Toniolo and Linder, Equation (13)

Percentage Accurate: 50.5% → 66.1%

Time: 8.8s

Alternatives: 15

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: 50.5% 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: 66.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \frac{{l\_m}^{2}}{Om}\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot l\_m, \frac{l\_m}{Om}, t\right) - \left(\left(n \cdot \frac{l\_m}{Om}\right) \cdot \frac{l\_m}{Om}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{\frac{U - U*}{Om}}{Om}, \frac{2}{Om}\right)\right)}\\ \end{array} \end{array} \]

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 (+ n n)) (sqrt (* U (+ t (* -2.0 (/ (pow l_m 2.0) Om))))))
     (if (<= t_1 2e+151)
       (sqrt
        (*
         (* (+ n n) U)
         (-
          (fma (* -2.0 l_m) (/ l_m Om) t)
          (* (* (* n (/ l_m Om)) (/ l_m Om)) (- U U*)))))
       (*
        l_m
        (sqrt
         (* -2.0 (* (* U n) (fma n (/ (/ (- U U*) Om) 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((n + n)) * sqrt((U * (t + (-2.0 * (pow(l_m, 2.0) / Om)))));
	} else if (t_1 <= 2e+151) {
		tmp = sqrt((((n + n) * U) * (fma((-2.0 * l_m), (l_m / Om), t) - (((n * (l_m / Om)) * (l_m / Om)) * (U - U_42_)))));
	} else {
		tmp = l_m * sqrt((-2.0 * ((U * n) * fma(n, (((U - U_42_) / Om) / 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(n + n)) * sqrt(Float64(U * Float64(t + Float64(-2.0 * Float64((l_m ^ 2.0) / Om))))));
	elseif (t_1 <= 2e+151)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * Float64(fma(Float64(-2.0 * l_m), Float64(l_m / Om), t) - Float64(Float64(Float64(n * Float64(l_m / Om)) * Float64(l_m / Om)) * Float64(U - U_42_)))));
	else
		tmp = Float64(l_m * sqrt(Float64(-2.0 * Float64(Float64(U * n) * fma(n, Float64(Float64(Float64(U - U_42_) / Om) / Om), Float64(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[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t + N[(-2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+151], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(N[(-2.0 * l$95$m), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + t), $MachinePrecision] - N[(N[(N[(n * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-2.0 * N[(N[(U * n), $MachinePrecision] * N[(n * N[(N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 / 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 50.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. *-commutative N/A

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

   3. Applied rewrites 27.1%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 26.5

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

   6. Applied rewrites 26.5%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2.00000000000000003e151

   1. Initial program 50.5%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 54.3

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

   3. Applied rewrites 54.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 54.3

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

   5. Applied rewrites 54.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 55.4

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

   7. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 55.4

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

   9. Applied rewrites 55.4%

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

   if 2.00000000000000003e151 < (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 50.5%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 27.8

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

   4. Applied rewrites 27.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 28.7

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

      6. lift-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. mult-flip-rev N/A

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

      18. lower-/.f64 28.4

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

   6. Applied rewrites 28.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 30.0

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

   8. Applied rewrites 30.0%

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

Alternative 2: 65.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \frac{{l\_m}^{2}}{Om}\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(-2 \cdot l\_m, \frac{l\_m}{Om}, t\right) - \left(n \cdot \left(\frac{l\_m}{Om} \cdot \frac{l\_m}{Om}\right)\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{\frac{U - U*}{Om}}{Om}, \frac{2}{Om}\right)\right)}\\ \end{array} \end{array} \]

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 (+ n n)) (sqrt (* U (+ t (* -2.0 (/ (pow l_m 2.0) Om))))))
     (if (<= t_1 2e+151)
       (sqrt
        (*
         (* (+ n n) U)
         (-
          (fma (* -2.0 l_m) (/ l_m Om) t)
          (* (* n (* (/ l_m Om) (/ l_m Om))) (- U U*)))))
       (*
        l_m
        (sqrt
         (* -2.0 (* (* U n) (fma n (/ (/ (- U U*) Om) 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((n + n)) * sqrt((U * (t + (-2.0 * (pow(l_m, 2.0) / Om)))));
	} else if (t_1 <= 2e+151) {
		tmp = sqrt((((n + n) * U) * (fma((-2.0 * l_m), (l_m / Om), t) - ((n * ((l_m / Om) * (l_m / Om))) * (U - U_42_)))));
	} else {
		tmp = l_m * sqrt((-2.0 * ((U * n) * fma(n, (((U - U_42_) / Om) / 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(n + n)) * sqrt(Float64(U * Float64(t + Float64(-2.0 * Float64((l_m ^ 2.0) / Om))))));
	elseif (t_1 <= 2e+151)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * Float64(fma(Float64(-2.0 * l_m), Float64(l_m / Om), t) - Float64(Float64(n * Float64(Float64(l_m / Om) * Float64(l_m / Om))) * Float64(U - U_42_)))));
	else
		tmp = Float64(l_m * sqrt(Float64(-2.0 * Float64(Float64(U * n) * fma(n, Float64(Float64(Float64(U - U_42_) / Om) / Om), Float64(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[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t + N[(-2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+151], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(N[(-2.0 * l$95$m), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + t), $MachinePrecision] - N[(N[(n * N[(N[(l$95$m / Om), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-2.0 * N[(N[(U * n), $MachinePrecision] * N[(n * N[(N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 / 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 50.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. *-commutative N/A

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

   3. Applied rewrites 27.1%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 26.5

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

   6. Applied rewrites 26.5%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2.00000000000000003e151

   1. Initial program 50.5%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 54.3

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

   3. Applied rewrites 54.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 54.3

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

   5. Applied rewrites 54.3%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 54.3

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

   7. Applied rewrites 54.3%

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

   if 2.00000000000000003e151 < (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 50.5%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 27.8

