Toniolo and Linder, Equation (13)

Percentage Accurate: 49.0% → 66.7%

Time: 9.1s

Alternatives: 17

Speedup: 0.4×

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: 49.0% 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: 66.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{\left(t - \frac{\mathsf{fma}\left(l\_m \cdot l\_m, \frac{\left(U - U*\right) \cdot n}{Om}, \left(l\_m \cdot l\_m\right) \cdot 2\right)}{Om}\right) \cdot n} \cdot \sqrt{U}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(\frac{l\_m}{Om}, l\_m \cdot -2, t\right) - \left(\frac{l\_m}{Om} \cdot \left(\frac{l\_m}{Om} \cdot n\right)\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \frac{\mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right)}{Om}\right)\right)}\\ \end{array} \end{array} \]

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*)))))))
   (if (<= t_1 0.0)
     (*
      (sqrt 2.0)
      (*
       (sqrt
        (*
         (-
          t
          (/ (fma (* l_m l_m) (/ (* (- U U*) n) Om) (* (* l_m l_m) 2.0)) Om))
         n))
       (sqrt U)))
     (if (<= t_1 2e+146)
       (sqrt
        (*
         (* (+ n n) U)
         (-
          (fma (/ l_m Om) (* l_m -2.0) t)
          (* (* (/ l_m Om) (* (/ l_m Om) n)) (- U U*)))))
       (*
        l_m
        (sqrt (* -2.0 (* U (* n (/ (fma (- U U*) (/ n Om) 2.0) Om))))))))))

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(2.0) * (sqrt(((t - (fma((l_m * l_m), (((U - U_42_) * n) / Om), ((l_m * l_m) * 2.0)) / Om)) * n)) * sqrt(U));
	} else if (t_1 <= 2e+146) {
		tmp = sqrt((((n + n) * U) * (fma((l_m / Om), (l_m * -2.0), t) - (((l_m / Om) * ((l_m / Om) * n)) * (U - U_42_)))));
	} else {
		tmp = l_m * sqrt((-2.0 * (U * (n * (fma((U - U_42_), (n / Om), 2.0) / Om)))));
	}
	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(2.0) * Float64(sqrt(Float64(Float64(t - Float64(fma(Float64(l_m * l_m), Float64(Float64(Float64(U - U_42_) * n) / Om), Float64(Float64(l_m * l_m) * 2.0)) / Om)) * n)) * sqrt(U)));
	elseif (t_1 <= 2e+146)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * Float64(fma(Float64(l_m / Om), Float64(l_m * -2.0), t) - Float64(Float64(Float64(l_m / Om) * Float64(Float64(l_m / Om) * n)) * Float64(U - U_42_)))));
	else
		tmp = Float64(l_m * sqrt(Float64(-2.0 * Float64(U * Float64(n * Float64(fma(Float64(U - U_42_), Float64(n / Om), 2.0) / Om))))));
	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]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(N[(t - N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(N[(U - U$42$), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+146], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(l$95$m * -2.0), $MachinePrecision] + t), $MachinePrecision] - N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-2.0 * N[(U * N[(n * N[(N[(N[(U - U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision] + 2.0), $MachinePrecision] / Om), $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 49.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 52.6

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

   3. Applied rewrites 52.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*r/ N/A

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

      15. lift-*.f64 N/A

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

      16. frac-times N/A

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

      17. lift-/.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. associate-*l* N/A

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

      20. lower-*.f64 N/A

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

      21. lower-*.f64 53.6

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

   5. Applied rewrites 53.6%

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

   7. Applied rewrites 50.6%

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

   9. Applied rewrites 26.1%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1.99999999999999987e146

   1. Initial program 49.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 52.6

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

   3. Applied rewrites 52.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*r/ N/A

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

      15. lift-*.f64 N/A

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

      16. frac-times N/A

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

      17. lift-/.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. associate-*l* N/A

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

      20. lower-*.f64 N/A

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

      21. lower-*.f64 53.6

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

   5. Applied rewrites 53.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 53.6

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

   7. Applied rewrites 53.6%

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

   if 1.99999999999999987e146 < (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 49.0%

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

   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 28.4

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

   4. Applied rewrites 28.4%

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

   5. Taylor expanded in Om around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 31.6

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

   7. Applied rewrites 31.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 32.7

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

   9. Applied rewrites 32.7%

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

Alternative 2: 66.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-4, \frac{{l\_m}^{2} \cdot n}{Om}, 2 \cdot \left(n \cdot t\right)\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(\frac{l\_m}{Om}, l\_m \cdot -2, t\right) - \left(\frac{l\_m}{Om} \cdot \left(\frac{l\_m}{Om} \cdot n\right)\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \frac{\mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right)}{Om}\right)\right)}\\ \end{array} \end{array} \]

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*)))))))
   (if (<= t_1 0.0)
     (* (sqrt (fma -4.0 (/ (* (pow l_m 2.0) n) Om) (* 2.0 (* n t)))) (sqrt U))
     (if (<= t_1 2e+146)
       (sqrt
        (*
         (* (+ n n) U)
         (-
          (fma (/ l_m Om) (* l_m -2.0) t)
          (* (* (/ l_m Om) (* (/ l_m Om) n)) (- U U*)))))
       (*
        l_m
        (sqrt (* -2.0 (* U (* n (/ (fma (- U U*) (/ n Om) 2.0) Om))))))))))

