Toniolo and Linder, Equation (13)

Percentage Accurate: 50.2% → 62.8%

Time: 11.6s

Alternatives: 22

Speedup: 1.2×

Specification

Math

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

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: 50.2% accurate, 1.0× speedup

Math

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

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: 62.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\left|\ell\right| \cdot \left|\ell\right|}{Om}\right) - \left(n \cdot {\left(\frac{\left|\ell\right|}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{\left(\frac{\left|\ell\right| \cdot \mathsf{fma}\left(-2, \left|\ell\right|, \frac{\left(n \cdot \left(U* - U\right)\right) \cdot \left|\ell\right|}{Om}\right)}{Om} + t\right) \cdot \left(n + n\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\sqrt{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* (* 2.0 n) U)
          (-
           (- t (* 2.0 (/ (* (fabs l) (fabs l)) Om)))
           (* (* n (pow (/ (fabs l) Om) 2.0)) (- U U*))))))
   (if (<= t_1 5e-300)
     (*
      (sqrt
       (*
        (+
         (/
          (* (fabs l) (fma -2.0 (fabs l) (/ (* (* n (- U* U)) (fabs l)) Om)))
          Om)
         t)
        (+ n n)))
      (sqrt U))
     (if (<= t_1 2e+289)
       (sqrt t_1)
       (*
        (fabs l)
        (sqrt
         (*
          -2.0
          (*
           U
           (* n (fma 2.0 (/ 1.0 Om) (/ (* n (- U U*)) (pow Om 2.0))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((fabs(l) * fabs(l)) / Om))) - ((n * pow((fabs(l) / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 5e-300) {
		tmp = sqrt(((((fabs(l) * fma(-2.0, fabs(l), (((n * (U_42_ - U)) * fabs(l)) / Om))) / Om) + t) * (n + n))) * sqrt(U);
	} else if (t_1 <= 2e+289) {
		tmp = sqrt(t_1);
	} else {
		tmp = fabs(l) * sqrt((-2.0 * (U * (n * fma(2.0, (1.0 / Om), ((n * (U - U_42_)) / pow(Om, 2.0)))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(abs(l) * abs(l)) / Om))) - Float64(Float64(n * (Float64(abs(l) / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_1 <= 5e-300)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(abs(l) * fma(-2.0, abs(l), Float64(Float64(Float64(n * Float64(U_42_ - U)) * abs(l)) / Om))) / Om) + t) * Float64(n + n))) * sqrt(U));
	elseif (t_1 <= 2e+289)
		tmp = sqrt(t_1);
	else
		tmp = Float64(abs(l) * 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

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(N[Abs[l], $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-300], N[(N[Sqrt[N[(N[(N[(N[(N[Abs[l], $MachinePrecision] * N[(-2.0 * N[Abs[l], $MachinePrecision] + N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+289], N[Sqrt[t$95$1], $MachinePrecision], N[(N[Abs[l], $MachinePrecision] * 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*)))) < 4.99999999999999996e-300

   1. Initial program 50.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 51.8%

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

   5. Applied rewrites 50.9%

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

   7. Applied rewrites 29.5%

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

   if 4.99999999999999996e-300 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 2.0000000000000001e289

   1. Initial program 50.2%

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

   if 2.0000000000000001e289 < (*.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 50.2%

