <= 1e+14)
		tmp = t_3;
	elseif (t_2 <= 1e+280)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * t));
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(l_m * l_m) * n) * Float64(Float64(2.0 + Float64(Float64(n * Float64(U - U_42_)) / Om)) / Om)) * U) * -2.0));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(n + n), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+14], t$95$3, If[LessEqual[t$95$2, 1e+280], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$3, N[Sqrt[N[(N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision] * N[(N[(2.0 + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 0.0

   1. Initial program 49.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      5. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot n}} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot n}} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. count-2-rev N/A

         \[\leadsto \sqrt{\color{blue}{n + n}} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \sqrt{\color{blue}{n + n}} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{n + n} \cdot \color{blue}{\sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot U}} \]

   3. Applied rewrites 25.9%

      \[\leadsto \color{blue}{\sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(U* - U, \frac{\ell \cdot \ell}{Om \cdot Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right) \cdot U}} \]

   4. Taylor expanded in n around 0

      \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \left(t + \color{blue}{-2 \cdot \frac{{\ell}^{2}}{Om}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{\color{blue}{Om}}\right)} \]

      5. lower-pow.f64 25.3

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)} \]

   6. Applied rewrites 25.3%

      \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot \color{blue}{U}} \]

      3. lift-+.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U} \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(t - \left(\mathsf{neg}\left(-2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U} \]

      6. metadata-eval N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U} \]

      7. lift-/.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U} \]

      8. lift-pow.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U} \]

      9. pow2 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot U} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot \color{blue}{U}} \]

      11. fp-cancel-sub-sign-inv N/A

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot U} \]

      12. metadata-eval N/A

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot U} \]

      13. +-commutative N/A

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(-2 \cdot \frac{\ell \cdot \ell}{Om} + t\right) \cdot U} \]

      14. *-commutative N/A

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(\frac{\ell \cdot \ell}{Om} \cdot -2 + t\right) \cdot U} \]

      15. lower-fma.f64 N/A

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right) \cdot U} \]

      16. associate-/l* N/A

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right) \cdot U} \]

      17. lift-/.f64 N/A

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right) \cdot U} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot U} \]

      19. lower-*.f64 27.4

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot U} \]

   8. Applied rewrites 27.4%

      \[\leadsto \sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot \color{blue}{U}} \]

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1e14 or 1e280 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 49.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      5. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot n}} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot n}} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. count-2-rev N/A

         \[\leadsto \sqrt{\color{blue}{n + n}} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \sqrt{\color{blue}{n + n}} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{n + n} \cdot \color{blue}{\sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot U}} \]

   3. Applied rewrites 25.9%

      \[\leadsto \color{blue}{\sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(U* - U, \frac{\ell \cdot \ell}{Om \cdot Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right) \cdot U}} \]

   4. Taylor expanded in n around 0

      \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \left(t + \color{blue}{-2 \cdot \frac{{\ell}^{2}}{Om}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{\color{blue}{Om}}\right)} \]

      5. lower-pow.f64 25.3

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)} \]

   6. Applied rewrites 25.3%

      \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{n + n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{n + n}} \cdot \sqrt{U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \color{blue}{\sqrt{U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]

      4. sqrt-unprod N/A

         \[\leadsto \color{blue}{\sqrt{\left(n + n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(n + n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      6. lower-*.f64 43.9

         \[\leadsto \sqrt{\color{blue}{\left(n + n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

   8. Applied rewrites 47.5%

      \[\leadsto \color{blue}{\sqrt{\left(n + n\right) \cdot \left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot U\right)}} \]

   if 1e14 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1e280

   1. Initial program 49.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 36.2

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 36.2%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      5. lower-*.f64 35.9

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

   6. Applied rewrites 35.9%

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \color{blue}{t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      9. lower-*.f64 35.9

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      11. count-2-rev N/A

          \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \]

      12. lift-+.f64 35.9

          \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \]

   8. Applied rewrites 35.9%

      \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \color{blue}{t}} \]

   if +inf.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 49.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in l around inf