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(fma(-4.0, ((pow(l_m, 2.0) * n) / Om), (2.0 * (n * t)))) * sqrt(U);
	} else if (t_1 <= 2e+146) {
		tmp = sqrt((((n + n) * U) * (fma((l_m / Om), (l_m * -2.0), t) - (((l_m / Om) * ((l_m / Om) * n)) * (U - U_42_)))));
	} else {
		tmp = l_m * sqrt((-2.0 * (U * (n * (fma((U - U_42_), (n / Om), 2.0) / Om)))));
	}
	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(fma(-4.0, Float64(Float64((l_m ^ 2.0) * n) / Om), Float64(2.0 * Float64(n * t)))) * sqrt(U));
	elseif (t_1 <= 2e+146)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * Float64(fma(Float64(l_m / Om), Float64(l_m * -2.0), t) - Float64(Float64(Float64(l_m / Om) * Float64(Float64(l_m / Om) * n)) * Float64(U - U_42_)))));
	else
		tmp = Float64(l_m * sqrt(Float64(-2.0 * Float64(U * Float64(n * Float64(fma(Float64(U - U_42_), Float64(n / Om), 2.0) / Om))))));
	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]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[(-4.0 * N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+146], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(l$95$m * -2.0), $MachinePrecision] + t), $MachinePrecision] - N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-2.0 * N[(U * N[(n * N[(N[(N[(U - U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision] + 2.0), $MachinePrecision] / Om), $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 49.0%

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

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 24.0%

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

   4. Taylor expanded in Om around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 25.2

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

   6. Applied rewrites 25.2%

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

   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*))))) < 1.99999999999999987e146

   1. Initial program 49.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 52.6

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

   3. Applied rewrites 52.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*r/ N/A

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

      15. lift-*.f64 N/A

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

      16. frac-times N/A

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

      17. lift-/.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. associate-*l* N/A

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

      20. lower-*.f64 N/A

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

      21. lower-*.f64 53.6

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

   5. Applied rewrites 53.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 53.6

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

   7. Applied rewrites 53.6%

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

   if 1.99999999999999987e146 < (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 49.0%

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

   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 28.4

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

   4. Applied rewrites 28.4%

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

   5. Taylor expanded in Om around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 31.6

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

   7. Applied rewrites 31.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 32.7

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

   9. Applied rewrites 32.7%

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

Alternative 3: 66.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \sqrt{t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-4, \frac{{l\_m}^{2} \cdot n}{Om}, 2 \cdot \left(n \cdot t\right)\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(\frac{l\_m}{Om}, \left(\frac{l\_m}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(-2, t\_1, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \frac{\mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right)}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (sqrt
          (*
           t_2
           (- (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))))
   (if (<= t_3 0.0)
     (* (sqrt (fma -4.0 (/ (* (pow l_m 2.0) n) Om) (* 2.0 (* n t)))) (sqrt U))
     (if (<= t_3 2e+146)
       (sqrt
        (*
         t_2
         (fma (/ l_m Om) (* (* (/ l_m Om) n) (- U* U)) (fma -2.0 t_1 t))))
       (*
        l_m
        (sqrt (* -2.0 (* U (* n (/ (fma (- U U*) (/ n Om) 2.0) Om))))))))))

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 * l_m) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = sqrt((t_2 * ((t - (2.0 * t_1)) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt(fma(-4.0, ((pow(l_m, 2.0) * n) / Om), (2.0 * (n * t)))) * sqrt(U);
	} else if (t_3 <= 2e+146) {
		tmp = sqrt((t_2 * fma((l_m / Om), (((l_m / Om) * n) * (U_42_ - U)), fma(-2.0, t_1, t))));
	} else {
		tmp = l_m * sqrt((-2.0 * (U * (n * (fma((U - U_42_), (n / Om), 2.0) / Om)))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m * l_m) / Om)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(fma(-4.0, Float64(Float64((l_m ^ 2.0) * n) / Om), Float64(2.0 * Float64(n * t)))) * sqrt(U));
	elseif (t_3 <= 2e+146)
		tmp = sqrt(Float64(t_2 * fma(Float64(l_m / Om), Float64(Float64(Float64(l_m / Om) * n) * Float64(U_42_ - U)), fma(-2.0, t_1, t))));
	else
		tmp = Float64(l_m * sqrt(Float64(-2.0 * Float64(U * Float64(n * Float64(fma(Float64(U - U_42_), Float64(n / Om), 2.0) / Om))))));
	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 * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $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$3, 0.0], N[(N[Sqrt[N[(-4.0 * N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+146], N[Sqrt[N[(t$95$2 * N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-2.0 * N[(U * N[(n * N[(N[(N[(U - U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision] + 2.0), $MachinePrecision] / Om), $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 49.0%

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

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 24.0%

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

   4. Taylor expanded in Om around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 25.2

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

   6. Applied rewrites 25.2%

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

   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*))))) < 1.99999999999999987e146

   1. Initial program 49.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. distribute-rgt-neg-in N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 50.4%

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

   if 1.99999999999999987e146 < (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 49.0%