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

   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 15.1%

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

   4. Applied rewrites 15.1%

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

Alternative 2: 62.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := \frac{\left|\ell\right|}{Om}\\ t_3 := \left(n \cdot {t\_2}^{2}\right) \cdot \left(U - U*\right)\\ t_4 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{\left|\ell\right| \cdot \left|\ell\right|}{Om}\right) - t\_3\right)\\ \mathbf{if}\;t\_4 \leq 5 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{\left(\frac{\left|\ell\right| \cdot \mathsf{fma}\left(-2, \left|\ell\right|, \frac{\left(n \cdot \left(U* - U\right)\right) \cdot \left|\ell\right|}{Om}\right)}{Om} + t\right) \cdot \left(n + n\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(\mathsf{fma}\left(t\_2, \left|\ell\right| \cdot -2, t\right) - t\_3\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2 (/ (fabs l) Om))
        (t_3 (* (* n (pow t_2 2.0)) (- U U*)))
        (t_4 (* t_1 (- (- t (* 2.0 (/ (* (fabs l) (fabs l)) Om))) t_3))))
   (if (<= t_4 5e-300)
     (*
      (sqrt
       (*
        (+
         (/
          (* (fabs l) (fma -2.0 (fabs l) (/ (* (* n (- U* U)) (fabs l)) Om)))
          Om)
         t)
        (+ n n)))
      (sqrt U))
     (if (<= t_4 2e+289)
       (sqrt (* t_1 (- (fma t_2 (* (fabs l) -2.0) t) t_3)))
       (*
        (fabs l)
        (sqrt
         (*
          -2.0
          (*
           U
           (* n (fma 2.0 (/ 1.0 Om) (/ (* n (- U U*)) (pow Om 2.0))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = fabs(l) / Om;
	double t_3 = (n * pow(t_2, 2.0)) * (U - U_42_);
	double t_4 = t_1 * ((t - (2.0 * ((fabs(l) * fabs(l)) / Om))) - t_3);
	double tmp;
	if (t_4 <= 5e-300) {
		tmp = sqrt(((((fabs(l) * fma(-2.0, fabs(l), (((n * (U_42_ - U)) * fabs(l)) / Om))) / Om) + t) * (n + n))) * sqrt(U);
	} else if (t_4 <= 2e+289) {
		tmp = sqrt((t_1 * (fma(t_2, (fabs(l) * -2.0), t) - t_3)));
	} else {
		tmp = fabs(l) * sqrt((-2.0 * (U * (n * fma(2.0, (1.0 / Om), ((n * (U - U_42_)) / pow(Om, 2.0)))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(abs(l) / Om)
	t_3 = Float64(Float64(n * (t_2 ^ 2.0)) * Float64(U - U_42_))
	t_4 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(abs(l) * abs(l)) / Om))) - t_3))
	tmp = 0.0
	if (t_4 <= 5e-300)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(abs(l) * fma(-2.0, abs(l), Float64(Float64(Float64(n * Float64(U_42_ - U)) * abs(l)) / Om))) / Om) + t) * Float64(n + n))) * sqrt(U));
	elseif (t_4 <= 2e+289)
		tmp = sqrt(Float64(t_1 * Float64(fma(t_2, Float64(abs(l) * -2.0), t) - t_3)));
	else
		tmp = Float64(abs(l) * 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

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[l], $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[(N[(n * N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 5e-300], N[(N[Sqrt[N[(N[(N[(N[(N[Abs[l], $MachinePrecision] * N[(-2.0 * N[Abs[l], $MachinePrecision] + N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e+289], N[Sqrt[N[(t$95$1 * N[(N[(t$95$2 * N[(N[Abs[l], $MachinePrecision] * -2.0), $MachinePrecision] + t), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[l], $MachinePrecision] * 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*)))) < 4.99999999999999996e-300

   1. Initial program 50.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 51.8%

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

   5. Applied rewrites 50.9%

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

   7. Applied rewrites 29.5%

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

   if 4.99999999999999996e-300 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 2.0000000000000001e289

   1. Initial program 50.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 54.0%

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

   3. Applied rewrites 54.0%

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

   if 2.0000000000000001e289 < (*.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 50.2%

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

   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 15.1%

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

   4. Applied rewrites 15.1%

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

Alternative 3: 62.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|\ell\right| \cdot \left|\ell\right|}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \frac{\left|\ell\right|}{Om}\\ t_4 := t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {t\_3}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_4 \leq 5 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{\left(\frac{\left|\ell\right| \cdot \mathsf{fma}\left(-2, \left|\ell\right|, \frac{\left(n \cdot \left(U* - U\right)\right) \cdot \left|\ell\right|}{Om}\right)}{Om} + t\right) \cdot \left(n + n\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(t\_3, \left(t\_3 \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(-2, t\_1, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* (fabs l) (fabs l)) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3 (/ (fabs l) Om))
        (t_4 (* t_2 (- (- t (* 2.0 t_1)) (* (* n (pow t_3 2.0)) (- U U*))))))
   (if (<= t_4 5e-300)
     (*
      (sqrt
       (*
        (+
         (/
          (* (fabs l) (fma -2.0 (fabs l) (/ (* (* n (- U* U)) (fabs l)) Om)))
          Om)
         t)
        (+ n n)))
      (sqrt U))
     (if (<= t_4 2e+289)
       (sqrt (* t_2 (fma t_3 (* (* t_3 n) (- U* U)) (fma -2.0 t_1 t))))
       (*
        (fabs l)
        (sqrt
         (*
          -2.0
          (*
           U
           (* n (fma 2.0 (/ 1.0 Om) (/ (* n (- U U*)) (pow Om 2.0))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (fabs(l) * fabs(l)) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = fabs(l) / Om;
	double t_4 = t_2 * ((t - (2.0 * t_1)) - ((n * pow(t_3, 2.0)) * (U - U_42_)));
	double tmp;
	if (t_4 <= 5e-300) {
		tmp = sqrt(((((fabs(l) * fma(-2.0, fabs(l), (((n * (U_42_ - U)) * fabs(l)) / Om))) / Om) + t) * (n + n))) * sqrt(U);
	} else if (t_4 <= 2e+289) {
		tmp = sqrt((t_2 * fma(t_3, ((t_3 * n) * (U_42_ - U)), fma(-2.0, t_1, t))));
	} else {
		tmp = fabs(l) * sqrt((-2.0 * (U * (n * fma(2.0, (1.0 / Om), ((n * (U - U_42_)) / pow(Om, 2.0)))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(abs(l) * abs(l)) / Om)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(abs(l) / Om)
	t_4 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (t_3 ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_4 <= 5e-300)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(abs(l) * fma(-2.0, abs(l), Float64(Float64(Float64(n * Float64(U_42_ - U)) * abs(l)) / Om))) / Om) + t) * Float64(n + n))) * sqrt(U));
	elseif (t_4 <= 2e+289)
		tmp = sqrt(Float64(t_2 * fma(t_3, Float64(Float64(t_3 * n) * Float64(U_42_ - U)), fma(-2.0, t_1, t))));
	else
		tmp = Float64(abs(l) * 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

