Toniolo and Linder, Equation (13)

Percentage Accurate: 50.3% → 65.2%

Time: 9.3s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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: 50.3% 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: 65.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (*.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 50.3%

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

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

      4. lower-*.f64 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. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 50.1

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

   3. Applied rewrites 45.0%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 44.1

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

   6. Applied rewrites 44.1%

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

   8. Applied rewrites 27.3%

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

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

   1. Initial program 50.3%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 53.9

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

   3. Applied rewrites 53.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 55.0

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

   5. Applied rewrites 55.0%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 55.0

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

   7. Applied rewrites 55.0%

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

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

   1. Initial program 50.3%

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

   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 27.5

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

   4. Applied rewrites 27.5%

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

Alternative 2: 60.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], 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[(l$95$m / Om), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], (-N[((-N[(N[(l$95$m * N[Sqrt[N[(-2.0 * N[(U * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]) * n), $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 50.3%

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

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

      4. lower-*.f64 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. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 50.1

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

   3. Applied rewrites 45.0%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 44.1

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

   6. Applied rewrites 44.1%

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

   8. Applied rewrites 27.3%

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

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

   1. Initial program 50.3%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 53.9

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

   3. Applied rewrites 53.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 55.0

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

   5. Applied rewrites 55.0%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 55.0

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

   7. Applied rewrites 55.0%

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

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

   1. Initial program 50.3%

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

   2. Taylor expanded in n around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-pow.f64 9.6

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

   4. Applied rewrites 9.6%

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

   5. Taylor expanded in l around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-pow.f64 11.3

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

   7. Applied rewrites 11.3%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 11.3

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 11.3

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

   9. Applied rewrites 11.3%

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

   10. Taylor expanded in Om around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower--.f64 13.7

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

   12. Applied rewrites 13.7%

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

Alternative 3: 56.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$1 * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(t$95$1 - N[(N[(U - U$42$), $MachinePrecision] * N[(l$95$m * N[(N[(l$95$m / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], (-N[((-N[(N[(l$95$m * N[Sqrt[N[(-2.0 * N[(U * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]) * n), $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 50.3%

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

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

      4. lower-*.f64 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. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 50.1

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

   3. Applied rewrites 45.0%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 44.1

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

   6. Applied rewrites 44.1%

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

   8. Applied rewrites 27.3%

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

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

   1. Initial program 50.3%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 53.9

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

   3. Applied rewrites 53.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 55.0

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

   5. Applied rewrites 55.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 55.0

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

   7. Applied rewrites 50.3%

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

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

   1. Initial program 50.3%

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

   2. Taylor expanded in n around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-pow.f64 9.6

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

   4. Applied rewrites 9.6%

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

   5. Taylor expanded in l around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-pow.f64 11.3

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

   7. Applied rewrites 11.3%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 11.3

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 11.3

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

   9. Applied rewrites 11.3%

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

   10. Taylor expanded in Om around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower--.f64 13.7

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

   12. Applied rewrites 13.7%

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

Alternative 4: 54.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right) \cdot U}\\ \mathbf{elif}\;t\_1 \leq 10^{+231}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{{l\_m}^{2}}{Om}\right)}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(\mathsf{fma}\left(U* - U, \left(l\_m \cdot \frac{l\_m}{Om \cdot Om}\right) \cdot n, \mathsf{fma}\left(l\_m \cdot \frac{l\_m}{Om}, -2, t\right)\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;-\left(-\frac{l\_m \cdot \sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}}{Om}\right) \cdot n\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* (* 2.0 n) U)
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_1 0.0)
     (* (sqrt (+ n n)) (sqrt (* (fma (* (/ l_m Om) l_m) -2.0 t) U)))
     (if (<= t_1 1e+231)
       (sqrt (* (* (+ n n) U) (+ t (* -2.0 (/ (pow l_m 2.0) Om)))))
       (if (<= t_1 INFINITY)
         (sqrt
          (*
           (+ n n)
           (*
            (fma
             (- U* U)
             (* (* l_m (/ l_m (* Om Om))) n)
             (fma (* l_m (/ l_m Om)) -2.0 t))
            U)))
         (- (* (- (/ (* l_m (sqrt (* -2.0 (* U (- U U*))))) Om)) n)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt((n + n)) * sqrt((fma(((l_m / Om) * l_m), -2.0, t) * U));
	} else if (t_1 <= 1e+231) {
		tmp = sqrt((((n + n) * U) * (t + (-2.0 * (pow(l_m, 2.0) / Om)))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt(((n + n) * (fma((U_42_ - U), ((l_m * (l_m / (Om * Om))) * n), fma((l_m * (l_m / Om)), -2.0, t)) * U)));
	} else {
		tmp = -(-((l_m * sqrt((-2.0 * (U * (U - U_42_))))) / Om) * n);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(Float64(fma(Float64(Float64(l_m / Om) * l_m), -2.0, t) * U)));
	elseif (t_1 <= 1e+231)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * Float64(t + Float64(-2.0 * Float64((l_m ^ 2.0) / Om)))));
	elseif (t_1 <= Inf)
		tmp = sqrt(Float64(Float64(n + n) * Float64(fma(Float64(U_42_ - U), Float64(Float64(l_m * Float64(l_m / Float64(Om * Om))) * n), fma(Float64(l_m * Float64(l_m / Om)), -2.0, t)) * U)));
	else
		tmp = Float64(-Float64(Float64(-Float64(Float64(l_m * sqrt(Float64(-2.0 * Float64(U * Float64(U - U_42_))))) / Om)) * n));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+231], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(-2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[(l$95$m * N[(l$95$m / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] + N[(N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], (-N[((-N[(N[(l$95$m * N[Sqrt[N[(-2.0 * N[(U * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]) * n), $MachinePrecision])]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 50.3%

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

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

      4. lower-*.f64 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. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 50.1