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

   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 28.4

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

   4. Applied rewrites 28.4%

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

   5. Taylor expanded in Om around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 31.6

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

   7. Applied rewrites 31.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 32.7

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

   9. Applied rewrites 32.7%

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

Alternative 4: 66.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \sqrt{t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-4, \frac{{l\_m}^{2} \cdot n}{Om}, 2 \cdot \left(n \cdot t\right)\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(\left(U* - U\right) \cdot \frac{l\_m}{Om}, \frac{l\_m}{Om} \cdot n, \mathsf{fma}\left(-2, t\_1, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \frac{\mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right)}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (sqrt
          (*
           t_2
           (- (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))))
   (if (<= t_3 0.0)
     (* (sqrt (fma -4.0 (/ (* (pow l_m 2.0) n) Om) (* 2.0 (* n t)))) (sqrt U))
     (if (<= t_3 2e+146)
       (sqrt
        (*
         t_2
         (fma (* (- U* U) (/ l_m Om)) (* (/ l_m Om) n) (fma -2.0 t_1 t))))
       (*
        l_m
        (sqrt (* -2.0 (* U (* n (/ (fma (- U U*) (/ n Om) 2.0) Om))))))))))

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 * l_m) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = sqrt((t_2 * ((t - (2.0 * t_1)) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt(fma(-4.0, ((pow(l_m, 2.0) * n) / Om), (2.0 * (n * t)))) * sqrt(U);
	} else if (t_3 <= 2e+146) {
		tmp = sqrt((t_2 * fma(((U_42_ - U) * (l_m / Om)), ((l_m / Om) * n), fma(-2.0, t_1, t))));
	} else {
		tmp = l_m * sqrt((-2.0 * (U * (n * (fma((U - U_42_), (n / Om), 2.0) / Om)))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m * l_m) / Om)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(fma(-4.0, Float64(Float64((l_m ^ 2.0) * n) / Om), Float64(2.0 * Float64(n * t)))) * sqrt(U));
	elseif (t_3 <= 2e+146)
		tmp = sqrt(Float64(t_2 * fma(Float64(Float64(U_42_ - U) * Float64(l_m / Om)), Float64(Float64(l_m / Om) * n), fma(-2.0, t_1, t))));
	else
		tmp = Float64(l_m * sqrt(Float64(-2.0 * Float64(U * Float64(n * Float64(fma(Float64(U - U_42_), Float64(n / Om), 2.0) / Om))))));
	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 * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $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$3, 0.0], N[(N[Sqrt[N[(-4.0 * N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+146], N[Sqrt[N[(t$95$2 * N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision] + N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-2.0 * N[(U * N[(n * N[(N[(N[(U - U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision] + 2.0), $MachinePrecision] / Om), $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 49.0%

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

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 24.0%

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

   4. Taylor expanded in Om around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 25.2

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

   6. Applied rewrites 25.2%

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

   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*))))) < 1.99999999999999987e146

   1. Initial program 49.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. associate-*r* N/A

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

   3. Applied rewrites 50.3%

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

   if 1.99999999999999987e146 < (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 49.0%

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

   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 28.4

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

   4. Applied rewrites 28.4%

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

   5. Taylor expanded in Om around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 31.6

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

   7. Applied rewrites 31.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 32.7

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

   9. Applied rewrites 32.7%

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

Alternative 5: 66.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \sqrt{t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-4, \frac{{l\_m}^{2} \cdot n}{Om}, 2 \cdot \left(n \cdot t\right)\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(U*, \left(\frac{l\_m}{Om} \cdot \frac{l\_m}{Om}\right) \cdot n, \mathsf{fma}\left(-2, t\_1, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \frac{\mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right)}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (sqrt
          (*
           t_2
           (- (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))))
   (if (<= t_3 0.0)
     (* (sqrt (fma -4.0 (/ (* (pow l_m 2.0) n) Om) (* 2.0 (* n t)))) (sqrt U))
     (if (<= t_3 2e+146)
       (sqrt (* t_2 (fma U* (* (* (/ l_m Om) (/ l_m Om)) n) (fma -2.0 t_1 t))))
       (*
        l_m
        (sqrt (* -2.0 (* U (* n (/ (fma (- U U*) (/ n Om) 2.0) Om))))))))))

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 * l_m) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = sqrt((t_2 * ((t - (2.0 * t_1)) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt(fma(-4.0, ((pow(l_m, 2.0) * n) / Om), (2.0 * (n * t)))) * sqrt(U);
	} else if (t_3 <= 2e+146) {
		tmp = sqrt((t_2 * fma(U_42_, (((l_m / Om) * (l_m / Om)) * n), fma(-2.0, t_1, t))));
	} else {
		tmp = l_m * sqrt((-2.0 * (U * (n * (fma((U - U_42_), (n / Om), 2.0) / Om)))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m * l_m) / Om)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(fma(-4.0, Float64(Float64((l_m ^ 2.0) * n) / Om), Float64(2.0 * Float64(n * t)))) * sqrt(U));
	elseif (t_3 <= 2e+146)
		tmp = sqrt(Float64(t_2 * fma(U_42_, Float64(Float64(Float64(l_m / Om) * Float64(l_m / Om)) * n), fma(-2.0, t_1, t))));
	else
		tmp = Float64(l_m * sqrt(Float64(-2.0 * Float64(U * Float64(n * Float64(fma(Float64(U - U_42_), Float64(n / Om), 2.0) / Om))))));
	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 * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $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$3, 0.0], N[(N[Sqrt[N[(-4.0 * N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+146], N[Sqrt[N[(t$95$2 * N[(U$42$ * N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] + N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-2.0 * N[(U * N[(n * N[(N[(N[(U - U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision] + 2.0), $MachinePrecision] / Om), $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 49.0%