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[l], $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 5e-300], N[(N[Sqrt[N[(N[(N[(N[(N[Abs[l], $MachinePrecision] * N[(-2.0 * N[Abs[l], $MachinePrecision] + N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e+289], N[Sqrt[N[(t$95$2 * N[(t$95$3 * N[(N[(t$95$3 * n), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[l], $MachinePrecision] * 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*)))) < 4.99999999999999996e-300

   1. Initial program 50.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 51.8%

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

   5. Applied rewrites 50.9%

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

   7. Applied rewrites 29.5%

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

   if 4.99999999999999996e-300 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 2.0000000000000001e289

   1. Initial program 50.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 51.8%

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

   if 2.0000000000000001e289 < (*.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 50.2%

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

   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 15.1%

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

   4. Applied rewrites 15.1%

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

Alternative 4: 59.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \frac{\ell}{Om} \cdot \ell\\ t_4 := \sqrt{t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ t_5 := n \cdot \left(U* - U\right)\\ \mathbf{if}\;t\_4 \leq 2 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{\left(\frac{\ell \cdot \mathsf{fma}\left(-2, \ell, \frac{t\_5 \cdot \ell}{Om}\right)}{Om} + t\right) \cdot \left(n + n\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(-2, t\_1, t\right)\right)}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(\frac{\left(U* - U\right) \cdot n}{Om}, t\_3, \mathsf{fma}\left(t\_3, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\ell \cdot \mathsf{fma}\left(t\_5, \frac{\ell}{Om}, -2 \cdot \ell\right)}{Om} + t\right) \cdot \left(U \cdot \left(n + n\right)\right)}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3 (* (/ l Om) l))
        (t_4
         (sqrt
          (* t_2 (- (- t (* 2.0 t_1)) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
        (t_5 (* n (- U* U))))
   (if (<= t_4 2e-150)
     (*
      (sqrt (* (+ (/ (* l (fma -2.0 l (/ (* t_5 l) Om))) Om) t) (+ n n)))
      (sqrt U))
     (if (<= t_4 5e+102)
       (sqrt
        (* t_2 (fma (/ l Om) (* (* (/ l Om) n) (- U* U)) (fma -2.0 t_1 t))))
       (if (<= t_4 INFINITY)
         (sqrt (* t_2 (fma (/ (* (- U* U) n) Om) t_3 (fma t_3 -2.0 t))))
         (sqrt
          (*
           (+ (/ (* l (fma t_5 (/ l Om) (* -2.0 l))) Om) t)
           (* U (+ n n)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l * l) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = (l / Om) * l;
	double t_4 = sqrt((t_2 * ((t - (2.0 * t_1)) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double t_5 = n * (U_42_ - U);
	double tmp;
	if (t_4 <= 2e-150) {
		tmp = sqrt(((((l * fma(-2.0, l, ((t_5 * l) / Om))) / Om) + t) * (n + n))) * sqrt(U);
	} else if (t_4 <= 5e+102) {
		tmp = sqrt((t_2 * fma((l / Om), (((l / Om) * n) * (U_42_ - U)), fma(-2.0, t_1, t))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * fma((((U_42_ - U) * n) / Om), t_3, fma(t_3, -2.0, t))));
	} else {
		tmp = sqrt(((((l * fma(t_5, (l / Om), (-2.0 * l))) / Om) + t) * (U * (n + n))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * l) / Om)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(Float64(l / Om) * l)
	t_4 = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
	t_5 = Float64(n * Float64(U_42_ - U))
	tmp = 0.0
	if (t_4 <= 2e-150)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(l * fma(-2.0, l, Float64(Float64(t_5 * l) / Om))) / Om) + t) * Float64(n + n))) * sqrt(U));
	elseif (t_4 <= 5e+102)
		tmp = sqrt(Float64(t_2 * fma(Float64(l / Om), Float64(Float64(Float64(l / Om) * n) * Float64(U_42_ - U)), fma(-2.0, t_1, t))));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(t_2 * fma(Float64(Float64(Float64(U_42_ - U) * n) / Om), t_3, fma(t_3, -2.0, t))));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(l * fma(t_5, Float64(l / Om), Float64(-2.0 * l))) / Om) + t) * Float64(U * Float64(n + n))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 2e-150], N[(N[Sqrt[N[(N[(N[(N[(l * N[(-2.0 * l + N[(N[(t$95$5 * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 5e+102], N[Sqrt[N[(t$95$2 * N[(N[(l / Om), $MachinePrecision] * N[(N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(t$95$2 * N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] * t$95$3 + N[(t$95$3 * -2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(l * N[(t$95$5 * N[(l / Om), $MachinePrecision] + N[(-2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2.00000000000000001e-150