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

   4. Applied rewrites 27.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 28.7

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

      6. lift-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. mult-flip-rev N/A

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

      18. lower-/.f64 28.4

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

   6. Applied rewrites 28.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 30.0

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

   8. Applied rewrites 30.0%

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

Alternative 3: 63.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := \sqrt{t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \frac{{l\_m}^{2}}{Om}\right)}\\ \mathbf{elif}\;t\_2 \leq 10^{+147}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t - \mathsf{fma}\left(\left(U - U*\right) \cdot n, \frac{l\_m}{Om} \cdot \frac{l\_m}{Om}, \frac{l\_m}{Om} \cdot \left(l\_m + l\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{\frac{U - U*}{Om}}{Om}, \frac{2}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (sqrt
          (*
           t_1
           (-
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))))
   (if (<= t_2 0.0)
     (* (sqrt (+ n n)) (sqrt (* U (+ t (* -2.0 (/ (pow l_m 2.0) Om))))))
     (if (<= t_2 1e+147)
       (sqrt
        (*
         t_1
         (-
          t
          (fma
           (* (- U U*) n)
           (* (/ l_m Om) (/ l_m Om))
           (* (/ l_m Om) (+ l_m l_m))))))
       (*
        l_m
        (sqrt
         (* -2.0 (* (* U n) (fma n (/ (/ (- U U*) Om) 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 = (2.0 * n) * U;
	double t_2 = sqrt((t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((n + n)) * sqrt((U * (t + (-2.0 * (pow(l_m, 2.0) / Om)))));
	} else if (t_2 <= 1e+147) {
		tmp = sqrt((t_1 * (t - fma(((U - U_42_) * n), ((l_m / Om) * (l_m / Om)), ((l_m / Om) * (l_m + l_m))))));
	} else {
		tmp = l_m * sqrt((-2.0 * ((U * n) * fma(n, (((U - U_42_) / Om) / 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(2.0 * n) * U)
	t_2 = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(Float64(U * Float64(t + Float64(-2.0 * Float64((l_m ^ 2.0) / Om))))));
	elseif (t_2 <= 1e+147)
		tmp = sqrt(Float64(t_1 * Float64(t - fma(Float64(Float64(U - U_42_) * n), Float64(Float64(l_m / Om) * Float64(l_m / Om)), Float64(Float64(l_m / Om) * Float64(l_m + l_m))))));
	else
		tmp = Float64(l_m * sqrt(Float64(-2.0 * Float64(Float64(U * n) * fma(n, Float64(Float64(Float64(U - U_42_) / Om) / Om), Float64(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[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t + N[(-2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+147], N[Sqrt[N[(t$95$1 * N[(t - N[(N[(N[(U - U$42$), $MachinePrecision] * n), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + N[(N[(l$95$m / Om), $MachinePrecision] * N[(l$95$m + l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-2.0 * N[(N[(U * n), $MachinePrecision] * N[(n * N[(N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 / 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 50.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. *-commutative N/A

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

   3. Applied rewrites 27.1%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 26.5

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

   6. Applied rewrites 26.5%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 9.9999999999999998e146

   1. Initial program 50.5%

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

      1. 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

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 47.7

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. frac-times N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 42.1

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

   3. Applied rewrites 42.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lower-*.f64 51.4

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

   5. Applied rewrites 51.4%

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

   if 9.9999999999999998e146 < (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 50.5%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 27.8

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

   4. Applied rewrites 27.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 28.7

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

      6. lift-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. mult-flip-rev N/A

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

      18. lower-/.f64 28.4

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

   6. Applied rewrites 28.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 30.0

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

   8. Applied rewrites 30.0%

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

Alternative 4: 61.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{U \cdot \left(t + -2 \cdot \frac{{l\_m}^{2}}{Om}\right)}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U* - U, \frac{l\_m \cdot l\_m}{Om \cdot Om} \cdot n, \mathsf{fma}\left(-2, t\_1, t\right)\right) \cdot \left(U \cdot \left(n + n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{\frac{U - U*}{Om}}{Om}, \frac{2}{Om}\right)\right)}\\ \end{array} \end{array} \]

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
         (sqrt
          (*
           (* (* 2.0 n) U)
           (- (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))))
   (if (<= t_2 0.0)
     (* (sqrt (+ n n)) (sqrt (* U (+ t (* -2.0 (/ (pow l_m 2.0) Om))))))
     (if (<= t_2 2e+151)
       (sqrt
        (*
         (fma (- U* U) (* (/ (* l_m l_m) (* Om Om)) n) (fma -2.0 t_1 t))
         (* U (+ n n))))
       (*
        l_m
        (sqrt
         (* -2.0 (* (* U n) (fma n (/ (/ (- U U*) Om) 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 = sqrt((((2.0 * n) * U) * ((t - (2.0 * t_1)) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((n + n)) * sqrt((U * (t + (-2.0 * (pow(l_m, 2.0) / Om)))));
	} else if (t_2 <= 2e+151) {
		tmp = sqrt((fma((U_42_ - U), (((l_m * l_m) / (Om * Om)) * n), fma(-2.0, t_1, t)) * (U * (n + n))));
	} else {
		tmp = l_m * sqrt((-2.0 * ((U * n) * fma(n, (((U - U_42_) / Om) / 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 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * 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_2 <= 0.0)
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(Float64(U * Float64(t + Float64(-2.0 * Float64((l_m ^ 2.0) / Om))))));
	elseif (t_2 <= 2e+151)
		tmp = sqrt(Float64(fma(Float64(U_42_ - U), Float64(Float64(Float64(l_m * l_m) / Float64(Om * Om)) * n), fma(-2.0, t_1, t)) * Float64(U * Float64(n + n))));
	else
		tmp = Float64(l_m * sqrt(Float64(-2.0 * Float64(Float64(U * n) * fma(n, Float64(Float64(Float64(U - U_42_) / Om) / Om), Float64(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[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * 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$2, 0.0], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t + N[(-2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+151], N[Sqrt[N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] + N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision] * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-2.0 * N[(N[(U * n), $MachinePrecision] * N[(n * N[(N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 / 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 50.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. *-commutative N/A