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

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 24.0%

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

   4. Taylor expanded in Om around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 25.2

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

   6. Applied rewrites 25.2%

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

   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*))))) < 1.99999999999999987e146

   1. Initial program 49.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-fma.f64 N/A

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

   3. Applied rewrites 45.3%

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

   4. Taylor expanded in U around 0

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

   6. Applied rewrites 45.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. frac-times N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 49.0

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

   8. Applied rewrites 49.0%

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

   if 1.99999999999999987e146 < (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 49.0%

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

   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 28.4

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

   4. Applied rewrites 28.4%

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

   5. Taylor expanded in Om around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 31.6

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

   7. Applied rewrites 31.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 32.7

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

   9. Applied rewrites 32.7%

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

Alternative 6: 66.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \sqrt{t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(t + -2 \cdot \frac{{l\_m}^{2}}{Om}\right)\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(U*, \left(\frac{l\_m}{Om} \cdot \frac{l\_m}{Om}\right) \cdot n, \mathsf{fma}\left(-2, t\_1, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \frac{\mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right)}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (sqrt
          (*
           t_2
           (- (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))))
   (if (<= t_3 0.0)
     (* (sqrt (* 2.0 (* n (+ t (* -2.0 (/ (pow l_m 2.0) Om)))))) (sqrt U))
     (if (<= t_3 2e+146)
       (sqrt (* t_2 (fma U* (* (* (/ l_m Om) (/ l_m Om)) n) (fma -2.0 t_1 t))))
       (*
        l_m
        (sqrt (* -2.0 (* U (* n (/ (fma (- U U*) (/ n Om) 2.0) Om))))))))))

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 * l_m) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = sqrt((t_2 * ((t - (2.0 * t_1)) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((2.0 * (n * (t + (-2.0 * (pow(l_m, 2.0) / Om)))))) * sqrt(U);
	} else if (t_3 <= 2e+146) {
		tmp = sqrt((t_2 * fma(U_42_, (((l_m / Om) * (l_m / Om)) * n), fma(-2.0, t_1, t))));
	} else {
		tmp = l_m * sqrt((-2.0 * (U * (n * (fma((U - U_42_), (n / Om), 2.0) / Om)))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m * l_m) / Om)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * Float64(n * Float64(t + Float64(-2.0 * Float64((l_m ^ 2.0) / Om)))))) * sqrt(U));
	elseif (t_3 <= 2e+146)
		tmp = sqrt(Float64(t_2 * fma(U_42_, Float64(Float64(Float64(l_m / Om) * Float64(l_m / Om)) * n), fma(-2.0, t_1, t))));
	else
		tmp = Float64(l_m * sqrt(Float64(-2.0 * Float64(U * Float64(n * Float64(fma(Float64(U - U_42_), Float64(n / Om), 2.0) / Om))))));
	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 * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $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$3, 0.0], N[(N[Sqrt[N[(2.0 * N[(n * N[(t + N[(-2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+146], N[Sqrt[N[(t$95$2 * N[(U$42$ * N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] + N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-2.0 * N[(U * N[(n * N[(N[(N[(U - U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision] + 2.0), $MachinePrecision] / Om), $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 49.0%

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

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 24.0%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 25.7

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

   6. Applied rewrites 25.7%

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

   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*))))) < 1.99999999999999987e146

   1. Initial program 49.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-fma.f64 N/A

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

   3. Applied rewrites 45.3%

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

   4. Taylor expanded in U around 0

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

   6. Applied rewrites 45.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. frac-times N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 49.0

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

   8. Applied rewrites 49.0%

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

   if 1.99999999999999987e146 < (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 49.0%

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

   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 28.4

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

   4. Applied rewrites 28.4%

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

   5. Taylor expanded in Om around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 31.6

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

   7. Applied rewrites 31.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 32.7

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

   9. Applied rewrites 32.7%

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

Alternative 7: 62.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := t + -2 \cdot \frac{{l\_m}^{2}}{Om}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot t\_1\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \frac{\mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right)}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (+ t (* -2.0 (/ (pow l_m 2.0) Om))))
        (t_2
         (sqrt
          (*
           (* (* 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 (* 2.0 (* n t_1))) (sqrt U))
     (if (<= t_2 2e+146)
       (sqrt (* (* (+ n n) U) t_1))
       (*
        l_m
        (sqrt (* -2.0 (* U (* n (/ (fma (- U U*) (/ n Om) 2.0) Om))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = t + (-2.0 * (pow(l_m, 2.0) / Om));
	double t_2 = 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_2 <= 0.0) {
		tmp = sqrt((2.0 * (n * t_1))) * sqrt(U);
	} else if (t_2 <= 2e+146) {
		tmp = sqrt((((n + n) * U) * t_1));
	} else {
		tmp = l_m * sqrt((-2.0 * (U * (n * (fma((U - U_42_), (n / Om), 2.0) / Om)))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(t + Float64(-2.0 * Float64((l_m ^ 2.0) / Om)))
	t_2 = 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_2 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * Float64(n * t_1))) * sqrt(U));
	elseif (t_2 <= 2e+146)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * t_1));
	else
		tmp = Float64(l_m * sqrt(Float64(-2.0 * Float64(U * Float64(n * Float64(fma(Float64(U - U_42_), Float64(n / Om), 2.0) / Om))))));
	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[(t + N[(-2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$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$2, 0.0], N[(N[Sqrt[N[(2.0 * N[(n * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+146], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-2.0 * N[(U * N[(n * N[(N[(N[(U - U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision] + 2.0), $MachinePrecision] / Om), $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 49.0%