   1. Initial program 50.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 51.8%

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

   5. Applied rewrites 50.9%

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

   7. Applied rewrites 29.5%

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

   if 2.00000000000000001e-150 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5e102

   1. Initial program 50.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 51.8%

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

   if 5e102 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 50.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 54.0%

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

   3. Applied rewrites 54.0%

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

   5. Applied rewrites 49.8%

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

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

   1. Initial program 50.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 51.8%

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

   5. Applied rewrites 50.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 51.8%

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

   7. Applied rewrites 51.8%

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

Alternative 5: 58.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \sqrt{t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ t_4 := n \cdot \left(U* - U\right)\\ \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{\left(\frac{\ell \cdot \mathsf{fma}\left(-2, \ell, \frac{t\_4 \cdot \ell}{Om}\right)}{Om} + t\right) \cdot \left(n + n\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(-2, t\_1, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\ell \cdot \mathsf{fma}\left(t\_4, \frac{\ell}{Om}, -2 \cdot \ell\right)}{Om} + t\right) \cdot \left(U \cdot \left(n + n\right)\right)}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (sqrt
          (* t_2 (- (- t (* 2.0 t_1)) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
        (t_4 (* n (- U* U))))
   (if (<= t_3 2e-150)
     (*
      (sqrt (* (+ (/ (* l (fma -2.0 l (/ (* t_4 l) Om))) Om) t) (+ n n)))
      (sqrt U))
     (if (<= t_3 INFINITY)
       (sqrt
        (* t_2 (fma (/ l Om) (* (* (/ l Om) n) (- U* U)) (fma -2.0 t_1 t))))
       (sqrt
        (* (+ (/ (* l (fma t_4 (/ l Om) (* -2.0 l))) Om) t) (* U (+ n n))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l * l) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = sqrt((t_2 * ((t - (2.0 * t_1)) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double t_4 = n * (U_42_ - U);
	double tmp;
	if (t_3 <= 2e-150) {
		tmp = sqrt(((((l * fma(-2.0, l, ((t_4 * l) / Om))) / Om) + t) * (n + n))) * sqrt(U);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * fma((l / Om), (((l / Om) * n) * (U_42_ - U)), fma(-2.0, t_1, t))));
	} else {
		tmp = sqrt(((((l * fma(t_4, (l / Om), (-2.0 * l))) / Om) + t) * (U * (n + n))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * l) / Om)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
	t_4 = Float64(n * Float64(U_42_ - U))
	tmp = 0.0
	if (t_3 <= 2e-150)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(l * fma(-2.0, l, Float64(Float64(t_4 * l) / Om))) / Om) + t) * Float64(n + n))) * sqrt(U));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * fma(Float64(l / Om), Float64(Float64(Float64(l / Om) * n) * Float64(U_42_ - U)), fma(-2.0, t_1, t))));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(l * fma(t_4, Float64(l / Om), Float64(-2.0 * l))) / Om) + t) * Float64(U * Float64(n + n))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 2e-150], N[(N[Sqrt[N[(N[(N[(N[(l * N[(-2.0 * l + N[(N[(t$95$4 * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * N[(N[(l / Om), $MachinePrecision] * N[(N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(l * N[(t$95$4 * N[(l / Om), $MachinePrecision] + N[(-2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2.00000000000000001e-150

   1. Initial program 50.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 51.8%

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

   5. Applied rewrites 50.9%

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

   7. Applied rewrites 29.5%

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

   if 2.00000000000000001e-150 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 50.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 51.8%