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

   3. Applied rewrites 45.0%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 44.1

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

   6. Applied rewrites 44.1%

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

   8. Applied rewrites 27.3%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1.0000000000000001e231

   1. Initial program 50.3%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 53.9

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

   3. Applied rewrites 53.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 55.0

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

   5. Applied rewrites 55.0%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 55.0

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

   7. Applied rewrites 55.0%

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

   8. Taylor expanded in n around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 44.1

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

   10. Applied rewrites 44.1%

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

   if 1.0000000000000001e231 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 50.3%

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

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

      4. lower-*.f64 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. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 50.1

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

   3. Applied rewrites 45.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 46.4

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 46.4

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lift-/.f64 N/A

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

      13. lower-*.f64 49.6

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

   5. Applied rewrites 49.6%

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

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

   1. Initial program 50.3%

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

   2. Taylor expanded in n around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-pow.f64 9.6

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

   4. Applied rewrites 9.6%

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

   5. Taylor expanded in l around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-pow.f64 11.3

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

   7. Applied rewrites 11.3%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 11.3

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 11.3

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

   9. Applied rewrites 11.3%

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

   10. Taylor expanded in Om around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower--.f64 13.7

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

   12. Applied rewrites 13.7%

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

Alternative 5: 51.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq -5.2 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{{l\_m}^{2}}{Om}\right)}\\ \mathbf{elif}\;n \leq -3.4 \cdot 10^{-15}:\\ \;\;\;\;-l\_m \cdot \left(n \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-4, \frac{U \cdot \left({l\_m}^{2} \cdot n\right)}{Om}, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right) \cdot U}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n -5.2e+124)
   (sqrt (* (* (+ n n) U) (+ t (* -2.0 (/ (pow l_m 2.0) Om)))))
   (if (<= n -3.4e-15)
     (- (* l_m (* n (sqrt (* -2.0 (/ (* U (- U U*)) (pow Om 2.0)))))))
     (if (<= n -1e-310)
       (sqrt (fma -4.0 (/ (* U (* (pow l_m 2.0) n)) Om) (* 2.0 (* U (* n t)))))
       (* (sqrt (+ n n)) (sqrt (* (fma (* (/ l_m Om) l_m) -2.0 t) U)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= -5.2e+124) {
		tmp = sqrt((((n + n) * U) * (t + (-2.0 * (pow(l_m, 2.0) / Om)))));
	} else if (n <= -3.4e-15) {
		tmp = -(l_m * (n * sqrt((-2.0 * ((U * (U - U_42_)) / pow(Om, 2.0))))));
	} else if (n <= -1e-310) {
		tmp = sqrt(fma(-4.0, ((U * (pow(l_m, 2.0) * n)) / Om), (2.0 * (U * (n * t)))));
	} else {
		tmp = sqrt((n + n)) * sqrt((fma(((l_m / Om) * l_m), -2.0, t) * U));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= -5.2e+124)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * Float64(t + Float64(-2.0 * Float64((l_m ^ 2.0) / Om)))));
	elseif (n <= -3.4e-15)
		tmp = Float64(-Float64(l_m * Float64(n * sqrt(Float64(-2.0 * Float64(Float64(U * Float64(U - U_42_)) / (Om ^ 2.0)))))));
	elseif (n <= -1e-310)
		tmp = sqrt(fma(-4.0, Float64(Float64(U * Float64((l_m ^ 2.0) * n)) / Om), Float64(2.0 * Float64(U * Float64(n * t)))));
	else
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(Float64(fma(Float64(Float64(l_m / Om) * l_m), -2.0, t) * U)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, -5.2e+124], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(-2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, -3.4e-15], (-N[(l$95$m * N[(n * N[Sqrt[N[(-2.0 * N[(N[(U * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), If[LessEqual[n, -1e-310], N[Sqrt[N[(-4.0 * N[(N[(U * N[(N[Power[l$95$m, 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -5.2000000000000001e124

   1. Initial program 50.3%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 53.9

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

   3. Applied rewrites 53.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 55.0

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

   5. Applied rewrites 55.0%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 55.0

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

   7. Applied rewrites 55.0%

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

   8. Taylor expanded in n around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 44.1

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

   10. Applied rewrites 44.1%

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

   if -5.2000000000000001e124 < n < -3.4e-15

   1. Initial program 50.3%

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

   2. Taylor expanded in n around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-pow.f64 9.6

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

   4. Applied rewrites 9.6%

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

   5. Taylor expanded in l around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-pow.f64 11.3

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

   7. Applied rewrites 11.3%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 11.3

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 11.3

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

   9. Applied rewrites 11.3%

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

   10. Taylor expanded in l around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-pow.f64 10.5

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

   12. Applied rewrites 10.5%

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

   if -3.4e-15 < n < -9.999999999999969e-311

   1. Initial program 50.3%

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

   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 42.8

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

   4. Applied rewrites 42.8%

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

   if -9.999999999999969e-311 < n

   1. Initial program 50.3%

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

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

      4. lower-*.f64 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. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 50.1