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

   3. Applied rewrites 27.1%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 26.5

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

   6. Applied rewrites 26.5%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2.00000000000000003e151

   1. Initial program 50.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.5

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

   3. Applied rewrites 44.9%

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

   if 2.00000000000000003e151 < (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 50.5%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 27.8

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

   4. Applied rewrites 27.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 28.7

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

      6. lift-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. mult-flip-rev N/A

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

      18. lower-/.f64 28.4

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

   6. Applied rewrites 28.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 30.0

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

   8. Applied rewrites 30.0%

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

Alternative 5: 61.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{{l\_m}^{2}}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \sqrt{t\_2 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{U \cdot \left(t + -2 \cdot t\_1\right)}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{t\_2 \cdot \left(t - 2 \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{\frac{U - U*}{Om}}{Om}, \frac{2}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (pow l_m 2.0) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (sqrt
          (*
           t_2
           (-
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))))
   (if (<= t_3 0.0)
     (* (sqrt (+ n n)) (sqrt (* U (+ t (* -2.0 t_1)))))
     (if (<= t_3 2e+151)
       (sqrt (* t_2 (- t (* 2.0 t_1))))
       (*
        l_m
        (sqrt
         (* -2.0 (* (* U n) (fma n (/ (/ (- U U*) Om) 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 = pow(l_m, 2.0) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = sqrt((t_2 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((n + n)) * sqrt((U * (t + (-2.0 * t_1))));
	} else if (t_3 <= 2e+151) {
		tmp = sqrt((t_2 * (t - (2.0 * t_1))));
	} else {
		tmp = l_m * sqrt((-2.0 * ((U * n) * fma(n, (((U - U_42_) / Om) / Om), (2.0 / Om)))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64((l_m ^ 2.0) / Om)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = sqrt(Float64(t_2 * 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_3 <= 0.0)
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(Float64(U * Float64(t + Float64(-2.0 * t_1)))));
	elseif (t_3 <= 2e+151)
		tmp = sqrt(Float64(t_2 * Float64(t - Float64(2.0 * t_1))));
	else
		tmp = Float64(l_m * sqrt(Float64(-2.0 * Float64(Float64(U * n) * fma(n, Float64(Float64(Float64(U - U_42_) / Om) / Om), Float64(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[Power[l$95$m, 2.0], $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 * 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$3, 0.0], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t + N[(-2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+151], N[Sqrt[N[(t$95$2 * N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-2.0 * N[(N[(U * n), $MachinePrecision] * N[(n * N[(N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 / 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 50.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. *-commutative N/A

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

   3. Applied rewrites 27.1%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 26.5

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

   6. Applied rewrites 26.5%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2.00000000000000003e151

   1. Initial program 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 44.4

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

   4. Applied rewrites 44.4%

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

   if 2.00000000000000003e151 < (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 50.5%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 27.8

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

   4. Applied rewrites 27.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 28.7

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

      6. lift-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. mult-flip-rev N/A

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

      18. lower-/.f64 28.4

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

   6. Applied rewrites 28.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 30.0

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

   8. Applied rewrites 30.0%

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

Alternative 6: 61.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := \sqrt{t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t - 2 \cdot \frac{{l\_m}^{2}}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{\frac{U - U*}{Om}}{Om}, \frac{2}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (sqrt
          (*
           t_1
           (-
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))))
   (if (<= t_2 0.0)
     (sqrt (* U (* (+ n n) t)))
     (if (<= t_2 2e+151)
       (sqrt (* t_1 (- t (* 2.0 (/ (pow l_m 2.0) Om)))))
       (*
        l_m
        (sqrt
         (* -2.0 (* (* U n) (fma n (/ (/ (- U U*) Om) 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 = (2.0 * n) * U;
	double t_2 = sqrt((t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((U * ((n + n) * t)));
	} else if (t_2 <= 2e+151) {
		tmp = sqrt((t_1 * (t - (2.0 * (pow(l_m, 2.0) / Om)))));
	} else {
		tmp = l_m * sqrt((-2.0 * ((U * n) * fma(n, (((U - U_42_) / Om) / 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(2.0 * n) * U)
	t_2 = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = sqrt(Float64(U * Float64(Float64(n + n) * t)));
	elseif (t_2 <= 2e+151)
		tmp = sqrt(Float64(t_1 * Float64(t - Float64(2.0 * Float64((l_m ^ 2.0) / Om)))));
	else
		tmp = Float64(l_m * sqrt(Float64(-2.0 * Float64(Float64(U * n) * fma(n, Float64(Float64(Float64(U - U_42_) / Om) / Om), Float64(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[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[Sqrt[N[(U * N[(N[(n + n), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 2e+151], N[Sqrt[N[(t$95$1 * N[(t - N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-2.0 * N[(N[(U * n), $MachinePrecision] * N[(n * N[(N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 / 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 50.5%

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

   2. 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 37.1

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

   4. Applied rewrites 37.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 36.6

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

   6. Applied rewrites 36.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 36.6

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 36.6

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

      13. lift-*.f64 N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f64 36.6

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

   8. Applied rewrites 36.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 37.1

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 37.1

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

   10. Applied rewrites 37.1%

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

   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*))))) < 2.00000000000000003e151

   1. Initial program 50.5%

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

   2. Taylor expanded in n around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 44.4

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

   4. Applied rewrites 44.4%

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

   if 2.00000000000000003e151 < (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 50.5%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 27.8

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

   4. Applied rewrites 27.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 28.7

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

      6. lift-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. mult-flip-rev N/A

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

      18. lower-/.f64 28.4

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

   6. Applied rewrites 28.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 30.0

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

   8. Applied rewrites 30.0%

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

Alternative 7: 60.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\left(l\_m \cdot l\_m\right) \cdot U\right) \cdot n}{Om}, -4, \left(U \cdot \left(n + n\right)\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{\frac{U - U*}{Om}}{Om}, \frac{2}{Om}\right)\right)}\\ \end{array} \end{array} \]