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

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

   1. Initial program 50.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 51.8%

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

   5. Applied rewrites 50.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 51.8%

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

   7. Applied rewrites 51.8%

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

Alternative 6: 58.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := U \cdot \left(n + n\right)\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ t_3 := n \cdot \left(U* - U\right)\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{\left(\frac{\ell \cdot \mathsf{fma}\left(-2, \ell, \frac{t\_3 \cdot \ell}{Om}\right)}{Om} + t\right) \cdot \left(n + n\right)} \cdot \sqrt{U}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(\frac{\mathsf{fma}\left(\ell \cdot U*, \frac{\ell}{Om} \cdot n, \left(\ell \cdot \ell\right) \cdot -2\right)}{Om} + t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\ell \cdot \mathsf{fma}\left(t\_3, \frac{\ell}{Om}, -2 \cdot \ell\right)}{Om} + t\right) \cdot t\_1}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* U (+ n n)))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
        (t_3 (* n (- U* U))))
   (if (<= t_2 2e-150)
     (*
      (sqrt (* (+ (/ (* l (fma -2.0 l (/ (* t_3 l) Om))) Om) t) (+ n n)))
      (sqrt U))
     (if (<= t_2 INFINITY)
       (sqrt
        (* (+ (/ (fma (* l U*) (* (/ l Om) n) (* (* l l) -2.0)) Om) t) t_1))
       (sqrt (* (+ (/ (* l (fma t_3 (/ l Om) (* -2.0 l))) Om) t) t_1))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = U * (n + n);
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double t_3 = n * (U_42_ - U);
	double tmp;
	if (t_2 <= 2e-150) {
		tmp = sqrt(((((l * fma(-2.0, l, ((t_3 * l) / Om))) / Om) + t) * (n + n))) * sqrt(U);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((((fma((l * U_42_), ((l / Om) * n), ((l * l) * -2.0)) / Om) + t) * t_1));
	} else {
		tmp = sqrt(((((l * fma(t_3, (l / Om), (-2.0 * l))) / Om) + t) * t_1));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(U * Float64(n + n))
	t_2 = 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_)))))
	t_3 = Float64(n * Float64(U_42_ - U))
	tmp = 0.0
	if (t_2 <= 2e-150)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(l * fma(-2.0, l, Float64(Float64(t_3 * l) / Om))) / Om) + t) * Float64(n + n))) * sqrt(U));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(Float64(Float64(fma(Float64(l * U_42_), Float64(Float64(l / Om) * n), Float64(Float64(l * l) * -2.0)) / Om) + t) * t_1));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(l * fma(t_3, Float64(l / Om), Float64(-2.0 * l))) / Om) + t) * t_1));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * 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]}, Block[{t$95$3 = N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 2e-150], N[(N[Sqrt[N[(N[(N[(N[(l * N[(-2.0 * l + N[(N[(t$95$3 * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(N[(N[(N[(N[(l * U$42$), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(l * N[(t$95$3 * N[(l / Om), $MachinePrecision] + N[(-2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2.00000000000000001e-150

   1. Initial program 50.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 51.8%

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

   5. Applied rewrites 50.9%

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

   7. Applied rewrites 29.5%

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

   if 2.00000000000000001e-150 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 50.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 51.8%

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in U around 0

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

   8. Applied rewrites 51.5%

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

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

   1. Initial program 50.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 51.8%

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

   5. Applied rewrites 50.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 51.8%

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

   7. Applied rewrites 51.8%

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

Alternative 7: 57.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{\ell}{Om} \cdot n\\ t_2 := \left(\ell \cdot \ell\right) \cdot -2\\ \mathbf{if}\;U \leq 2.15 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{\left(\frac{\mathsf{fma}\left(\ell \cdot U*, t\_1, t\_2\right)}{Om} + t\right) \cdot \left(U \cdot \left(n + n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\mathsf{fma}\left(\ell \cdot \left(U* - U\right), t\_1, t\_2\right)}{Om} + t\right) \cdot \left(n + n\right)} \cdot \sqrt{U}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (/ l Om) n)) (t_2 (* (* l l) -2.0)))
   (if (<= U 2.15e-287)
     (sqrt (* (+ (/ (fma (* l U*) t_1 t_2) Om) t) (* U (+ n n))))
     (*
      (sqrt (* (+ (/ (fma (* l (- U* U)) t_1 t_2) Om) t) (+ n n)))
      (sqrt U)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l / Om) * n;
	double t_2 = (l * l) * -2.0;
	double tmp;
	if (U <= 2.15e-287) {
		tmp = sqrt((((fma((l * U_42_), t_1, t_2) / Om) + t) * (U * (n + n))));
	} else {
		tmp = sqrt((((fma((l * (U_42_ - U)), t_1, t_2) / Om) + t) * (n + n))) * sqrt(U);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l / Om) * n)
	t_2 = Float64(Float64(l * l) * -2.0)
	tmp = 0.0
	if (U <= 2.15e-287)
		tmp = sqrt(Float64(Float64(Float64(fma(Float64(l * U_42_), t_1, t_2) / Om) + t) * Float64(U * Float64(n + n))));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(fma(Float64(l * Float64(U_42_ - U)), t_1, t_2) / Om) + t) * Float64(n + n))) * sqrt(U));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] * -2.0), $MachinePrecision]}, If[LessEqual[U, 2.15e-287], N[Sqrt[N[(N[(N[(N[(N[(l * U$42$), $MachinePrecision] * t$95$1 + t$95$2), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(N[(l * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] * t$95$1 + t$95$2), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if U < 2.14999999999999995e-287