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

   3. Applied rewrites 45.0%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 44.1

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

   6. Applied rewrites 44.1%

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

   8. Applied rewrites 27.3%

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

Alternative 6: 49.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right) \cdot U\\ \mathbf{if}\;n \leq -5.2 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{{l\_m}^{2}}{Om}\right)}\\ \mathbf{elif}\;n \leq -3.4 \cdot 10^{-15}:\\ \;\;\;\;-l\_m \cdot \left(n \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-301}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{t\_1}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (fma (* (/ l_m Om) l_m) -2.0 t) U)))
   (if (<= n -5.2e+124)
     (sqrt (* (* (+ n n) U) (+ t (* -2.0 (/ (pow l_m 2.0) Om)))))
     (if (<= n -3.4e-15)
       (- (* l_m (* n (sqrt (* -2.0 (/ (* U (- U U*)) (pow Om 2.0)))))))
       (if (<= n 1.5e-301)
         (sqrt (* (+ n n) t_1))
         (* (sqrt (+ n n)) (sqrt t_1)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = fma(((l_m / Om) * l_m), -2.0, t) * U;
	double tmp;
	if (n <= -5.2e+124) {
		tmp = sqrt((((n + n) * U) * (t + (-2.0 * (pow(l_m, 2.0) / Om)))));
	} else if (n <= -3.4e-15) {
		tmp = -(l_m * (n * sqrt((-2.0 * ((U * (U - U_42_)) / pow(Om, 2.0))))));
	} else if (n <= 1.5e-301) {
		tmp = sqrt(((n + n) * t_1));
	} else {
		tmp = sqrt((n + n)) * sqrt(t_1);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(fma(Float64(Float64(l_m / Om) * l_m), -2.0, t) * U)
	tmp = 0.0
	if (n <= -5.2e+124)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * Float64(t + Float64(-2.0 * Float64((l_m ^ 2.0) / Om)))));
	elseif (n <= -3.4e-15)
		tmp = Float64(-Float64(l_m * Float64(n * sqrt(Float64(-2.0 * Float64(Float64(U * Float64(U - U_42_)) / (Om ^ 2.0)))))));
	elseif (n <= 1.5e-301)
		tmp = sqrt(Float64(Float64(n + n) * t_1));
	else
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(t_1));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision] * U), $MachinePrecision]}, If[LessEqual[n, -5.2e+124], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(-2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, -3.4e-15], (-N[(l$95$m * N[(n * N[Sqrt[N[(-2.0 * N[(N[(U * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), If[LessEqual[n, 1.5e-301], N[Sqrt[N[(N[(n + n), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -5.2000000000000001e124

   1. Initial program 50.3%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 53.9

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

   3. Applied rewrites 53.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 55.0

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

   5. Applied rewrites 55.0%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 55.0

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

   7. Applied rewrites 55.0%

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

   8. Taylor expanded in n around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 44.1

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

   10. Applied rewrites 44.1%

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

   if -5.2000000000000001e124 < n < -3.4e-15

   1. Initial program 50.3%

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

   2. Taylor expanded in n around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-pow.f64 9.6

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

   4. Applied rewrites 9.6%

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

   5. Taylor expanded in l around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-pow.f64 11.3

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

   7. Applied rewrites 11.3%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 11.3

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 11.3

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

   9. Applied rewrites 11.3%

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

   10. Taylor expanded in l around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-pow.f64 10.5

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

   12. Applied rewrites 10.5%

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

   if -3.4e-15 < n < 1.5e-301

   1. Initial program 50.3%

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

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

      4. lower-*.f64 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. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 50.1

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

   3. Applied rewrites 45.0%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 44.1

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

   6. Applied rewrites 44.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. associate-/l* N/A

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

      17. lift-/.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 47.3

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

   8. Applied rewrites 47.3%

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

   if 1.5e-301 < n

   1. Initial program 50.3%

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

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

      4. lower-*.f64 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. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 50.1

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

   3. Applied rewrites 45.0%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 44.1

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

   6. Applied rewrites 44.1%

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

   8. Applied rewrites 27.3%

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

Alternative 7: 49.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right) \cdot U\\ \mathbf{if}\;n \leq -5.2 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{{l\_m}^{2}}{Om}\right)}\\ \mathbf{elif}\;n \leq -3.4 \cdot 10^{-15}:\\ \;\;\;\;-\left(l\_m \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right) \cdot n\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-301}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{t\_1}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (fma (* (/ l_m Om) l_m) -2.0 t) U)))
   (if (<= n -5.2e+124)
     (sqrt (* (* (+ n n) U) (+ t (* -2.0 (/ (pow l_m 2.0) Om)))))
     (if (<= n -3.4e-15)
       (- (* (* l_m (sqrt (* -2.0 (/ (* U (- U U*)) (pow Om 2.0))))) n))
       (if (<= n 1.5e-301)
         (sqrt (* (+ n n) t_1))
         (* (sqrt (+ n n)) (sqrt t_1)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = fma(((l_m / Om) * l_m), -2.0, t) * U;
	double tmp;
	if (n <= -5.2e+124) {
		tmp = sqrt((((n + n) * U) * (t + (-2.0 * (pow(l_m, 2.0) / Om)))));
	} else if (n <= -3.4e-15) {
		tmp = -((l_m * sqrt((-2.0 * ((U * (U - U_42_)) / pow(Om, 2.0))))) * n);
	} else if (n <= 1.5e-301) {
		tmp = sqrt(((n + n) * t_1));
	} else {
		tmp = sqrt((n + n)) * sqrt(t_1);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(fma(Float64(Float64(l_m / Om) * l_m), -2.0, t) * U)
	tmp = 0.0
	if (n <= -5.2e+124)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * Float64(t + Float64(-2.0 * Float64((l_m ^ 2.0) / Om)))));
	elseif (n <= -3.4e-15)
		tmp = Float64(-Float64(Float64(l_m * sqrt(Float64(-2.0 * Float64(Float64(U * Float64(U - U_42_)) / (Om ^ 2.0))))) * n));
	elseif (n <= 1.5e-301)
		tmp = sqrt(Float64(Float64(n + n) * t_1));
	else
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(t_1));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision] * U), $MachinePrecision]}, If[LessEqual[n, -5.2e+124], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(-2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, -3.4e-15], (-N[(N[(l$95$m * N[Sqrt[N[(-2.0 * N[(N[(U * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]), If[LessEqual[n, 1.5e-301], N[Sqrt[N[(N[(n + n), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -5.2000000000000001e124

   1. Initial program 50.3%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 53.9

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

   3. Applied rewrites 53.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 55.0

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

   5. Applied rewrites 55.0%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 55.0

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

   7. Applied rewrites 55.0%

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

   8. Taylor expanded in n around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 44.1

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

   10. Applied rewrites 44.1%

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

   if -5.2000000000000001e124 < n < -3.4e-15

   1. Initial program 50.3%

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

   2. Taylor expanded in n around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-pow.f64 9.6

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

   4. Applied rewrites 9.6%

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

   5. Taylor expanded in l around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-pow.f64 11.3

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

   7. Applied rewrites 11.3%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 11.3

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 11.3

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

   9. Applied rewrites 11.3%

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

   10. Taylor expanded in l around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-pow.f64 10.5

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

   12. Applied rewrites 10.5%

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

   if -3.4e-15 < n < 1.5e-301

   1. Initial program 50.3%

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

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

      4. lower-*.f64 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. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 50.1