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 (* U (* (+ n n) t)))
     (if (<= t_1 2e+151)
       (sqrt (fma (/ (* (* (* l_m l_m) U) n) Om) -4.0 (* (* U (+ n n)) t)))
       (*
        l_m
        (sqrt
         (* -2.0 (* (* U n) (fma n (/ (/ (- U U*) Om) 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((U * ((n + n) * t)));
	} else if (t_1 <= 2e+151) {
		tmp = sqrt(fma(((((l_m * l_m) * U) * n) / Om), -4.0, ((U * (n + n)) * t)));
	} else {
		tmp = l_m * sqrt((-2.0 * ((U * n) * fma(n, (((U - U_42_) / Om) / 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 = sqrt(Float64(U * Float64(Float64(n + n) * t)));
	elseif (t_1 <= 2e+151)
		tmp = sqrt(fma(Float64(Float64(Float64(Float64(l_m * l_m) * U) * n) / Om), -4.0, Float64(Float64(U * Float64(n + n)) * t)));
	else
		tmp = Float64(l_m * sqrt(Float64(-2.0 * Float64(Float64(U * n) * fma(n, Float64(Float64(Float64(U - U_42_) / Om) / Om), Float64(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[Sqrt[N[(U * N[(N[(n + n), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 2e+151], N[Sqrt[N[(N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * U), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] * -4.0 + N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-2.0 * N[(N[(U * n), $MachinePrecision] * N[(n * N[(N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 / 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 50.5%

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

   2. 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 37.1

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

   4. Applied rewrites 37.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 36.6

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

   6. Applied rewrites 36.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 36.6

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 36.6

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

      13. lift-*.f64 N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f64 36.6

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

   8. Applied rewrites 36.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 37.1

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 37.1

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

   10. Applied rewrites 37.1%

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

   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*))))) < 2.00000000000000003e151

   1. Initial program 50.5%

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

   2. Taylor expanded in Om around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 43.9

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

   4. Applied rewrites 43.9%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 43.9

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 43.3

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 43.3

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

   6. Applied rewrites 43.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 43.3

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lower-*.f64 43.1

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 43.1

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

      24. lift-*.f64 N/A

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

      25. count-2-rev N/A

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

      26. lift-+.f64 43.1

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

   8. Applied rewrites 43.1%

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

   if 2.00000000000000003e151 < (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 50.5%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 27.8

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

   4. Applied rewrites 27.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 28.7

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

      6. lift-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. mult-flip-rev N/A

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

      18. lower-/.f64 28.4

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

   6. Applied rewrites 28.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 30.0

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

   8. Applied rewrites 30.0%

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

Alternative 8: 60.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\left(l\_m \cdot l\_m\right) \cdot U\right) \cdot n}{Om}, -4, \left(U \cdot \left(n + n\right)\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{-2 \cdot \left(\left(U \cdot n\right) \cdot \frac{2 + \frac{n \cdot \left(U - U*\right)}{Om}}{Om}\right)}\\ \end{array} \end{array} \]

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 (* U (* (+ n n) t)))
     (if (<= t_1 2e+151)
       (sqrt (fma (/ (* (* (* l_m l_m) U) n) Om) -4.0 (* (* U (+ n n)) t)))
       (*
        l_m
        (sqrt (* -2.0 (* (* U n) (/ (+ 2.0 (/ (* n (- U U*)) Om)) 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((U * ((n + n) * t)));
	} else if (t_1 <= 2e+151) {
		tmp = sqrt(fma(((((l_m * l_m) * U) * n) / Om), -4.0, ((U * (n + n)) * t)));
	} else {
		tmp = l_m * sqrt((-2.0 * ((U * n) * ((2.0 + ((n * (U - U_42_)) / Om)) / 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 = sqrt(Float64(U * Float64(Float64(n + n) * t)));
	elseif (t_1 <= 2e+151)
		tmp = sqrt(fma(Float64(Float64(Float64(Float64(l_m * l_m) * U) * n) / Om), -4.0, Float64(Float64(U * Float64(n + n)) * t)));
	else
		tmp = Float64(l_m * sqrt(Float64(-2.0 * Float64(Float64(U * n) * Float64(Float64(2.0 + Float64(Float64(n * Float64(U - U_42_)) / Om)) / 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[Sqrt[N[(U * N[(N[(n + n), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 2e+151], N[Sqrt[N[(N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * U), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] * -4.0 + N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-2.0 * N[(N[(U * n), $MachinePrecision] * N[(N[(2.0 + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / Om), $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 50.5%

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

   2. 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 37.1

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

   4. Applied rewrites 37.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 36.6

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

   6. Applied rewrites 36.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 36.6

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 36.6

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

      13. lift-*.f64 N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f64 36.6

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

   8. Applied rewrites 36.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 37.1

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 37.1

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

   10. Applied rewrites 37.1%

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

   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*))))) < 2.00000000000000003e151

   1. Initial program 50.5%

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

   2. Taylor expanded in Om around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 43.9

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

   4. Applied rewrites 43.9%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 43.9

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 43.3

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 43.3

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

   6. Applied rewrites 43.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 43.3

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lower-*.f64 43.1

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 43.1

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

      24. lift-*.f64 N/A

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

      25. count-2-rev N/A

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

      26. lift-+.f64 43.1

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

   8. Applied rewrites 43.1%

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

   if 2.00000000000000003e151 < (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 50.5%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 27.8

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

   4. Applied rewrites 27.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 28.7

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

      6. lift-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. mult-flip-rev N/A

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

      18. lower-/.f64 28.4

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

   6. Applied rewrites 28.4%

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

   7. Taylor expanded in Om around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 30.9

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

   9. Applied rewrites 30.9%

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

Alternative 9: 59.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\left(l\_m \cdot l\_m\right) \cdot U\right) \cdot n}{Om}, -4, \left(U \cdot \left(n + n\right)\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \frac{2 + \frac{n \cdot \left(U - U*\right)}{Om}}{Om}\right)\right)}\\ \end{array} \end{array} \]