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \color{blue}{\left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \left(U \cdot \color{blue}{\left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \color{blue}{\left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)}\right)\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(\color{blue}{n} \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \color{blue}{\left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)}\right)\right)\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \mathsf{fma}\left(2, \color{blue}{\frac{1}{Om}}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{\color{blue}{Om}}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right)} \]

      11. lower-pow.f64 20.7

          \[\leadsto \sqrt{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right)} \]

   4. Applied rewrites 20.7%

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{-2 \cdot \color{blue}{\left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right) \cdot \color{blue}{-2}} \]

      3. lower-*.f64 20.7

         \[\leadsto \sqrt{\left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right) \cdot \color{blue}{-2}} \]

   6. Applied rewrites 21.3%

      \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right)\right) \cdot U\right) \cdot -2}} \]

   7. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\left(\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \frac{2 + \frac{n \cdot \left(U - U*\right)}{Om}}{Om}\right) \cdot U\right) \cdot -2} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \frac{2 + \frac{n \cdot \left(U - U*\right)}{Om}}{Om}\right) \cdot U\right) \cdot -2} \]

      2. lower-+.f64 N/A

         \[\leadsto \sqrt{\left(\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \frac{2 + \frac{n \cdot \left(U - U*\right)}{Om}}{Om}\right) \cdot U\right) \cdot -2} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \frac{2 + \frac{n \cdot \left(U - U*\right)}{Om}}{Om}\right) \cdot U\right) \cdot -2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \frac{2 + \frac{n \cdot \left(U - U*\right)}{Om}}{Om}\right) \cdot U\right) \cdot -2} \]

      5. lower--.f64 23.3

         \[\leadsto \sqrt{\left(\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \frac{2 + \frac{n \cdot \left(U - U*\right)}{Om}}{Om}\right) \cdot U\right) \cdot -2} \]

   9. Applied rewrites 23.3%

      \[\leadsto \sqrt{\left(\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \frac{2 + \frac{n \cdot \left(U - U*\right)}{Om}}{Om}\right) \cdot U\right) \cdot -2} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 10: 51.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right) \cdot U\\ \mathbf{if}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{t\_1}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (fma (* (/ l_m Om) l_m) -2.0 t) U)))
   (if (<= n -5e-310) (sqrt (* (+ n n) t_1)) (* (sqrt (+ n n)) (sqrt t_1)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = fma(((l_m / Om) * l_m), -2.0, t) * U;
	double tmp;
	if (n <= -5e-310) {
		tmp = sqrt(((n + n) * t_1));
	} else {
		tmp = sqrt((n + n)) * sqrt(t_1);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(fma(Float64(Float64(l_m / Om) * l_m), -2.0, t) * U)
	tmp = 0.0
	if (n <= -5e-310)
		tmp = sqrt(Float64(Float64(n + n) * t_1));
	else
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(t_1));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision] * U), $MachinePrecision]}, If[LessEqual[n, -5e-310], N[Sqrt[N[(N[(n + n), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < -4.999999999999985e-310

   1. Initial program 49.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      5. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot n}} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot n}} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. count-2-rev N/A

         \[\leadsto \sqrt{\color{blue}{n + n}} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \sqrt{\color{blue}{n + n}} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{n + n} \cdot \color{blue}{\sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot U}} \]

   3. Applied rewrites 25.9%

      \[\leadsto \color{blue}{\sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(U* - U, \frac{\ell \cdot \ell}{Om \cdot Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right) \cdot U}} \]

   4. Taylor expanded in n around 0

      \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \left(t + \color{blue}{-2 \cdot \frac{{\ell}^{2}}{Om}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{\color{blue}{Om}}\right)} \]

      5. lower-pow.f64 25.3

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)} \]

   6. Applied rewrites 25.3%

      \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{n + n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{n + n}} \cdot \sqrt{U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \color{blue}{\sqrt{U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]

      4. sqrt-unprod N/A

         \[\leadsto \color{blue}{\sqrt{\left(n + n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(n + n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      6. lower-*.f64 43.9

         \[\leadsto \sqrt{\color{blue}{\left(n + n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