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

   3. Applied rewrites 45.0%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 44.1

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

   6. Applied rewrites 44.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. associate-/l* N/A

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

      17. lift-/.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 47.3

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

   8. Applied rewrites 47.3%

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

   if 1.5e-301 < n

   1. Initial program 50.3%

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

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

      4. lower-*.f64 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. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 50.1

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

   3. Applied rewrites 45.0%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 44.1

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

   6. Applied rewrites 44.1%

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

   8. Applied rewrites 27.3%

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

Alternative 8: 48.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right) \cdot U}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{{l\_m}^{2}}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\left(-\frac{l\_m \cdot \sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}}{Om}\right) \cdot n\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* (* 2.0 n) U)
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_1 0.0)
     (* (sqrt (+ n n)) (sqrt (* (fma (* (/ l_m Om) l_m) -2.0 t) U)))
     (if (<= t_1 4e+306)
       (sqrt (* (* (+ n n) U) (+ t (* -2.0 (/ (pow l_m 2.0) Om)))))
       (- (* (- (/ (* l_m (sqrt (* -2.0 (* U (- U U*))))) Om)) n))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt((n + n)) * sqrt((fma(((l_m / Om) * l_m), -2.0, t) * U));
	} else if (t_1 <= 4e+306) {
		tmp = sqrt((((n + n) * U) * (t + (-2.0 * (pow(l_m, 2.0) / Om)))));
	} else {
		tmp = -(-((l_m * sqrt((-2.0 * (U * (U - U_42_))))) / Om) * n);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(Float64(fma(Float64(Float64(l_m / Om) * l_m), -2.0, t) * U)));
	elseif (t_1 <= 4e+306)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * Float64(t + Float64(-2.0 * Float64((l_m ^ 2.0) / Om)))));
	else
		tmp = Float64(-Float64(Float64(-Float64(Float64(l_m * sqrt(Float64(-2.0 * Float64(U * Float64(U - U_42_))))) / Om)) * n));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+306], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * N[(t + N[(-2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], (-N[((-N[(N[(l$95$m * N[Sqrt[N[(-2.0 * N[(U * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]) * n), $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 50.3%

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

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

      4. lower-*.f64 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. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 50.1

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

   3. Applied rewrites 45.0%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 44.1

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

   6. Applied rewrites 44.1%

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

   8. Applied rewrites 27.3%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.00000000000000007e306

   1. Initial program 50.3%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 53.9

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

   3. Applied rewrites 53.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 55.0

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

   5. Applied rewrites 55.0%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 55.0

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

   7. Applied rewrites 55.0%

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

   8. Taylor expanded in n around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 44.1

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

   10. Applied rewrites 44.1%

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

   if 4.00000000000000007e306 < (*.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 50.3%

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

   2. Taylor expanded in n around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-pow.f64 9.6

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

   4. Applied rewrites 9.6%

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

   5. Taylor expanded in l around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-pow.f64 11.3

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

   7. Applied rewrites 11.3%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 11.3

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 11.3

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

   9. Applied rewrites 11.3%

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

   10. Taylor expanded in Om around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower--.f64 13.7

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

   12. Applied rewrites 13.7%

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

Alternative 9: 48.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right) \cdot U\\ \mathbf{if}\;n \leq 1.5 \cdot 10^{-301}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{t\_1}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (fma (* (/ l_m Om) l_m) -2.0 t) U)))
   (if (<= n 1.5e-301) (sqrt (* (+ n n) t_1)) (* (sqrt (+ n n)) (sqrt t_1)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = fma(((l_m / Om) * l_m), -2.0, t) * U;
	double tmp;
	if (n <= 1.5e-301) {
		tmp = sqrt(((n + n) * t_1));
	} else {
		tmp = sqrt((n + n)) * sqrt(t_1);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(fma(Float64(Float64(l_m / Om) * l_m), -2.0, t) * U)
	tmp = 0.0
	if (n <= 1.5e-301)
		tmp = sqrt(Float64(Float64(n + n) * t_1));
	else
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(t_1));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision] * U), $MachinePrecision]}, If[LessEqual[n, 1.5e-301], N[Sqrt[N[(N[(n + n), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < 1.5e-301

   1. Initial program 50.3%

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

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

      4. lower-*.f64 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. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 50.1

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

   3. Applied rewrites 45.0%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 44.1

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

   6. Applied rewrites 44.1%

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

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(U \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot \color{blue}{U}\right)} \]

      3. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)} \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\left(t - \left(\mathsf{neg}\left(-2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)} \]

      6. metadata-eval N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)} \]

      8. lift-pow.f64 N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)} \]

      9. pow2 N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot U\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot \color{blue}{U}\right)} \]

      11. fp-cancel-sub-sign-inv N/A

          \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot U\right)} \]

      12. metadata-eval N/A

          \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot U\right)} \]

      13. +-commutative N/A

          \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om} + t\right) \cdot U\right)} \]

      14. *-commutative N/A

          \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\left(\frac{\ell \cdot \ell}{Om} \cdot -2 + t\right) \cdot U\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right) \cdot U\right)} \]

      16. associate-/l* N/A

          \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right) \cdot U\right)} \]

      17. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right) \cdot U\right)} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot U\right)} \]

      19. lower-*.f64 47.3

          \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot U\right)} \]

   8. Applied rewrites 47.3%

      \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot \color{blue}{U}\right)} \]

   if 1.5e-301 < n

   1. Initial program 50.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. 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. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      4. lower-*.f64 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. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right)} \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      6. count-2-rev N/A

         \[\leadsto \sqrt{\color{blue}{\left(n + n\right)} \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n + n\right)} \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot U\right)}} \]

      9. lower-*.f64 50.1

         \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot U\right)}} \]

   3. Applied rewrites 45.0%

      \[\leadsto \sqrt{\color{blue}{\left(n + n\right) \cdot \left(\mathsf{fma}\left(U* - U, \frac{\ell \cdot \ell}{Om \cdot Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right) \cdot U\right)}} \]