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

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 24.0%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 25.7

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

   6. Applied rewrites 25.7%

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

   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*))))) < 1.99999999999999987e146

   1. Initial program 49.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 52.6

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

   3. Applied rewrites 52.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*r/ N/A

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

      15. lift-*.f64 N/A

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

      16. frac-times N/A

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

      17. lift-/.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. associate-*l* N/A

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

      20. lower-*.f64 N/A

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

      21. lower-*.f64 53.6

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

   5. Applied rewrites 53.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 53.6

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

   7. Applied rewrites 53.6%

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

   8. Taylor expanded in n around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 42.6

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

   10. Applied rewrites 42.6%

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

   if 1.99999999999999987e146 < (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 49.0%

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

   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 28.4

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

   4. Applied rewrites 28.4%

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

   5. Taylor expanded in Om around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 31.6

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

   7. Applied rewrites 31.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 32.7

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

   9. Applied rewrites 32.7%

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

Alternative 8: 61.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot t\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{{l\_m}^{2}}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \frac{\mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right)}{Om}\right)\right)}\\ \end{array} \end{array} \]

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*)))))))
   (if (<= t_1 0.0)
     (* (sqrt (* 2.0 (* n t))) (sqrt U))
     (if (<= t_1 2e+146)
       (sqrt (* (* (+ n n) U) (+ t (* -2.0 (/ (pow l_m 2.0) Om)))))
       (*
        l_m
        (sqrt (* -2.0 (* U (* n (/ (fma (- U U*) (/ n Om) 2.0) Om))))))))))

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((2.0 * (n * t))) * sqrt(U);
	} else if (t_1 <= 2e+146) {
		tmp = sqrt((((n + n) * U) * (t + (-2.0 * (pow(l_m, 2.0) / Om)))));
	} else {
		tmp = l_m * sqrt((-2.0 * (U * (n * (fma((U - U_42_), (n / Om), 2.0) / Om)))));
	}
	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(2.0 * Float64(n * t))) * sqrt(U));
	elseif (t_1 <= 2e+146)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * Float64(t + Float64(-2.0 * Float64((l_m ^ 2.0) / Om)))));
	else
		tmp = Float64(l_m * sqrt(Float64(-2.0 * Float64(U * Float64(n * Float64(fma(Float64(U - U_42_), Float64(n / Om), 2.0) / Om))))));
	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]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[(2.0 * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+146], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(-2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-2.0 * N[(U * N[(n * N[(N[(N[(U - U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision] + 2.0), $MachinePrecision] / Om), $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 49.0%

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

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 24.0%

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

   4. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 21.0

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

   6. Applied rewrites 21.0%

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

   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*))))) < 1.99999999999999987e146

   1. Initial program 49.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 52.6

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

   3. Applied rewrites 52.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*r/ N/A

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

      15. lift-*.f64 N/A

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

      16. frac-times N/A

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

      17. lift-/.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. associate-*l* N/A

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

      20. lower-*.f64 N/A

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

      21. lower-*.f64 53.6

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

   5. Applied rewrites 53.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 53.6

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

   7. Applied rewrites 53.6%

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

   8. Taylor expanded in n around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 42.6

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

   10. Applied rewrites 42.6%

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

   if 1.99999999999999987e146 < (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 49.0%

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

   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 28.4

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

   4. Applied rewrites 28.4%

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

   5. Taylor expanded in Om around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 31.6

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

   7. Applied rewrites 31.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 32.7

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

   9. Applied rewrites 32.7%

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

Alternative 9: 57.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot t\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{\left|t \cdot \left(U \cdot \left(n + n\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \frac{\mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right)}{Om}\right)\right)}\\ \end{array} \end{array} \]

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*)))))))
   (if (<= t_1 0.0)
     (* (sqrt (* 2.0 (* n t))) (sqrt U))
     (if (<= t_1 2e+146)
       (sqrt (fabs (* t (* U (+ n n)))))
       (*
        l_m
        (sqrt (* -2.0 (* U (* n (/ (fma (- U U*) (/ n Om) 2.0) Om))))))))))

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((2.0 * (n * t))) * sqrt(U);
	} else if (t_1 <= 2e+146) {
		tmp = sqrt(fabs((t * (U * (n + n)))));
	} else {
		tmp = l_m * sqrt((-2.0 * (U * (n * (fma((U - U_42_), (n / Om), 2.0) / Om)))));
	}
	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(2.0 * Float64(n * t))) * sqrt(U));
	elseif (t_1 <= 2e+146)
		tmp = sqrt(abs(Float64(t * Float64(U * Float64(n + n)))));
	else
		tmp = Float64(l_m * sqrt(Float64(-2.0 * Float64(U * Float64(n * Float64(fma(Float64(U - U_42_), Float64(n / Om), 2.0) / Om))))));
	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]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[(2.0 * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+146], N[Sqrt[N[Abs[N[(t * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-2.0 * N[(U * N[(n * N[(N[(N[(U - U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision] + 2.0), $MachinePrecision] / Om), $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 49.0%

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

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 24.0%

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

   4. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 21.0

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

   6. Applied rewrites 21.0%

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

   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*))))) < 1.99999999999999987e146