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 (* U (* (+ n n) t)))
     (if (<= t_1 2e+151)
       (sqrt (fma (/ (* (* (* l_m l_m) U) n) Om) -4.0 (* (* U (+ n n)) t)))
       (*
        l_m
        (sqrt (* -2.0 (* U (* n (/ (+ 2.0 (/ (* n (- U U*)) Om)) 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((U * ((n + n) * t)));
	} else if (t_1 <= 2e+151) {
		tmp = sqrt(fma(((((l_m * l_m) * U) * n) / Om), -4.0, ((U * (n + n)) * t)));
	} else {
		tmp = l_m * sqrt((-2.0 * (U * (n * ((2.0 + ((n * (U - U_42_)) / Om)) / 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 = sqrt(Float64(U * Float64(Float64(n + n) * t)));
	elseif (t_1 <= 2e+151)
		tmp = sqrt(fma(Float64(Float64(Float64(Float64(l_m * l_m) * U) * n) / Om), -4.0, Float64(Float64(U * Float64(n + n)) * t)));
	else
		tmp = Float64(l_m * sqrt(Float64(-2.0 * Float64(U * Float64(n * Float64(Float64(2.0 + Float64(Float64(n * Float64(U - U_42_)) / Om)) / 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[Sqrt[N[(U * N[(N[(n + n), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 2e+151], N[Sqrt[N[(N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * U), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] * -4.0 + N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-2.0 * N[(U * N[(n * N[(N[(2.0 + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $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 50.5%

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

   2. 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 37.1

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

   4. Applied rewrites 37.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 36.6

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

   6. Applied rewrites 36.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 36.6

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 36.6

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

      13. lift-*.f64 N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f64 36.6

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

   8. Applied rewrites 36.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 37.1

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 37.1

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

   10. Applied rewrites 37.1%

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

   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*))))) < 2.00000000000000003e151

   1. Initial program 50.5%

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

   2. Taylor expanded in Om around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 43.9

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

   4. Applied rewrites 43.9%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 43.9

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 43.3

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 43.3

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

   6. Applied rewrites 43.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot n}{Om}, -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)} \]

      5. lift-*.f64 43.3

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, 2 \cdot \left(\left(t \cdot n\right) \cdot U\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, 2 \cdot \left(\left(t \cdot n\right) \cdot U\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, 2 \cdot \left(\left(t \cdot n\right) \cdot U\right)\right)} \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, 2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, 2 \cdot \left(t \cdot \left(U \cdot n\right)\right)\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, 2 \cdot \left(t \cdot \left(U \cdot n\right)\right)\right)} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

      14. associate-*r* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(2 \cdot \left(U \cdot n\right)\right) \cdot t\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(2 \cdot \left(U \cdot n\right)\right) \cdot t\right)} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)} \]

      17. associate-*l* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)} \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)} \]

      19. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)} \]

      20. lower-*.f64 43.1

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)} \]

      21. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)} \]

      22. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(U \cdot \left(2 \cdot n\right)\right) \cdot t\right)} \]

      23. lower-*.f64 43.1

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(U \cdot \left(2 \cdot n\right)\right) \cdot t\right)} \]

      24. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(U \cdot \left(2 \cdot n\right)\right) \cdot t\right)} \]

      25. count-2-rev N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(U \cdot \left(n + n\right)\right) \cdot t\right)} \]

      26. lift-+.f64 43.1

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(U \cdot \left(n + n\right)\right) \cdot t\right)} \]

   8. Applied rewrites 43.1%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, \color{blue}{-4}, \left(U \cdot \left(n + n\right)\right) \cdot t\right)} \]

   if 2.00000000000000003e151 < (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 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      11. lower-pow.f64 27.8

          \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

   4. Applied rewrites 27.8%

      \[\leadsto \color{blue}{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]

   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 31.1

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \frac{2 + \frac{n \cdot \left(U - U*\right)}{Om}}{Om}\right)\right)} \]

   7. Applied rewrites 31.1%

      \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \frac{2 + \frac{n \cdot \left(U - U*\right)}{Om}}{Om}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 49.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\left(\left(l\_m \cdot l\_m\right) \cdot U\right) \cdot n}{Om}, -4, \left(U \cdot \left(n + n\right)\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \frac{2}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* (* 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 (* U (* (+ n n) t)))
     (if (<= t_1 2e+302)
       (sqrt (fma (/ (* (* (* l_m l_m) U) n) Om) -4.0 (* (* U (+ n n)) t)))
       (* l_m (sqrt (* -2.0 (* U (* n (/ 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 = ((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((U * ((n + n) * t)));
	} else if (t_1 <= 2e+302) {
		tmp = sqrt(fma(((((l_m * l_m) * U) * n) / Om), -4.0, ((U * (n + n)) * t)));
	} else {
		tmp = l_m * sqrt((-2.0 * (U * (n * (2.0 / Om)))));
	}
	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 <= 0.0)
		tmp = sqrt(Float64(U * Float64(Float64(n + n) * t)));
	elseif (t_1 <= 2e+302)
		tmp = sqrt(fma(Float64(Float64(Float64(Float64(l_m * l_m) * U) * n) / Om), -4.0, Float64(Float64(U * Float64(n + n)) * t)));
	else
		tmp = Float64(l_m * sqrt(Float64(-2.0 * Float64(U * Float64(n * Float64(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[(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, 0.0], N[Sqrt[N[(U * N[(N[(n + n), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 2e+302], N[Sqrt[N[(N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * U), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] * -4.0 + N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-2.0 * N[(U * N[(n * N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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*)))) < 0.0

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. 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 37.1

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 37.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      5. lower-*.f64 36.6

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

   6. Applied rewrites 36.6%

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \color{blue}{t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      9. lower-*.f64 36.6

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      12. lower-*.f64 36.6

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      14. count-2-rev N/A

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      15. lift-+.f64 36.6

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   8. Applied rewrites 36.6%

      \[\leadsto \color{blue}{\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot \color{blue}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      3. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      4. count-2-rev N/A

         \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{U \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot t\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot t\right)}} \]

      8. lower-*.f64 37.1

         \[\leadsto \sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \color{blue}{t}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot t\right)} \]

      10. count-2-rev N/A

          \[\leadsto \sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)} \]

      11. lift-+.f64 37.1

          \[\leadsto \sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)} \]