   4. Taylor expanded in n around 0

      \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(U \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(U \cdot \left(t + \color{blue}{-2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(U \cdot \left(t + -2 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}}\right)\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{\color{blue}{Om}}\right)\right)} \]

      5. lower-pow.f64 44.1

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

   6. Applied rewrites 44.1%

      \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

   8. Applied rewrites 27.3%

      \[\leadsto \color{blue}{\sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot U}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 48.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(\mathsf{fma}\left(\frac{l\_m}{Om} \cdot l\_m, -2, t\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;-\left(-\frac{l\_m \cdot \sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}}{Om}\right) \cdot n\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<=
      (*
       (* (* 2.0 n) U)
       (-
        (- t (* 2.0 (/ (* l_m l_m) Om)))
        (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))
      INFINITY)
   (sqrt (* (+ n n) (* (fma (* (/ l_m Om) l_m) -2.0 t) U)))
   (- (* (- (/ (* l_m (sqrt (* -2.0 (* U (- U U*))))) Om)) n))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))) <= ((double) INFINITY)) {
		tmp = sqrt(((n + n) * (fma(((l_m / Om) * l_m), -2.0, t) * U)));
	} else {
		tmp = -(-((l_m * sqrt((-2.0 * (U * (U - U_42_))))) / Om) * n);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_)))) <= Inf)
		tmp = sqrt(Float64(Float64(n + n) * Float64(fma(Float64(Float64(l_m / Om) * l_m), -2.0, t) * U)));
	else
		tmp = Float64(-Float64(Float64(-Float64(Float64(l_m * sqrt(Float64(-2.0 * Float64(U * Float64(U - U_42_))))) / Om)) * n));
	end
	return 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], Infinity], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(N[(N[(N[(l$95$m / Om), $MachinePrecision] * l$95$m), $MachinePrecision] * -2.0 + t), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], (-N[((-N[(N[(l$95$m * N[Sqrt[N[(-2.0 * N[(U * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]) * n), $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*)))) < +inf.0

   1. Initial program 50.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. 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. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      4. lower-*.f64 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. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right)} \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      6. count-2-rev N/A

         \[\leadsto \sqrt{\color{blue}{\left(n + n\right)} \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n + n\right)} \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot U\right)}} \]

      9. lower-*.f64 50.1

         \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot U\right)}} \]

   3. Applied rewrites 45.0%

      \[\leadsto \sqrt{\color{blue}{\left(n + n\right) \cdot \left(\mathsf{fma}\left(U* - U, \frac{\ell \cdot \ell}{Om \cdot Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right) \cdot U\right)}} \]

   4. Taylor expanded in n around 0

      \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(U \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(U \cdot \left(t + \color{blue}{-2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(U \cdot \left(t + -2 \cdot \color{blue}{\frac{{\ell}^{2}}{Om}}\right)\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{\color{blue}{Om}}\right)\right)} \]

      5. lower-pow.f64 44.1

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

   6. Applied rewrites 44.1%

      \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(U \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot \color{blue}{U}\right)} \]

      3. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)} \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\left(t - \left(\mathsf{neg}\left(-2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)} \]

      6. metadata-eval N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)} \]

      8. lift-pow.f64 N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot U\right)} \]

      9. pow2 N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot U\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot \color{blue}{U}\right)} \]

      11. fp-cancel-sub-sign-inv N/A

          \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot U\right)} \]

      12. metadata-eval N/A

          \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\left(t + -2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot U\right)} \]

      13. +-commutative N/A

          \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om} + t\right) \cdot U\right)} \]

      14. *-commutative N/A

          \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\left(\frac{\ell \cdot \ell}{Om} \cdot -2 + t\right) \cdot U\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right) \cdot U\right)} \]

      16. associate-/l* N/A

          \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right) \cdot U\right)} \]

      17. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right) \cdot U\right)} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot U\right)} \]

      19. lower-*.f64 47.3

          \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot U\right)} \]

   8. Applied rewrites 47.3%

      \[\leadsto \sqrt{\left(n + n\right) \cdot \left(\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right) \cdot \color{blue}{U}\right)} \]

   if +inf.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 50.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(n \cdot \sqrt{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(n \cdot \sqrt{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \color{blue}{\sqrt{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \sqrt{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \sqrt{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \sqrt{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \sqrt{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \sqrt{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \sqrt{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}\right) \]

      9. lower--.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \sqrt{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}\right) \]

      10. lower-pow.f64 9.6

          \[\leadsto -1 \cdot \left(n \cdot \sqrt{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}\right) \]

   4. Applied rewrites 9.6%

      \[\leadsto \color{blue}{-1 \cdot \left(n \cdot \sqrt{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}\right)} \]

   5. Taylor expanded in l around -inf

      \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \color{blue}{\left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \left(\ell \cdot \color{blue}{\sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}}\right)\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right)\right) \]

      7. lower--.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right)\right) \]

      8. lower-pow.f64 11.3

         \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right)\right) \]

   7. Applied rewrites 11.3%

      \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \color{blue}{\left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)}\right)\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(n \cdot \left(-1 \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right)\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(n \cdot \left(-1 \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right)\right) \]

      3. lower-neg.f64 11.3

         \[\leadsto -n \cdot \left(-1 \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto -n \cdot \left(-1 \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto -\left(-1 \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right) \cdot n \]

      6. lower-*.f64 11.3

         \[\leadsto -\left(-1 \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right) \cdot n \]

   9. Applied rewrites 11.3%

      \[\leadsto -\left(-\sqrt{\frac{-2 \cdot \left(\left(U - U*\right) \cdot U\right)}{Om \cdot Om}} \cdot \ell\right) \cdot n \]

   10. Taylor expanded in Om around 0

       \[\leadsto -\left(-\frac{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}}{Om}\right) \cdot n \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto -\left(-\frac{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}}{Om}\right) \cdot n \]