   8. Applied rewrites 47.5%

      \[\leadsto \color{blue}{\sqrt{\left(n + n\right) \cdot \left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot U\right)}} \]

   if -4.999999999999985e-310 < n

   1. Initial program 49.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      5. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot n}} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot n}} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. count-2-rev N/A

         \[\leadsto \sqrt{\color{blue}{n + n}} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \sqrt{\color{blue}{n + n}} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{n + n} \cdot \color{blue}{\sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot U}} \]

   3. Applied rewrites 25.9%

      \[\leadsto \color{blue}{\sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(U* - U, \frac{\ell \cdot \ell}{Om \cdot Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right) \cdot U}} \]

   4. Taylor expanded in n around 0

      \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \left(t + \color{blue}{-2 \cdot \frac{{\ell}^{2}}{Om}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{\color{blue}{Om}}\right)} \]

      5. lower-pow.f64 25.3

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)} \]

   6. Applied rewrites 25.3%

      \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot \color{blue}{U}} \]

      3. lift-+.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U} \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(t - \left(\mathsf{neg}\left(-2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U} \]

      6. metadata-eval N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U} \]

      7. lift-/.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U} \]

      8. lift-pow.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U} \]

      9. pow2 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot U} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot \color{blue}{U}} \]

      11. fp-cancel-sub-sign-inv N/A

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot U} \]

      12. metadata-eval N/A

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot U} \]

      13. +-commutative N/A

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(-2 \cdot \frac{\ell \cdot \ell}{Om} + t\right) \cdot U} \]

      14. *-commutative N/A

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\left(\frac{\ell \cdot \ell}{Om} \cdot -2 + t\right) \cdot U} \]

      15. lower-fma.f64 N/A

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right) \cdot U} \]

      16. associate-/l* N/A

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right) \cdot U} \]

      17. lift-/.f64 N/A

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right) \cdot U} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot U} \]

      19. lower-*.f64 27.4

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot U} \]

   8. Applied rewrites 27.4%

      \[\leadsto \sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot \color{blue}{U}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 47.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 5.5 \cdot 10^{-100}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(\mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right) \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 5.5e-100)
   (sqrt (* 2.0 (* U (* n t))))
   (sqrt (* (+ n n) (* (fma (* (/ l_m Om) l_m) -2.0 t) U)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 5.5e-100) {
		tmp = sqrt((2.0 * (U * (n * t))));
	} else {
		tmp = sqrt(((n + n) * (fma(((l_m / Om) * l_m), -2.0, t) * U)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 5.5e-100)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	else
		tmp = sqrt(Float64(Float64(n + n) * Float64(fma(Float64(Float64(l_m / Om) * l_m), -2.0, t) * U)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 5.5e-100], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.50000000000000011e-100

   1. Initial program 49.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 36.2

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 36.2%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

   if 5.50000000000000011e-100 < l

   1. Initial program 49.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      5. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot n}} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot n}} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. count-2-rev N/A

         \[\leadsto \sqrt{\color{blue}{n + n}} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \sqrt{\color{blue}{n + n}} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{n + n} \cdot \color{blue}{\sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot U}} \]

   3. Applied rewrites 25.9%

      \[\leadsto \color{blue}{\sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(U* - U, \frac{\ell \cdot \ell}{Om \cdot Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right) \cdot U}} \]

   4. Taylor expanded in n around 0

      \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \left(t + \color{blue}{-2 \cdot \frac{{\ell}^{2}}{Om}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{\color{blue}{Om}}\right)} \]

      5. lower-pow.f64 25.3

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)} \]

   6. Applied rewrites 25.3%

      \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{n + n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{n + n}} \cdot \sqrt{U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{n + n} \cdot \color{blue}{\sqrt{U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]

      4. sqrt-unprod N/A

         \[\leadsto \color{blue}{\sqrt{\left(n + n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(n + n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      6. lower-*.f64 43.9

         \[\leadsto \sqrt{\color{blue}{\left(n + n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