   10. Applied rewrites 37.1%

       \[\leadsto \sqrt{U \cdot \color{blue}{\left(\left(n + n\right) \cdot t\right)}} \]

   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*)))) < 2.0000000000000002e302

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(-4, \color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(-4, \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{\color{blue}{Om}}, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(-4, \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(-4, \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(-4, \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(-4, \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(-4, \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      8. lower-*.f64 43.9

         \[\leadsto \sqrt{\mathsf{fma}\left(-4, \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

   4. Applied rewrites 43.9%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-4, \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \sqrt{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + \color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} \cdot -4 + \color{blue}{2} \cdot \left(U \cdot \left(n \cdot t\right)\right)} \]

      3. lower-fma.f64 43.9

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}, \color{blue}{-4}, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      7. pow2 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      9. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot n}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot n}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      11. lower-*.f64 43.3

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot n}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot n}{Om}, -4, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot n}{Om}, -4, \left(U \cdot \left(n \cdot t\right)\right) \cdot 2\right)} \]

      14. lower-*.f64 43.3

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot n}{Om}, -4, \left(U \cdot \left(n \cdot t\right)\right) \cdot 2\right)} \]

   6. Applied rewrites 43.3%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\frac{\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot n}{Om}, -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot n}{Om}, -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot n}{Om}, -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)} \]

      5. lift-*.f64 43.3

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, 2 \cdot \left(\left(t \cdot n\right) \cdot U\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, 2 \cdot \left(\left(t \cdot n\right) \cdot U\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, 2 \cdot \left(\left(t \cdot n\right) \cdot U\right)\right)} \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, 2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, 2 \cdot \left(t \cdot \left(U \cdot n\right)\right)\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, 2 \cdot \left(t \cdot \left(U \cdot n\right)\right)\right)} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

      14. associate-*r* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(2 \cdot \left(U \cdot n\right)\right) \cdot t\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(2 \cdot \left(U \cdot n\right)\right) \cdot t\right)} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(2 \cdot \left(n \cdot U\right)\right) \cdot t\right)} \]

      17. associate-*l* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)} \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)} \]

      19. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)} \]

      20. lower-*.f64 43.1

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)} \]

      21. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(\left(2 \cdot n\right) \cdot U\right) \cdot t\right)} \]

      22. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(U \cdot \left(2 \cdot n\right)\right) \cdot t\right)} \]

      23. lower-*.f64 43.1

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(U \cdot \left(2 \cdot n\right)\right) \cdot t\right)} \]

      24. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(U \cdot \left(2 \cdot n\right)\right) \cdot t\right)} \]

      25. count-2-rev N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(U \cdot \left(n + n\right)\right) \cdot t\right)} \]

      26. lift-+.f64 43.1

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, -4, \left(U \cdot \left(n + n\right)\right) \cdot t\right)} \]

   8. Applied rewrites 43.1%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot n}{Om}, \color{blue}{-4}, \left(U \cdot \left(n + n\right)\right) \cdot t\right)} \]

   if 2.0000000000000002e302 < (*.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 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      11. lower-pow.f64 27.8

          \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

   4. Applied rewrites 27.8%

      \[\leadsto \color{blue}{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]

   5. Taylor expanded in n around 0

      \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \frac{2}{Om}\right)\right)} \]
   6. Step-by-step derivation

      1. lower-/.f64 17.8

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \frac{2}{Om}\right)\right)} \]

   7. Applied rewrites 17.8%

      \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \frac{2}{Om}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 47.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \frac{2}{Om}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* (* 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 (* U (* (+ n n) t)))
     (if (<= t_1 2e+302)
       (sqrt (* (* U (+ n n)) t))
       (* l_m (sqrt (* -2.0 (* U (* n (/ 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 = ((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((U * ((n + n) * t)));
	} else if (t_1 <= 2e+302) {
		tmp = sqrt(((U * (n + n)) * t));
	} else {
		tmp = l_m * sqrt((-2.0 * (U * (n * (2.0 / 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) :: t_1
    real(8) :: tmp
    t_1 = ((2.0d0 * n) * u) * ((t - (2.0d0 * ((l_m * l_m) / om))) - ((n * ((l_m / om) ** 2.0d0)) * (u - u_42)))
    if (t_1 <= 0.0d0) then
        tmp = sqrt((u * ((n + n) * t)))
    else if (t_1 <= 2d+302) then
        tmp = sqrt(((u * (n + n)) * t))
    else
        tmp = l_m * sqrt(((-2.0d0) * (u * (n * (2.0d0 / 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 t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * Math.pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.sqrt((U * ((n + n) * t)));
	} else if (t_1 <= 2e+302) {
		tmp = Math.sqrt(((U * (n + n)) * t));
	} else {
		tmp = l_m * Math.sqrt((-2.0 * (U * (n * (2.0 / Om)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * math.pow((l_m / Om), 2.0)) * (U - U_42_)))
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.sqrt((U * ((n + n) * t)))
	elif t_1 <= 2e+302:
		tmp = math.sqrt(((U * (n + n)) * t))
	else:
		tmp = l_m * math.sqrt((-2.0 * (U * (n * (2.0 / Om)))))
	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 <= 0.0)
		tmp = sqrt(Float64(U * Float64(Float64(n + n) * t)));
	elseif (t_1 <= 2e+302)
		tmp = sqrt(Float64(Float64(U * Float64(n + n)) * t));
	else
		tmp = Float64(l_m * sqrt(Float64(-2.0 * Float64(U * Float64(n * Float64(2.0 / Om))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * ((l_m / Om) ^ 2.0)) * (U - U_42_)));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = sqrt((U * ((n + n) * t)));
	elseif (t_1 <= 2e+302)
		tmp = sqrt(((U * (n + n)) * t));
	else
		tmp = l_m * sqrt((-2.0 * (U * (n * (2.0 / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(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, 0.0], N[Sqrt[N[(U * N[(N[(n + n), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 2e+302], N[Sqrt[N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[(-2.0 * N[(U * N[(n * N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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*)))) < 0.0