       2. lower-*.f64 N/A

          \[\leadsto -\left(-\frac{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}}{Om}\right) \cdot n \]

       3. lower-sqrt.f64 N/A

          \[\leadsto -\left(-\frac{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}}{Om}\right) \cdot n \]

       4. lower-*.f64 N/A

          \[\leadsto -\left(-\frac{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}}{Om}\right) \cdot n \]

       5. lower-*.f64 N/A

          \[\leadsto -\left(-\frac{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}}{Om}\right) \cdot n \]

       6. lower--.f64 13.7

          \[\leadsto -\left(-\frac{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}}{Om}\right) \cdot n \]

   12. Applied rewrites 13.7%

       \[\leadsto -\left(-\frac{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}}{Om}\right) \cdot n \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 44.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{t \cdot U}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-\left(-\frac{l\_m \cdot \sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}}{Om}\right) \cdot n\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* (* 2.0 n) U)
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_1 0.0)
     (* (sqrt (+ n n)) (sqrt (* t U)))
     (if (<= t_1 4e+306)
       (sqrt (* (* U (+ n n)) t))
       (- (* (- (/ (* l_m (sqrt (* -2.0 (* U (- U U*))))) Om)) n))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt((n + n)) * sqrt((t * U));
	} else if (t_1 <= 4e+306) {
		tmp = sqrt(((U * (n + n)) * t));
	} else {
		tmp = -(-((l_m * sqrt((-2.0 * (U * (U - U_42_))))) / Om) * n);
	}
	return tmp;
}

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l_m, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: 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((n + n)) * sqrt((t * u))
    else if (t_1 <= 4d+306) then
        tmp = sqrt(((u * (n + n)) * t))
    else
        tmp = -(-((l_m * sqrt(((-2.0d0) * (u * (u - u_42))))) / om) * n)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double 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((n + n)) * Math.sqrt((t * U));
	} else if (t_1 <= 4e+306) {
		tmp = Math.sqrt(((U * (n + n)) * t));
	} else {
		tmp = -(-((l_m * Math.sqrt((-2.0 * (U * (U - U_42_))))) / Om) * n);
	}
	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((n + n)) * math.sqrt((t * U))
	elif t_1 <= 4e+306:
		tmp = math.sqrt(((U * (n + n)) * t))
	else:
		tmp = -(-((l_m * math.sqrt((-2.0 * (U * (U - U_42_))))) / Om) * n)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(Float64(t * U)));
	elseif (t_1 <= 4e+306)
		tmp = sqrt(Float64(Float64(U * Float64(n + n)) * t));
	else
		tmp = Float64(-Float64(Float64(-Float64(Float64(l_m * sqrt(Float64(-2.0 * Float64(U * Float64(U - U_42_))))) / Om)) * n));
	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((n + n)) * sqrt((t * U));
	elseif (t_1 <= 4e+306)
		tmp = sqrt(((U * (n + n)) * t));
	else
		tmp = -(-((l_m * sqrt((-2.0 * (U * (U - U_42_))))) / Om) * n);
	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[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+306], N[Sqrt[N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], (-N[((-N[(N[(l$95$m * N[Sqrt[N[(-2.0 * N[(U * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]) * n), $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 50.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. 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. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      4. lower-*.f64 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. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right)} \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      6. count-2-rev N/A

         \[\leadsto \sqrt{\color{blue}{\left(n + n\right)} \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n + n\right)} \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot U\right)}} \]

      9. lower-*.f64 50.1

         \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot U\right)}} \]

   3. Applied rewrites 45.0%

      \[\leadsto \sqrt{\color{blue}{\left(n + n\right) \cdot \left(\mathsf{fma}\left(U* - U, \frac{\ell \cdot \ell}{Om \cdot Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right) \cdot U\right)}} \]

   4. Taylor expanded in t around inf

      \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(U \cdot t\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 35.4

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(U \cdot \color{blue}{t}\right)} \]

   6. Applied rewrites 35.4%

      \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(U \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(n + n\right) \cdot \left(U \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n + n\right) \cdot \left(U \cdot t\right)}} \]

      3. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{n + n} \cdot \sqrt{U \cdot t}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{n + n} \cdot \sqrt{U \cdot t}} \]

   8. Applied rewrites 20.7%

      \[\leadsto \color{blue}{\sqrt{n + n} \cdot \sqrt{t \cdot U}} \]

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.00000000000000007e306

   1. Initial program 50.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 36.1

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 36.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   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 36.1

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

   6. Applied rewrites 36.1%

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]
   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. count-2-rev N/A

         \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \]

      8. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \]

      10. lower-*.f64 36.1

          \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      13. lower-*.f64 36.1

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   8. Applied rewrites 36.1%

      \[\leadsto \color{blue}{\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}} \]

   if 4.00000000000000007e306 < (*.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 50.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in n around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(n \cdot \sqrt{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(n \cdot \sqrt{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \color{blue}{\sqrt{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \sqrt{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \sqrt{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \sqrt{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \sqrt{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \sqrt{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \sqrt{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}\right) \]

      9. lower--.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \sqrt{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}\right) \]

      10. lower-pow.f64 9.6

          \[\leadsto -1 \cdot \left(n \cdot \sqrt{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}\right) \]

   4. Applied rewrites 9.6%

      \[\leadsto \color{blue}{-1 \cdot \left(n \cdot \sqrt{-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}\right)} \]

   5. Taylor expanded in l around -inf

      \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \color{blue}{\left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \left(\ell \cdot \color{blue}{\sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}}\right)\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right)\right) \]

      7. lower--.f64 N/A

         \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right)\right) \]

      8. lower-pow.f64 11.3

         \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right)\right) \]

   7. Applied rewrites 11.3%

      \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \color{blue}{\left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)}\right)\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(n \cdot \left(-1 \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right)\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(n \cdot \left(-1 \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right)\right) \]

      3. lower-neg.f64 11.3

         \[\leadsto -n \cdot \left(-1 \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto -n \cdot \left(-1 \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto -\left(-1 \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right) \cdot n \]