   8. Applied rewrites 47.5%

      \[\leadsto \color{blue}{\sqrt{\left(n + n\right) \cdot \left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot U\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 38.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;t\_1 \leq 10^{+140}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))))
   (if (<= t_1 0.0)
     (* (sqrt (+ n n)) (sqrt (* U t)))
     (if (<= t_1 1e+140)
       (sqrt (* 2.0 (* (* U n) t)))
       (sqrt (* (+ n n) (* U t)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt((n + n)) * sqrt((U * t));
	} else if (t_1 <= 1e+140) {
		tmp = sqrt((2.0 * ((U * n) * t)));
	} else {
		tmp = sqrt(((n + n) * (U * t)));
	}
	return tmp;
}

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l_m, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l_m * l_m) / om))) - ((n * ((l_m / om) ** 2.0d0)) * (u - u_42)))))
    if (t_1 <= 0.0d0) then
        tmp = sqrt((n + n)) * sqrt((u * t))
    else if (t_1 <= 1d+140) then
        tmp = sqrt((2.0d0 * ((u * n) * t)))
    else
        tmp = sqrt(((n + n) * (u * t)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * Math.pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.sqrt((n + n)) * Math.sqrt((U * t));
	} else if (t_1 <= 1e+140) {
		tmp = Math.sqrt((2.0 * ((U * n) * t)));
	} else {
		tmp = Math.sqrt(((n + n) * (U * t)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * math.pow((l_m / Om), 2.0)) * (U - U_42_)))))
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.sqrt((n + n)) * math.sqrt((U * t))
	elif t_1 <= 1e+140:
		tmp = math.sqrt((2.0 * ((U * n) * t)))
	else:
		tmp = math.sqrt(((n + n) * (U * t)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(Float64(U * t)));
	elseif (t_1 <= 1e+140)
		tmp = sqrt(Float64(2.0 * Float64(Float64(U * n) * t)));
	else
		tmp = sqrt(Float64(Float64(n + n) * Float64(U * t)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * ((l_m / Om) ^ 2.0)) * (U - U_42_)))));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = sqrt((n + n)) * sqrt((U * t));
	elseif (t_1 <= 1e+140)
		tmp = sqrt((2.0 * ((U * n) * t)));
	else
		tmp = sqrt(((n + n) * (U * t)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+140], N[Sqrt[N[(2.0 * N[(N[(U * n), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 0.0

   1. Initial program 49.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      5. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot n}} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot n}} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. count-2-rev N/A

         \[\leadsto \sqrt{\color{blue}{n + n}} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \sqrt{\color{blue}{n + n}} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{n + n} \cdot \color{blue}{\sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot U}} \]

   3. Applied rewrites 25.9%

      \[\leadsto \color{blue}{\sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(U* - U, \frac{\ell \cdot \ell}{Om \cdot Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right) \cdot U}} \]

   4. Taylor expanded in t around inf

      \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{U \cdot t}} \]
   5. Step-by-step derivation

      1. lower-*.f64 20.8

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \color{blue}{t}} \]

   6. Applied rewrites 20.8%

      \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{U \cdot t}} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1.00000000000000006e140

   1. Initial program 49.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 36.2

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 36.2%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      5. lower-*.f64 35.9

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

   6. Applied rewrites 35.9%

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

   if 1.00000000000000006e140 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 49.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. sub-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + t\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\left(\mathsf{neg}\left(\color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right)\right) + t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell \cdot \ell}{Om}} + t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}} + t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} + t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} + t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{Om}}\right) + t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \ell\right) \cdot \frac{\ell}{Om}} + t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \ell, \frac{\ell}{Om}, t\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \ell}, \frac{\ell}{Om}, t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      13. metadata-eval 53.0

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(\color{blue}{-2} \cdot \ell, \frac{\ell}{Om}, t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.0%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} \cdot \left(U - U*\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \left(\color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}} \cdot n\right) \cdot \left(U - U*\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot n\right) \cdot \left(U - U*\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \cdot \left(U - U*\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot n\right)}\right) \cdot \left(U - U*\right)\right)} \]

      8. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)}\right)} \]

      10. lower-*.f64 54.6

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \frac{\ell}{Om} \cdot \color{blue}{\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)}\right)} \]

   5. Applied rewrites 54.6%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)}\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \frac{\ell}{Om} \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \frac{\ell}{Om} \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(2 \cdot n\right)} \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \frac{\ell}{Om} \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right)} \]