   1. Initial program 50.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 51.8%

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in U around 0

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

   8. Applied rewrites 51.5%

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

   if 2.14999999999999995e-287 < U

   1. Initial program 50.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 51.8%

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

   5. Applied rewrites 30.0%

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

Alternative 8: 57.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 50.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 51.8%

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

   5. Applied rewrites 50.9%

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

   7. Applied rewrites 28.8%

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

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

   1. Initial program 50.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 51.8%

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in U around 0

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

   8. Applied rewrites 51.5%

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

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

   1. Initial program 50.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 51.8%

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

   5. Applied rewrites 50.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 51.8%

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

   7. Applied rewrites 51.8%

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

Alternative 9: 55.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 50.2%

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

   3. Applied rewrites 23.7%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 25.6%

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

   6. Applied rewrites 25.6%

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

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

   1. Initial program 50.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 51.8%

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in U around 0

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

   8. Applied rewrites 51.5%

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

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

   1. Initial program 50.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 51.8%

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

   5. Applied rewrites 50.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 51.8%

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

   7. Applied rewrites 51.8%

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

Alternative 10: 53.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|\ell\right|}{Om}\\ t_2 := U \cdot \left(n + n\right)\\ \mathbf{if}\;\left|\ell\right| \leq 1.2 \cdot 10^{-96}:\\ \;\;\;\;\sqrt{\left(\frac{\mathsf{fma}\left(\left|\ell\right| \cdot U*, t\_1 \cdot n, \left(\left|\ell\right| \cdot \left|\ell\right|\right) \cdot -2\right)}{Om} + t\right) \cdot t\_2}\\ \mathbf{elif}\;\left|\ell\right| \leq 3.3 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\left(\left|\ell\right|\right)}^{2}}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\left|\ell\right| \cdot \mathsf{fma}\left(n \cdot \left(U* - U\right), t\_1, -2 \cdot \left|\ell\right|\right)}{Om} + t\right) \cdot t\_2}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (fabs l) Om)) (t_2 (* U (+ n n))))
   (if (<= (fabs l) 1.2e-96)
     (sqrt
      (*
       (+
        (/ (fma (* (fabs l) U*) (* t_1 n) (* (* (fabs l) (fabs l)) -2.0)) Om)
        t)
       t_2))
     (if (<= (fabs l) 3.3e-58)
       (sqrt (* 2.0 (* U (* n (- t (* 2.0 (/ (pow (fabs l) 2.0) Om)))))))
       (sqrt
        (*
         (+ (/ (* (fabs l) (fma (* n (- U* U)) t_1 (* -2.0 (fabs l)))) Om) t)
         t_2))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = fabs(l) / Om;
	double t_2 = U * (n + n);
	double tmp;
	if (fabs(l) <= 1.2e-96) {
		tmp = sqrt((((fma((fabs(l) * U_42_), (t_1 * n), ((fabs(l) * fabs(l)) * -2.0)) / Om) + t) * t_2));
	} else if (fabs(l) <= 3.3e-58) {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (pow(fabs(l), 2.0) / Om)))))));
	} else {
		tmp = sqrt(((((fabs(l) * fma((n * (U_42_ - U)), t_1, (-2.0 * fabs(l)))) / Om) + t) * t_2));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(abs(l) / Om)
	t_2 = Float64(U * Float64(n + n))
	tmp = 0.0
	if (abs(l) <= 1.2e-96)
		tmp = sqrt(Float64(Float64(Float64(fma(Float64(abs(l) * U_42_), Float64(t_1 * n), Float64(Float64(abs(l) * abs(l)) * -2.0)) / Om) + t) * t_2));
	elseif (abs(l) <= 3.3e-58)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64((abs(l) ^ 2.0) / Om)))))));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(abs(l) * fma(Float64(n * Float64(U_42_ - U)), t_1, Float64(-2.0 * abs(l)))) / Om) + t) * t_2));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[Abs[l], $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[l], $MachinePrecision], 1.2e-96], N[Sqrt[N[(N[(N[(N[(N[(N[Abs[l], $MachinePrecision] * U$42$), $MachinePrecision] * N[(t$95$1 * n), $MachinePrecision] + N[(N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Abs[l], $MachinePrecision], 3.3e-58], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(N[Power[N[Abs[l], $MachinePrecision], 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[Abs[l], $MachinePrecision] * N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(-2.0 * N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.2000000000000001e-96