      6. lower-*.f64 11.3

         \[\leadsto -\left(-1 \cdot \left(\ell \cdot \sqrt{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}\right)\right) \cdot n \]

   9. Applied rewrites 11.3%

      \[\leadsto -\left(-\sqrt{\frac{-2 \cdot \left(\left(U - U*\right) \cdot U\right)}{Om \cdot Om}} \cdot \ell\right) \cdot n \]

   10. Taylor expanded in Om around 0

       \[\leadsto -\left(-\frac{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}}{Om}\right) \cdot n \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto -\left(-\frac{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}}{Om}\right) \cdot n \]

       2. lower-*.f64 N/A

          \[\leadsto -\left(-\frac{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}}{Om}\right) \cdot n \]

       3. lower-sqrt.f64 N/A

          \[\leadsto -\left(-\frac{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}}{Om}\right) \cdot n \]

       4. lower-*.f64 N/A

          \[\leadsto -\left(-\frac{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}}{Om}\right) \cdot n \]

       5. lower-*.f64 N/A

          \[\leadsto -\left(-\frac{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}}{Om}\right) \cdot n \]

       6. lower--.f64 13.7

          \[\leadsto -\left(-\frac{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}}{Om}\right) \cdot n \]

   12. Applied rewrites 13.7%

       \[\leadsto -\left(-\frac{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(U - U*\right)\right)}}{Om}\right) \cdot n \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 39.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq -1.72 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\sqrt{2 \cdot \left(U \cdot n\right)} \cdot \sqrt{\frac{1}{t}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t -1.72e-307)
   (sqrt (* (* U (+ n n)) t))
   (* t (* (sqrt (* 2.0 (* U n))) (sqrt (/ 1.0 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 (t <= -1.72e-307) {
		tmp = sqrt(((U * (n + n)) * t));
	} else {
		tmp = t * (sqrt((2.0 * (U * n))) * sqrt((1.0 / 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 (t <= (-1.72d-307)) then
        tmp = sqrt(((u * (n + n)) * t))
    else
        tmp = t * (sqrt((2.0d0 * (u * n))) * sqrt((1.0d0 / 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 (t <= -1.72e-307) {
		tmp = Math.sqrt(((U * (n + n)) * t));
	} else {
		tmp = t * (Math.sqrt((2.0 * (U * n))) * Math.sqrt((1.0 / t)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if t <= -1.72e-307:
		tmp = math.sqrt(((U * (n + n)) * t))
	else:
		tmp = t * (math.sqrt((2.0 * (U * n))) * math.sqrt((1.0 / t)))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= -1.72e-307)
		tmp = sqrt(Float64(Float64(U * Float64(n + n)) * t));
	else
		tmp = Float64(t * Float64(sqrt(Float64(2.0 * Float64(U * n))) * sqrt(Float64(1.0 / 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 (t <= -1.72e-307)
		tmp = sqrt(((U * (n + n)) * t));
	else
		tmp = t * (sqrt((2.0 * (U * n))) * sqrt((1.0 / 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[t, -1.72e-307], N[Sqrt[N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[(t * N[(N[Sqrt[N[(2.0 * N[(U * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.72000000000000008e-307

   1. Initial program 50.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 36.1

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 36.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   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 36.1

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

   6. Applied rewrites 36.1%

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]
   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. count-2-rev N/A

         \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \]

      8. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \]

      10. lower-*.f64 36.1

          \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      13. lower-*.f64 36.1

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   8. Applied rewrites 36.1%

      \[\leadsto \color{blue}{\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}} \]

   if -1.72000000000000008e-307 < t

   1. Initial program 50.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. sub-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + t\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\left(\mathsf{neg}\left(\color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right)\right) + t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell \cdot \ell}{Om}} + t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}} + t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} + t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} + t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{Om}}\right) + t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \ell\right) \cdot \frac{\ell}{Om}} + t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \ell, \frac{\ell}{Om}, t\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \ell}, \frac{\ell}{Om}, t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      13. metadata-eval 53.9

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(\color{blue}{-2} \cdot \ell, \frac{\ell}{Om}, t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.9%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   5. Applied rewrites 29.2%

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n + n\right)} \cdot \sqrt{\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right) - \left(\left(\ell \cdot \frac{\ell}{Om \cdot Om}\right) \cdot n\right) \cdot \left(U - U*\right)}} \]

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\sqrt{2 \cdot \left(U \cdot n\right)} \cdot \sqrt{\frac{1}{t}}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto t \cdot \color{blue}{\left(\sqrt{2 \cdot \left(U \cdot n\right)} \cdot \sqrt{\frac{1}{t}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto t \cdot \left(\sqrt{2 \cdot \left(U \cdot n\right)} \cdot \color{blue}{\sqrt{\frac{1}{t}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto t \cdot \left(\sqrt{2 \cdot \left(U \cdot n\right)} \cdot \sqrt{\color{blue}{\frac{1}{t}}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto t \cdot \left(\sqrt{2 \cdot \left(U \cdot n\right)} \cdot \sqrt{\frac{\color{blue}{1}}{t}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto t \cdot \left(\sqrt{2 \cdot \left(U \cdot n\right)} \cdot \sqrt{\frac{1}{t}}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto t \cdot \left(\sqrt{2 \cdot \left(U \cdot n\right)} \cdot \sqrt{\frac{1}{t}}\right) \]

      7. lower-/.f64 21.1

         \[\leadsto t \cdot \left(\sqrt{2 \cdot \left(U \cdot n\right)} \cdot \sqrt{\frac{1}{t}}\right) \]

   8. Applied rewrites 21.1%

      \[\leadsto \color{blue}{t \cdot \left(\sqrt{2 \cdot \left(U \cdot n\right)} \cdot \sqrt{\frac{1}{t}}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 39.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(n + n\right)\\ \mathbf{if}\;t \leq -1.72 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{t\_1 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (+ n n))))
   (if (<= t -1.72e-307) (sqrt (* t_1 t)) (* (sqrt t_1) (sqrt t)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (n + n);
	double tmp;
	if (t <= -1.72e-307) {
		tmp = sqrt((t_1 * t));
	} else {
		tmp = sqrt(t_1) * sqrt(t);
	}
	return tmp;
}