      4. count-2-rev N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(n + n\right)} \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \frac{\ell}{Om} \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(n + n\right)} \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \frac{\ell}{Om} \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(n + n\right) \cdot \left(U \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \frac{\ell}{Om} \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n + n\right) \cdot \left(U \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \frac{\ell}{Om} \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      8. lower-*.f64 55.5

         \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(U \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \frac{\ell}{Om} \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

   7. Applied rewrites 55.5%

      \[\leadsto \sqrt{\color{blue}{\left(n + n\right) \cdot \left(U \cdot \left(\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right) - \left(\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \frac{\ell}{Om}\right)\right)}} \]

   8. Taylor expanded in t around inf

      \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(U \cdot t\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f64 35.8

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(U \cdot \color{blue}{t}\right)} \]

   10. Applied rewrites 35.8%

       \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(U \cdot t\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 38.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ t_2 := \sqrt{\left(n + n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{-149}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+140}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
        (t_2 (sqrt (* (+ n n) (* U t)))))
   (if (<= t_1 4e-149)
     t_2
     (if (<= t_1 1e+140) (sqrt (* (* (+ n n) U) t)) t_2))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double t_2 = sqrt(((n + n) * (U * t)));
	double tmp;
	if (t_1 <= 4e-149) {
		tmp = t_2;
	} else if (t_1 <= 1e+140) {
		tmp = sqrt((((n + n) * U) * t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l_m, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l_m * l_m) / om))) - ((n * ((l_m / om) ** 2.0d0)) * (u - u_42)))))
    t_2 = sqrt(((n + n) * (u * t)))
    if (t_1 <= 4d-149) then
        tmp = t_2
    else if (t_1 <= 1d+140) then
        tmp = sqrt((((n + n) * u) * t))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * Math.pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double t_2 = Math.sqrt(((n + n) * (U * t)));
	double tmp;
	if (t_1 <= 4e-149) {
		tmp = t_2;
	} else if (t_1 <= 1e+140) {
		tmp = Math.sqrt((((n + n) * U) * t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * math.pow((l_m / Om), 2.0)) * (U - U_42_)))))
	t_2 = math.sqrt(((n + n) * (U * t)))
	tmp = 0
	if t_1 <= 4e-149:
		tmp = t_2
	elif t_1 <= 1e+140:
		tmp = math.sqrt((((n + n) * U) * t))
	else:
		tmp = t_2
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_)))))
	t_2 = sqrt(Float64(Float64(n + n) * Float64(U * t)))
	tmp = 0.0
	if (t_1 <= 4e-149)
		tmp = t_2;
	elseif (t_1 <= 1e+140)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * t));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * ((l_m / Om) ^ 2.0)) * (U - U_42_)))));
	t_2 = sqrt(((n + n) * (U * t)));
	tmp = 0.0;
	if (t_1 <= 4e-149)
		tmp = t_2;
	elseif (t_1 <= 1e+140)
		tmp = sqrt((((n + n) * U) * t));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 4e-149], t$95$2, If[LessEqual[t$95$1, 1e+140], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 3.99999999999999992e-149 or 1.00000000000000006e140 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 49.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. sub-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + t\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\left(\mathsf{neg}\left(\color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right)\right) + t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell \cdot \ell}{Om}} + t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}} + t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} + t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} + t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{Om}}\right) + t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \ell\right) \cdot \frac{\ell}{Om}} + t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \ell, \frac{\ell}{Om}, t\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \ell}, \frac{\ell}{Om}, t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      13. metadata-eval 53.0

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(\color{blue}{-2} \cdot \ell, \frac{\ell}{Om}, t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.0%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} \cdot \left(U - U*\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U - U*\right)\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \left(\color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}} \cdot n\right) \cdot \left(U - U*\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot n\right) \cdot \left(U - U*\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \color{blue}{\left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \cdot \left(U - U*\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot n\right)}\right) \cdot \left(U - U*\right)\right)} \]

      8. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)}\right)} \]