   1. Initial program 50.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 51.8%

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in U around 0

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

   8. Applied rewrites 51.5%

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

   if 1.2000000000000001e-96 < l < 3.30000000000000026e-58

   1. Initial program 50.2%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 45.0%

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

   4. Applied rewrites 45.0%

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

   if 3.30000000000000026e-58 < l

   1. Initial program 50.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 51.8%

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

   5. Applied rewrites 50.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 51.8%

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

   7. Applied rewrites 51.8%

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

Alternative 11: 50.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 3.3 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\left(\left|\ell\right|\right)}^{2}}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\left|\ell\right| \cdot \mathsf{fma}\left(n \cdot \left(U* - U\right), \frac{\left|\ell\right|}{Om}, -2 \cdot \left|\ell\right|\right)}{Om} + t\right) \cdot \left(U \cdot \left(n + n\right)\right)}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= (fabs l) 3.3e-58)
   (sqrt (* 2.0 (* U (* n (- t (* 2.0 (/ (pow (fabs l) 2.0) Om)))))))
   (sqrt
    (*
     (+
      (/
       (* (fabs l) (fma (* n (- U* U)) (/ (fabs l) Om) (* -2.0 (fabs l))))
       Om)
      t)
     (* U (+ n n))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (fabs(l) <= 3.3e-58) {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (pow(fabs(l), 2.0) / Om)))))));
	} else {
		tmp = sqrt(((((fabs(l) * fma((n * (U_42_ - U)), (fabs(l) / Om), (-2.0 * fabs(l)))) / Om) + t) * (U * (n + n))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (abs(l) <= 3.3e-58)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64((abs(l) ^ 2.0) / Om)))))));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(abs(l) * fma(Float64(n * Float64(U_42_ - U)), Float64(abs(l) / Om), Float64(-2.0 * abs(l)))) / Om) + t) * Float64(U * Float64(n + n))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[N[Abs[l], $MachinePrecision], 3.3e-58], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(N[Power[N[Abs[l], $MachinePrecision], 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[Abs[l], $MachinePrecision] * N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[l], $MachinePrecision] / Om), $MachinePrecision] + N[(-2.0 * N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.30000000000000026e-58

   1. Initial program 50.2%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 45.0%

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

   4. Applied rewrites 45.0%

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

   if 3.30000000000000026e-58 < l

   1. Initial program 50.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 51.8%

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

   5. Applied rewrites 50.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 51.8%

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

   7. Applied rewrites 51.8%

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

Alternative 12: 50.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 3.3 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\left(\left|\ell\right|\right)}^{2}}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\left|\ell\right| \cdot \mathsf{fma}\left(-2, \left|\ell\right|, \frac{\left(n \cdot \left(U* - U\right)\right) \cdot \left|\ell\right|}{Om}\right)}{Om} + t\right) \cdot \left(\left(n + n\right) \cdot U\right)}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= (fabs l) 3.3e-58)
   (sqrt (* 2.0 (* U (* n (- t (* 2.0 (/ (pow (fabs l) 2.0) Om)))))))
   (sqrt
    (*
     (+
      (/
       (* (fabs l) (fma -2.0 (fabs l) (/ (* (* n (- U* U)) (fabs l)) Om)))
       Om)
      t)
     (* (+ n n) U)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (fabs(l) <= 3.3e-58) {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * (pow(fabs(l), 2.0) / Om)))))));
	} else {
		tmp = sqrt(((((fabs(l) * fma(-2.0, fabs(l), (((n * (U_42_ - U)) * fabs(l)) / Om))) / Om) + t) * ((n + n) * U)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (abs(l) <= 3.3e-58)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64((abs(l) ^ 2.0) / Om)))))));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(abs(l) * fma(-2.0, abs(l), Float64(Float64(Float64(n * Float64(U_42_ - U)) * abs(l)) / Om))) / Om) + t) * Float64(Float64(n + n) * U)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[N[Abs[l], $MachinePrecision], 3.3e-58], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(N[Power[N[Abs[l], $MachinePrecision], 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[Abs[l], $MachinePrecision] * N[(-2.0 * N[Abs[l], $MachinePrecision] + N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.30000000000000026e-58

   1. Initial program 50.2%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 45.0%

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

   4. Applied rewrites 45.0%

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

   if 3.30000000000000026e-58 < l

   1. Initial program 50.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 51.8%

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

   5. Applied rewrites 50.9%

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

   7. Applied rewrites 51.4%

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

Alternative 13: 48.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n} \cdot \frac{-1 \cdot \left(\ell \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(U* - U\right)\right)\right)}\right)}{Om}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<=
      (*
       (* (* 2.0 n) U)
       (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))
      INFINITY)
   (sqrt (* 2.0 (* U (* n (- t (* 2.0 (/ (pow l 2.0) Om)))))))
   (* (sqrt n) (/ (* -1.0 (* l (sqrt (* 2.0 (* U (* n (- U* U))))))) Om))))