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l_m, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = u * (n + n)
    if (t <= (-1.72d-307)) then
        tmp = sqrt((t_1 * t))
    else
        tmp = sqrt(t_1) * sqrt(t)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (n + n);
	double tmp;
	if (t <= -1.72e-307) {
		tmp = Math.sqrt((t_1 * t));
	} else {
		tmp = Math.sqrt(t_1) * Math.sqrt(t);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = U * (n + n)
	tmp = 0
	if t <= -1.72e-307:
		tmp = math.sqrt((t_1 * t))
	else:
		tmp = math.sqrt(t_1) * math.sqrt(t)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(n + n))
	tmp = 0.0
	if (t <= -1.72e-307)
		tmp = sqrt(Float64(t_1 * t));
	else
		tmp = Float64(sqrt(t_1) * sqrt(t));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = U * (n + n);
	tmp = 0.0;
	if (t <= -1.72e-307)
		tmp = sqrt((t_1 * t));
	else
		tmp = sqrt(t_1) * sqrt(t);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.72e-307], N[Sqrt[N[(t$95$1 * t), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.72000000000000008e-307

   1. Initial program 50.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 36.1

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 36.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   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 36.1

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

   6. Applied rewrites 36.1%

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]
   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. count-2-rev N/A

         \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \]

      8. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \]

      10. lower-*.f64 36.1

          \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      13. lower-*.f64 36.1

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   8. Applied rewrites 36.1%

      \[\leadsto \color{blue}{\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}} \]

   if -1.72000000000000008e-307 < t

   1. Initial program 50.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      2. sub-flip N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + t\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\left(\mathsf{neg}\left(\color{blue}{2 \cdot \frac{\ell \cdot \ell}{Om}}\right)\right) + t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\ell \cdot \ell}{Om}} + t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}} + t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om} + t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      8. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right)} + t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{Om}}\right) + t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \ell\right) \cdot \frac{\ell}{Om}} + t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      11. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \ell, \frac{\ell}{Om}, t\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \ell}, \frac{\ell}{Om}, t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

      13. metadata-eval 53.9

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(\color{blue}{-2} \cdot \ell, \frac{\ell}{Om}, t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.9%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)} - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   5. Applied rewrites 29.2%

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n + n\right)} \cdot \sqrt{\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right) - \left(\left(\ell \cdot \frac{\ell}{Om \cdot Om}\right) \cdot n\right) \cdot \left(U - U*\right)}} \]

   6. Taylor expanded in l around 0

      \[\leadsto \sqrt{U \cdot \left(n + n\right)} \cdot \color{blue}{\sqrt{t}} \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 21.1

         \[\leadsto \sqrt{U \cdot \left(n + n\right)} \cdot \sqrt{t} \]

   8. Applied rewrites 21.1%

      \[\leadsto \sqrt{U \cdot \left(n + n\right)} \cdot \color{blue}{\sqrt{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 38.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 2 \cdot 10^{-322}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}\\ \end{array} \end{array} \]

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*))))
      2e-322)
   (sqrt (* (+ n n) (* U 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_)))) <= 2e-322) {
		tmp = sqrt(((n + n) * (U * 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)))) <= 2d-322) then
        tmp = sqrt(((n + n) * (u * 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_)))) <= 2e-322) {
		tmp = Math.sqrt(((n + n) * (U * 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_)))) <= 2e-322:
		tmp = math.sqrt(((n + n) * (U * 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_)))) <= 2e-322)
		tmp = sqrt(Float64(Float64(n + n) * Float64(U * 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_)))) <= 2e-322)
		tmp = sqrt(((n + n) * (U * 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], 2e-322], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(U * 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*)))) < 1.97626e-322

   1. Initial program 50.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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. 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. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}} \]

      4. lower-*.f64 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. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right)} \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      6. count-2-rev N/A

         \[\leadsto \sqrt{\color{blue}{\left(n + n\right)} \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n + n\right)} \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot U\right)}} \]

      9. lower-*.f64 50.1

         \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot U\right)}} \]

   3. Applied rewrites 45.0%

      \[\leadsto \sqrt{\color{blue}{\left(n + n\right) \cdot \left(\mathsf{fma}\left(U* - U, \frac{\ell \cdot \ell}{Om \cdot Om} \cdot n, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right) \cdot U\right)}} \]

   4. Taylor expanded in t around inf

      \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(U \cdot t\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 35.4

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(U \cdot \color{blue}{t}\right)} \]

   6. Applied rewrites 35.4%

      \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(U \cdot t\right)}} \]

   if 1.97626e-322 < (*.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 50.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 36.1

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 36.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   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 36.1

         \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

   6. Applied rewrites 36.1%

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]
   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. count-2-rev N/A

         \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \]

      8. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \]

      10. lower-*.f64 36.1

          \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      13. lower-*.f64 36.1

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   8. Applied rewrites 36.1%

      \[\leadsto \color{blue}{\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 36.1% accurate, 4.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* (* U (+ n n)) t)))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return sqrt(((U * (n + n)) * t));
}

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l_m, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt(((u * (n + n)) * t))
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return Math.sqrt(((U * (n + n)) * t));
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return math.sqrt(((U * (n + n)) * t))

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return sqrt(Float64(Float64(U * Float64(n + n)) * t))
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = sqrt(((U * (n + n)) * t));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.3%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

2. Taylor expanded in t around inf

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

   3. lower-*.f64 36.1

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

4. Applied rewrites 36.1%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
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 36.1

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot t\right)} \]

6. Applied rewrites 36.1%

   \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]
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. count-2-rev N/A

      \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \]

   8. lift-+.f64 N/A

      \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \]

   9. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \]

   10. lower-*.f64 36.1

       \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \color{blue}{t}} \]

   11. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t} \]

   12. *-commutative N/A

       \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   13. lower-*.f64 36.1

       \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

8. Applied rewrites 36.1%

   \[\leadsto \color{blue}{\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025142 
(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*))))))