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. 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 37.1

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 37.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      5. lower-*.f64 36.6

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

   6. Applied rewrites 36.6%

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \color{blue}{t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      9. lower-*.f64 36.6

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      12. lower-*.f64 36.6

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      14. count-2-rev N/A

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      15. lift-+.f64 36.6

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   8. Applied rewrites 36.6%

      \[\leadsto \color{blue}{\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot \color{blue}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      3. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      4. count-2-rev N/A

         \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{U \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot t\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot t\right)}} \]

      8. lower-*.f64 37.1

         \[\leadsto \sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \color{blue}{t}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot t\right)} \]

      10. count-2-rev N/A

          \[\leadsto \sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)} \]

      11. lift-+.f64 37.1

          \[\leadsto \sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)} \]

   10. Applied rewrites 37.1%

       \[\leadsto \sqrt{U \cdot \color{blue}{\left(\left(n + n\right) \cdot t\right)}} \]

   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*)))) < 2.0000000000000002e302

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. 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 37.1

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 37.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      5. lower-*.f64 36.6

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

   6. Applied rewrites 36.6%

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \color{blue}{t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      9. lower-*.f64 36.6

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      12. lower-*.f64 36.6

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      14. count-2-rev N/A

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      15. lift-+.f64 36.6

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   8. Applied rewrites 36.6%

      \[\leadsto \color{blue}{\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}} \]

   if 2.0000000000000002e302 < (*.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 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      11. lower-pow.f64 27.8

          \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

   4. Applied rewrites 27.8%

      \[\leadsto \color{blue}{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]

   5. Taylor expanded in n around 0

      \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \frac{2}{Om}\right)\right)} \]
   6. Step-by-step derivation

      1. lower-/.f64 17.8

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \frac{2}{Om}\right)\right)} \]

   7. Applied rewrites 17.8%

      \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \frac{2}{Om}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 46.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{-4 \cdot \frac{U \cdot n}{Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (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 0.0)
     (sqrt (* U (* (+ n n) t)))
     (if (<= t_1 2e+302)
       (sqrt (* (* U (+ n n)) t))
       (* 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 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 <= 0.0) {
		tmp = sqrt((U * ((n + n) * t)));
	} else if (t_1 <= 2e+302) {
		tmp = sqrt(((U * (n + n)) * t));
	} 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) :: t_1
    real(8) :: tmp
    t_1 = ((2.0d0 * n) * u) * ((t - (2.0d0 * ((l_m * l_m) / om))) - ((n * ((l_m / om) ** 2.0d0)) * (u - u_42)))
    if (t_1 <= 0.0d0) then
        tmp = sqrt((u * ((n + n) * t)))
    else if (t_1 <= 2d+302) then
        tmp = sqrt(((u * (n + n)) * t))
    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 t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * Math.pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.sqrt((U * ((n + n) * t)));
	} else if (t_1 <= 2e+302) {
		tmp = Math.sqrt(((U * (n + n)) * t));
	} 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_):
	t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * math.pow((l_m / Om), 2.0)) * (U - U_42_)))
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.sqrt((U * ((n + n) * t)))
	elif t_1 <= 2e+302:
		tmp = math.sqrt(((U * (n + n)) * t))
	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_)
	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 <= 0.0)
		tmp = sqrt(Float64(U * Float64(Float64(n + n) * t)));
	elseif (t_1 <= 2e+302)
		tmp = sqrt(Float64(Float64(U * Float64(n + n)) * t));
	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_)
	t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * ((l_m / Om) ^ 2.0)) * (U - U_42_)));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = sqrt((U * ((n + n) * t)));
	elseif (t_1 <= 2e+302)
		tmp = sqrt(((U * (n + n)) * t));
	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$42$_] := 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, 0.0], N[Sqrt[N[(U * N[(N[(n + n), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 2e+302], N[Sqrt[N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $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 (*.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 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. 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 37.1

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 37.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      5. lower-*.f64 36.6

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

   6. Applied rewrites 36.6%

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \color{blue}{t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      9. lower-*.f64 36.6

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      12. lower-*.f64 36.6

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      14. count-2-rev N/A

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      15. lift-+.f64 36.6

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   8. Applied rewrites 36.6%

      \[\leadsto \color{blue}{\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot \color{blue}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      3. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      4. count-2-rev N/A

         \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{U \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot t\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot t\right)}} \]

      8. lower-*.f64 37.1

         \[\leadsto \sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \color{blue}{t}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot t\right)} \]

      10. count-2-rev N/A

          \[\leadsto \sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)} \]

      11. lift-+.f64 37.1

          \[\leadsto \sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)} \]

   10. Applied rewrites 37.1%

       \[\leadsto \sqrt{U \cdot \color{blue}{\left(\left(n + n\right) \cdot t\right)}} \]

   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*)))) < 2.0000000000000002e302

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. 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 37.1

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 37.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      5. lower-*.f64 36.6

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

   6. Applied rewrites 36.6%

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \color{blue}{t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      9. lower-*.f64 36.6

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      12. lower-*.f64 36.6

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      14. count-2-rev N/A

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      15. lift-+.f64 36.6

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   8. Applied rewrites 36.6%

      \[\leadsto \color{blue}{\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}} \]

   if 2.0000000000000002e302 < (*.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 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      11. lower-pow.f64 27.8

          \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

   4. Applied rewrites 27.8%

      \[\leadsto \color{blue}{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]

   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 17.4

         \[\leadsto \ell \cdot \sqrt{-4 \cdot \frac{U \cdot n}{Om}} \]

   7. Applied rewrites 17.4%

      \[\leadsto \ell \cdot \sqrt{-4 \cdot \frac{U \cdot n}{Om}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 39.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq 3.1 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{U \cdot t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n 3.1e-280)
   (sqrt (* U (* (+ n n) t)))
   (* (sqrt (+ n n)) (sqrt (* U 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 (n <= 3.1e-280) {
		tmp = sqrt((U * ((n + n) * t)));
	} else {
		tmp = sqrt((n + n)) * sqrt((U * t));
	}
	return tmp;
}