   1. Initial program 49.0%

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

   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 35.5

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

   4. Applied rewrites 35.5%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

   6. Applied rewrites 37.5%

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

   if 1.99999999999999987e146 < (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 49.0%

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

   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 28.4

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

   4. Applied rewrites 28.4%

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

   5. Taylor expanded in Om around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 31.6

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

   7. Applied rewrites 31.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 32.7

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

   9. Applied rewrites 32.7%

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

Alternative 10: 57.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-242}:\\ \;\;\;\;\sqrt{\left|\left(\left(U + U\right) \cdot t\right) \cdot n\right|}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\sqrt{\left|t \cdot \left(U \cdot \left(n + n\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{\left(\left(-2 \cdot U\right) \cdot \frac{\mathsf{fma}\left(\frac{U - U*}{Om}, n, 2\right)}{Om}\right) \cdot n}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* (* 2.0 n) U)
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_1 5e-242)
     (sqrt (fabs (* (* (+ U U) t) n)))
     (if (<= t_1 2e+292)
       (sqrt (fabs (* t (* U (+ n n)))))
       (*
        l_m
        (sqrt (* (* (* -2.0 U) (/ (fma (/ (- U U*) Om) n 2.0) Om)) n)))))))

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) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 5e-242) {
		tmp = sqrt(fabs((((U + U) * t) * n)));
	} else if (t_1 <= 2e+292) {
		tmp = sqrt(fabs((t * (U * (n + n)))));
	} else {
		tmp = l_m * sqrt((((-2.0 * U) * (fma(((U - U_42_) / Om), n, 2.0) / Om)) * n));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = 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 <= 5e-242)
		tmp = sqrt(abs(Float64(Float64(Float64(U + U) * t) * n)));
	elseif (t_1 <= 2e+292)
		tmp = sqrt(abs(Float64(t * Float64(U * Float64(n + n)))));
	else
		tmp = Float64(l_m * sqrt(Float64(Float64(Float64(-2.0 * U) * Float64(fma(Float64(Float64(U - U_42_) / Om), n, 2.0) / Om)) * n)));
	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[(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$1, 5e-242], N[Sqrt[N[Abs[N[(N[(N[(U + U), $MachinePrecision] * t), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 2e+292], N[Sqrt[N[Abs[N[(t * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(N[(N[(-2.0 * U), $MachinePrecision] * N[(N[(N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] * n + 2.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.9999999999999998e-242

   1. Initial program 49.0%

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

   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 35.5

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

   4. Applied rewrites 35.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 35.5

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 35.5

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

   6. Applied rewrites 35.5%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

   8. Applied rewrites 37.4%

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

   if 4.9999999999999998e-242 < (*.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*)))) < 2e292

   1. Initial program 49.0%

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

   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 35.5

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

   4. Applied rewrites 35.5%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

   6. Applied rewrites 37.5%

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

   if 2e292 < (*.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 49.0%

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

   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 28.4

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

   4. Applied rewrites 28.4%

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

   5. Taylor expanded in Om around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 31.6

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

   7. Applied rewrites 31.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   9. Applied rewrites 33.2%

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

Alternative 11: 56.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot t\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{\left|t \cdot \left(U \cdot \left(n + n\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{\mathsf{fma}\left(\frac{U - U*}{Om}, n, 2\right)}{Om} \cdot n\right) \cdot U\right) \cdot -2} \cdot l\_m\\ \end{array} \end{array} \]

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 (* 2.0 (* n t))) (sqrt U))
     (if (<= t_1 2e+146)
       (sqrt (fabs (* t (* U (+ n n)))))
       (*
        (sqrt (* (* (* (/ (fma (/ (- U U*) Om) n 2.0) Om) n) U) -2.0))
        l_m)))))

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((2.0 * (n * t))) * sqrt(U);
	} else if (t_1 <= 2e+146) {
		tmp = sqrt(fabs((t * (U * (n + n)))));
	} else {
		tmp = sqrt(((((fma(((U - U_42_) / Om), n, 2.0) / Om) * n) * U) * -2.0)) * l_m;
	}
	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(2.0 * Float64(n * t))) * sqrt(U));
	elseif (t_1 <= 2e+146)
		tmp = sqrt(abs(Float64(t * Float64(U * Float64(n + n)))));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(fma(Float64(Float64(U - U_42_) / Om), n, 2.0) / Om) * n) * U) * -2.0)) * l_m);
	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]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[(2.0 * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+146], N[Sqrt[N[Abs[N[(t * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(N[(N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] * n + 2.0), $MachinePrecision] / Om), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] * l$95$m), $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 49.0%

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

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 24.0%

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

   4. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 21.0

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

   6. Applied rewrites 21.0%

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

   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*))))) < 1.99999999999999987e146

   1. Initial program 49.0%

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

   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 35.5

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

   4. Applied rewrites 35.5%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

   6. Applied rewrites 37.5%

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

   if 1.99999999999999987e146 < (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 49.0%