      10. lower-*.f64 54.6

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \frac{\ell}{Om} \cdot \color{blue}{\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)}\right)} \]

   5. Applied rewrites 54.6%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \color{blue}{\frac{\ell}{Om} \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)}\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \frac{\ell}{Om} \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \frac{\ell}{Om} \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(2 \cdot n\right)} \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \frac{\ell}{Om} \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right)} \]

      4. count-2-rev N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(n + n\right)} \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \frac{\ell}{Om} \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(n + n\right)} \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \frac{\ell}{Om} \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(n + n\right) \cdot \left(U \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \frac{\ell}{Om} \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n + n\right) \cdot \left(U \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \frac{\ell}{Om} \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      8. lower-*.f64 55.5

         \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(U \cdot \left(\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right) - \frac{\ell}{Om} \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

   7. Applied rewrites 55.5%

      \[\leadsto \sqrt{\color{blue}{\left(n + n\right) \cdot \left(U \cdot \left(\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right) - \left(\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \frac{\ell}{Om}\right)\right)}} \]

   8. Taylor expanded in t around inf

      \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(U \cdot t\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f64 35.8

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(U \cdot \color{blue}{t}\right)} \]

   10. Applied rewrites 35.8%

       \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(U \cdot t\right)}} \]

   if 3.99999999999999992e-149 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1.00000000000000006e140

   1. Initial program 49.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 36.2

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 36.2%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      5. lower-*.f64 35.9

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

   6. Applied rewrites 35.9%

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \color{blue}{t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      9. lower-*.f64 35.9

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      11. count-2-rev N/A

          \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \]

      12. lift-+.f64 35.9

          \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \]

   8. Applied rewrites 35.9%

      \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 36.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;Om \leq -4.7 \cdot 10^{+257}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= Om -4.7e+257)
   (sqrt (* (* (+ n n) U) t))
   (sqrt (* 2.0 (* U (* n t))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (Om <= -4.7e+257) {
		tmp = sqrt((((n + n) * U) * t));
	} else {
		tmp = sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l_m, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (om <= (-4.7d+257)) then
        tmp = sqrt((((n + n) * u) * t))
    else
        tmp = sqrt((2.0d0 * (u * (n * t))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (Om <= -4.7e+257) {
		tmp = Math.sqrt((((n + n) * U) * t));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if Om <= -4.7e+257:
		tmp = math.sqrt((((n + n) * U) * t))
	else:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (Om <= -4.7e+257)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * t));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (Om <= -4.7e+257)
		tmp = sqrt((((n + n) * U) * t));
	else
		tmp = sqrt((2.0 * (U * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[Om, -4.7e+257], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < -4.7e257

   1. Initial program 49.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 36.2

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 36.2%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      5. lower-*.f64 35.9

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

   6. Applied rewrites 35.9%

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \color{blue}{t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      9. lower-*.f64 35.9

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      11. count-2-rev N/A

          \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \]

      12. lift-+.f64 35.9

          \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \]

   8. Applied rewrites 35.9%

      \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \color{blue}{t}} \]

   if -4.7e257 < Om

   1. Initial program 49.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 36.2

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 36.2%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 35.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* (* (+ n n) U) t)))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return sqrt((((n + n) * U) * t));
}

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l_m, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((((n + n) * u) * t))
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return Math.sqrt((((n + n) * U) * t));
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return math.sqrt((((n + n) * U) * t))

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return sqrt(Float64(Float64(Float64(n + n) * U) * t))
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = sqrt((((n + n) * U) * t));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 49.1%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

2. Taylor expanded in t around inf

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

   3. lower-*.f64 36.2

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

4. Applied rewrites 36.2%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   3. associate-*r* N/A

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

   5. lower-*.f64 35.9

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

6. Applied rewrites 35.9%

   \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

   3. associate-*r* N/A

      \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \color{blue}{t}} \]

   4. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t} \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

   6. associate-*l* N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

   7. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

   8. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

   9. lower-*.f64 35.9

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

   10. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

   11. count-2-rev N/A

       \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \]

   12. lift-+.f64 35.9

       \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \]

8. Applied rewrites 35.9%

   \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \color{blue}{t}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025144 
(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*))))))