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l_m, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= 3.1d-280) then
        tmp = sqrt((u * ((n + n) * t)))
    else
        tmp = sqrt((n + n)) * sqrt((u * t))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= 3.1e-280) {
		tmp = Math.sqrt((U * ((n + n) * t)));
	} else {
		tmp = Math.sqrt((n + n)) * Math.sqrt((U * t));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if n <= 3.1e-280:
		tmp = math.sqrt((U * ((n + n) * t)))
	else:
		tmp = math.sqrt((n + n)) * math.sqrt((U * t))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= 3.1e-280)
		tmp = sqrt(Float64(U * Float64(Float64(n + n) * t)));
	else
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(Float64(U * 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 (n <= 3.1e-280)
		tmp = sqrt((U * ((n + n) * t)));
	else
		tmp = sqrt((n + n)) * sqrt((U * 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, 3.1e-280], N[Sqrt[N[(U * N[(N[(n + n), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 3.10000000000000021e-280

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. 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 37.1

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 37.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      5. lower-*.f64 36.6

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

   6. Applied rewrites 36.6%

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \color{blue}{t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      9. lower-*.f64 36.6

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      12. lower-*.f64 36.6

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      14. count-2-rev N/A

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      15. lift-+.f64 36.6

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   8. Applied rewrites 36.6%

      \[\leadsto \color{blue}{\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot \color{blue}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      3. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      4. count-2-rev N/A

         \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{U \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot t\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot t\right)}} \]

      8. lower-*.f64 37.1

         \[\leadsto \sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \color{blue}{t}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot t\right)} \]

      10. count-2-rev N/A

          \[\leadsto \sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)} \]

      11. lift-+.f64 37.1

          \[\leadsto \sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)} \]

   10. Applied rewrites 37.1%

       \[\leadsto \sqrt{U \cdot \color{blue}{\left(\left(n + n\right) \cdot t\right)}} \]

   if 3.10000000000000021e-280 < n

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      5. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot n}} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot n}} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. count-2-rev N/A

         \[\leadsto \sqrt{\color{blue}{n + n}} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \sqrt{\color{blue}{n + n}} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{n + n} \cdot \color{blue}{\sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot U}} \]

   3. Applied rewrites 27.1%

      \[\leadsto \color{blue}{\sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(U* - U, \frac{\ell \cdot \ell}{Om \cdot Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right) \cdot U}} \]

   4. Taylor expanded in t around inf

      \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{U \cdot t}} \]
   5. Step-by-step derivation

      1. lower-*.f64 21.6

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \color{blue}{t}} \]

   6. Applied rewrites 21.6%

      \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{U \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 39.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 0:\\ \;\;\;\;\sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}\\ \end{array} \end{array} \]

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 (* (+ n n) t)))
   (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 * ((n + n) * t)));
	} 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 * ((n + n) * t)))
    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 * ((n + n) * t)));
	} 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 * ((n + n) * t)))
	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(U * Float64(Float64(n + n) * t)));
	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 * ((n + n) * t)));
	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[(U * N[(N[(n + n), $MachinePrecision] * t), $MachinePrecision]), $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 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. 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 37.1

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 37.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      5. lower-*.f64 36.6

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

   6. Applied rewrites 36.6%

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \color{blue}{t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      9. lower-*.f64 36.6

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      12. lower-*.f64 36.6

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      14. count-2-rev N/A

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      15. lift-+.f64 36.6

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   8. Applied rewrites 36.6%

      \[\leadsto \color{blue}{\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot \color{blue}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      3. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      4. count-2-rev N/A

         \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{U \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot t\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot t\right)}} \]

      8. lower-*.f64 37.1

         \[\leadsto \sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \color{blue}{t}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot t\right)} \]

      10. count-2-rev N/A

          \[\leadsto \sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)} \]

      11. lift-+.f64 37.1

          \[\leadsto \sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)} \]

   10. Applied rewrites 37.1%

       \[\leadsto \sqrt{U \cdot \color{blue}{\left(\left(n + n\right) \cdot t\right)}} \]

   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 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. 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 37.1

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 37.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      5. lower-*.f64 36.6

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

   6. Applied rewrites 36.6%

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \color{blue}{t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      9. lower-*.f64 36.6

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      12. lower-*.f64 36.6

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      14. count-2-rev N/A

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      15. lift-+.f64 36.6

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   8. Applied rewrites 36.6%

      \[\leadsto \color{blue}{\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 37.1% accurate, 4.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)} \end{array} \]

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(U * Float64(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[(U * N[(N[(n + n), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.5%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

2. 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 37.1

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

4. Applied rewrites 37.1%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   3. associate-*r* N/A

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

   5. lower-*.f64 36.6

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

6. Applied rewrites 36.6%

   \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

   3. associate-*r* N/A

      \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \color{blue}{t}} \]

   4. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t} \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

   6. associate-*l* N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

   7. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

   8. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

   9. lower-*.f64 36.6

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

   10. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

   11. *-commutative N/A

       \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

   12. lower-*.f64 36.6

       \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

   13. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

   14. count-2-rev N/A

       \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   15. lift-+.f64 36.6

       \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

8. Applied rewrites 36.6%

   \[\leadsto \color{blue}{\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot \color{blue}{t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   3. lift-+.f64 N/A

      \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   4. count-2-rev N/A

      \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

   5. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

   6. associate-*l* N/A

      \[\leadsto \sqrt{U \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot t\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto \sqrt{U \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot t\right)}} \]

   8. lower-*.f64 37.1

      \[\leadsto \sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \color{blue}{t}\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot t\right)} \]

   10. count-2-rev N/A

       \[\leadsto \sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)} \]

   11. lift-+.f64 37.1

       \[\leadsto \sqrt{U \cdot \left(\left(n + n\right) \cdot t\right)} \]

10. Applied rewrites 37.1%

    \[\leadsto \sqrt{U \cdot \color{blue}{\left(\left(n + n\right) \cdot t\right)}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025143 
(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*))))))