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

   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 28.4

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

   4. Applied rewrites 28.4%

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

   5. Taylor expanded in Om around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 31.6

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

   7. Applied rewrites 31.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 31.6

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

   9. Applied rewrites 31.9%

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

Alternative 12: 42.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.75 \cdot 10^{-169}:\\ \;\;\;\;\sqrt{\left(U + U\right) \cdot \left(t \cdot n\right)}\\ \mathbf{elif}\;l\_m \leq 4 \cdot 10^{+96}:\\ \;\;\;\;\sqrt{\left|t \cdot \left(U \cdot \left(n + n\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{-4 \cdot \frac{U \cdot n}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.75e-169)
   (sqrt (* (+ U U) (* t n)))
   (if (<= l_m 4e+96)
     (sqrt (fabs (* t (* U (+ n n)))))
     (* l_m (sqrt (* -4.0 (/ (* U n) Om)))))))

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 <= 1.75e-169) {
		tmp = sqrt(((U + U) * (t * n)));
	} else if (l_m <= 4e+96) {
		tmp = sqrt(fabs((t * (U * (n + n)))));
	} else {
		tmp = l_m * sqrt((-4.0 * ((U * n) / Om)));
	}
	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 (l_m <= 1.75d-169) then
        tmp = sqrt(((u + u) * (t * n)))
    else if (l_m <= 4d+96) then
        tmp = sqrt(abs((t * (u * (n + n)))))
    else
        tmp = l_m * sqrt(((-4.0d0) * ((u * n) / om)))
    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 (l_m <= 1.75e-169) {
		tmp = Math.sqrt(((U + U) * (t * n)));
	} else if (l_m <= 4e+96) {
		tmp = Math.sqrt(Math.abs((t * (U * (n + n)))));
	} else {
		tmp = l_m * Math.sqrt((-4.0 * ((U * n) / Om)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if l_m <= 1.75e-169:
		tmp = math.sqrt(((U + U) * (t * n)))
	elif l_m <= 4e+96:
		tmp = math.sqrt(math.fabs((t * (U * (n + n)))))
	else:
		tmp = l_m * math.sqrt((-4.0 * ((U * n) / Om)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.75e-169)
		tmp = sqrt(Float64(Float64(U + U) * Float64(t * n)));
	elseif (l_m <= 4e+96)
		tmp = sqrt(abs(Float64(t * Float64(U * Float64(n + n)))));
	else
		tmp = Float64(l_m * sqrt(Float64(-4.0 * Float64(Float64(U * n) / Om))));
	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 (l_m <= 1.75e-169)
		tmp = sqrt(((U + U) * (t * n)));
	elseif (l_m <= 4e+96)
		tmp = sqrt(abs((t * (U * (n + n)))));
	else
		tmp = l_m * sqrt((-4.0 * ((U * n) / Om)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 1.75e-169], N[Sqrt[N[(N[(U + U), $MachinePrecision] * N[(t * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 4e+96], N[Sqrt[N[Abs[N[(t * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-4.0 * N[(N[(U * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.7500000000000001e-169

   1. Initial program 49.0%

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

   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 35.5

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

   4. Applied rewrites 35.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 35.5

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 35.5

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

   6. Applied rewrites 35.5%

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

   if 1.7500000000000001e-169 < l < 4.0000000000000002e96

   1. Initial program 49.0%

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

   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 35.5

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

   4. Applied rewrites 35.5%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

   6. Applied rewrites 37.5%

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

   if 4.0000000000000002e96 < l

   1. Initial program 49.0%

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

   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 28.4

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

   4. Applied rewrites 28.4%

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

   5. Taylor expanded in n around 0

      \[\leadsto \ell \cdot \sqrt{-4 \cdot \frac{U \cdot n}{Om}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-4 \cdot \frac{U \cdot n}{Om}} \]

      2. lower-/.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-4 \cdot \frac{U \cdot n}{Om}} \]

      3. lower-*.f64 17.5

         \[\leadsto \ell \cdot \sqrt{-4 \cdot \frac{U \cdot n}{Om}} \]

   7. Applied rewrites 17.5%

      \[\leadsto \ell \cdot \sqrt{-4 \cdot \frac{U \cdot n}{Om}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 40.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_))))) <= 0.0) {
		tmp = sqrt((2.0 * (n * t))) * sqrt(U);
	} else {
		tmp = sqrt(fabs((t * (U * (n + n)))));
	}
	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 (sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l_m * l_m) / om))) - ((n * ((l_m / om) ** 2.0d0)) * (u - u_42))))) <= 0.0d0) then
        tmp = sqrt((2.0d0 * (n * t))) * sqrt(u)
    else
        tmp = sqrt(abs((t * (u * (n + n)))))
    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 (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_))))) <= 0.0) {
		tmp = Math.sqrt((2.0 * (n * t))) * Math.sqrt(U);
	} else {
		tmp = Math.sqrt(Math.abs((t * (U * (n + n)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if 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_))))) <= 0.0:
		tmp = math.sqrt((2.0 * (n * t))) * math.sqrt(U)
	else:
		tmp = math.sqrt(math.fabs((t * (U * (n + n)))))
	return tmp

Julia

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[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], 0.0], N[(N[Sqrt[N[(2.0 * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[Abs[N[(t * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

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*))))) < 0.0

   1. Initial program 49.0%

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

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 24.0%

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

   4. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 21.0

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

   6. Applied rewrites 21.0%

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

   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*)))))

   1. Initial program 49.0%

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

   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 35.5

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

   4. Applied rewrites 35.5%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

   6. Applied rewrites 37.5%

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

Alternative 14: 40.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))) <= 0.0) {
		tmp = sqrt(fabs((((U + U) * t) * n)));
	} else {
		tmp = sqrt(fabs((t * (U * (n + n)))));
	}
	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 ((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l_m * l_m) / om))) - ((n * ((l_m / om) ** 2.0d0)) * (u - u_42)))) <= 0.0d0) then
        tmp = sqrt(abs((((u + u) * t) * n)))
    else
        tmp = sqrt(abs((t * (u * (n + n)))))
    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 ((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * Math.pow((l_m / Om), 2.0)) * (U - U_42_)))) <= 0.0) {
		tmp = Math.sqrt(Math.abs((((U + U) * t) * n)));
	} else {
		tmp = Math.sqrt(Math.abs((t * (U * (n + n)))));
	}
	return tmp;
}

Python

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

Julia

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[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], 0.0], N[Sqrt[N[Abs[N[(N[(N[(U + U), $MachinePrecision] * t), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(t * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 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 49.0%

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

   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 35.5

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

   4. Applied rewrites 35.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 35.5

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 35.5

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

   6. Applied rewrites 35.5%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

   8. Applied rewrites 37.4%

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

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

   1. Initial program 49.0%

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

   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 35.5

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

   4. Applied rewrites 35.5%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

   6. Applied rewrites 37.5%

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

Alternative 15: 40.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))) <= 0.0) {
		tmp = sqrt((((U + U) * t) * n));
	} else {
		tmp = sqrt(fabs((t * (U * (n + n)))));
	}
	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 ((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l_m * l_m) / om))) - ((n * ((l_m / om) ** 2.0d0)) * (u - u_42)))) <= 0.0d0) then
        tmp = sqrt((((u + u) * t) * n))
    else
        tmp = sqrt(abs((t * (u * (n + n)))))
    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 ((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * Math.pow((l_m / Om), 2.0)) * (U - U_42_)))) <= 0.0) {
		tmp = Math.sqrt((((U + U) * t) * n));
	} else {
		tmp = Math.sqrt(Math.abs((t * (U * (n + n)))));
	}
	return tmp;
}

Python

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

Julia

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[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], 0.0], N[Sqrt[N[(N[(N[(U + U), $MachinePrecision] * t), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(t * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 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 49.0%

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

   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 35.5

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

   4. Applied rewrites 35.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 35.1

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

   6. Applied rewrites 35.1%

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

   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. Initial program 49.0%

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

   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 35.5

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

   4. Applied rewrites 35.5%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

   6. Applied rewrites 37.5%

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

Alternative 16: 38.1% accurate, 0.8× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<=
      (*
       (* (* 2.0 n) U)
       (-
        (- t (* 2.0 (/ (* l_m l_m) Om)))
        (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))
      0.0)
   (sqrt (* (* (+ U U) t) n))
   (sqrt (* (* U (+ n 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 ((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))) <= 0.0) {
		tmp = sqrt((((U + U) * t) * n));
	} else {
		tmp = sqrt(((U * (n + 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 ((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l_m * l_m) / om))) - ((n * ((l_m / om) ** 2.0d0)) * (u - u_42)))) <= 0.0d0) then
        tmp = sqrt((((u + u) * t) * n))
    else
        tmp = sqrt(((u * (n + 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 ((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * Math.pow((l_m / Om), 2.0)) * (U - U_42_)))) <= 0.0) {
		tmp = Math.sqrt((((U + U) * t) * n));
	} else {
		tmp = Math.sqrt(((U * (n + n)) * t));
	}
	return tmp;
}

Python

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

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (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_)))) <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(U + U) * t) * n));
	else
		tmp = sqrt(Float64(Float64(U * Float64(n + 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 ((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * ((l_m / Om) ^ 2.0)) * (U - U_42_)))) <= 0.0)
		tmp = sqrt((((U + U) * t) * n));
	else
		tmp = sqrt(((U * (n + 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[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], 0.0], N[Sqrt[N[(N[(N[(U + U), $MachinePrecision] * t), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 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 49.0%

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

   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 35.5

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

   4. Applied rewrites 35.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 35.1

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

   6. Applied rewrites 35.1%

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

   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. Initial program 49.0%

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

   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 35.5

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

   4. Applied rewrites 35.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 35.0

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 35.0

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

      16. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      17. count-2-rev N/A

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      18. lift-+.f64 35.0

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   6. Applied rewrites 35.0%

      \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot \color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 35.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* (* U (+ n n)) t)))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return sqrt(((U * (n + n)) * 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(((u * (n + n)) * 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(((U * (n + n)) * t));
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return math.sqrt(((U * (n + n)) * t))

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return sqrt(Float64(Float64(U * Float64(n + n)) * t))
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = sqrt(((U * (n + n)) * t));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 49.0%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

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 35.5

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

4. Applied rewrites 35.5%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. associate-*r* N/A

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

   5. associate-*r* N/A

      \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \color{blue}{t}} \]

   6. *-commutative N/A

      \[\leadsto \sqrt{\left(\left(U \cdot n\right) \cdot 2\right) \cdot t} \]

   7. associate-*r* N/A

      \[\leadsto \sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot t} \]

   8. *-commutative N/A

      \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

   9. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

   10. *-commutative N/A

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

   11. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

   12. lower-*.f64 35.0

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

   13. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

   14. *-commutative N/A

       \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

   15. lower-*.f64 35.0

       \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

   16. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

   17. count-2-rev N/A

       \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   18. lift-+.f64 35.0

       \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

6. Applied rewrites 35.0%

   \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot \color{blue}{t}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025157 
(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*))))))