C

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

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))) <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * (Math.pow(l, 2.0) / Om)))))));
	} else {
		tmp = Math.sqrt(n) * ((-1.0 * (l * Math.sqrt((2.0 * (U * (n * (U_42_ - U))))))) / Om);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))) <= math.inf:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * (math.pow(l, 2.0) / Om)))))))
	else:
		tmp = math.sqrt(n) * ((-1.0 * (l * math.sqrt((2.0 * (U * (n * (U_42_ - U))))))) / Om)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (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_)))) <= Inf)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64((l ^ 2.0) / Om)))))));
	else
		tmp = Float64(sqrt(n) * Float64(Float64(-1.0 * Float64(l * sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(U_42_ - U))))))) / Om));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)))) <= Inf)
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * ((l ^ 2.0) / Om)))))));
	else
		tmp = sqrt(n) * ((-1.0 * (l * sqrt((2.0 * (U * (n * (U_42_ - U))))))) / Om);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[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], Infinity], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[n], $MachinePrecision] * N[(N[(-1.0 * N[(l * N[Sqrt[N[(2.0 * N[(U * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $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*)))) < +inf.0

   1. Initial program 50.2%

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

   2. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 45.0%

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

   4. Applied rewrites 45.0%

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

   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 50.2%

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

   3. Applied rewrites 23.7%

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

   4. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 5.5%

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

   6. Applied rewrites 5.5%

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

   7. Taylor expanded in l around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 6.7%

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

   9. Applied rewrites 6.7%

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

Alternative 14: 47.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{\left(\frac{-2 \cdot {\ell}^{2}}{Om} + t\right) \cdot \left(U \cdot \left(n + n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n} \cdot \frac{-1 \cdot \left(\ell \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(U* - U\right)\right)\right)}\right)}{Om}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* (* 2.0 n) U)
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_1 5e-300)
     (sqrt (* (+ n n) (* U (+ t (* -2.0 (/ (pow l 2.0) Om))))))
     (if (<= t_1 INFINITY)
       (sqrt (* (+ (/ (* -2.0 (pow l 2.0)) Om) t) (* U (+ n n))))
       (*
        (sqrt n)
        (/ (* -1.0 (* l (sqrt (* 2.0 (* U (* n (- U* U))))))) Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 5e-300) {
		tmp = sqrt(((n + n) * (U * (t + (-2.0 * (pow(l, 2.0) / Om))))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt(((((-2.0 * pow(l, 2.0)) / Om) + t) * (U * (n + n))));
	} else {
		tmp = sqrt(n) * ((-1.0 * (l * sqrt((2.0 * (U * (n * (U_42_ - U))))))) / Om);
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 5e-300) {
		tmp = Math.sqrt(((n + n) * (U * (t + (-2.0 * (Math.pow(l, 2.0) / Om))))));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((((-2.0 * Math.pow(l, 2.0)) / Om) + t) * (U * (n + n))));
	} else {
		tmp = Math.sqrt(n) * ((-1.0 * (l * Math.sqrt((2.0 * (U * (n * (U_42_ - U))))))) / Om);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))
	tmp = 0
	if t_1 <= 5e-300:
		tmp = math.sqrt(((n + n) * (U * (t + (-2.0 * (math.pow(l, 2.0) / Om))))))
	elif t_1 <= math.inf:
		tmp = math.sqrt(((((-2.0 * math.pow(l, 2.0)) / Om) + t) * (U * (n + n))))
	else:
		tmp = math.sqrt(n) * ((-1.0 * (l * math.sqrt((2.0 * (U * (n * (U_42_ - U))))))) / Om)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = 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_))))
	tmp = 0.0
	if (t_1 <= 5e-300)
		tmp = sqrt(Float64(Float64(n + n) * Float64(U * Float64(t + Float64(-2.0 * Float64((l ^ 2.0) / Om))))));
	elseif (t_1 <= Inf)
		tmp = sqrt(Float64(Float64(Float64(Float64(-2.0 * (l ^ 2.0)) / Om) + t) * Float64(U * Float64(n + n))));
	else
		tmp = Float64(sqrt(n) * Float64(Float64(-1.0 * Float64(l * sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(U_42_ - U))))))) / Om));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)));
	tmp = 0.0;
	if (t_1 <= 5e-300)
		tmp = sqrt(((n + n) * (U * (t + (-2.0 * ((l ^ 2.0) / Om))))));
	elseif (t_1 <= Inf)
		tmp = sqrt(((((-2.0 * (l ^ 2.0)) / Om) + t) * (U * (n + n))));
	else
		tmp = sqrt(n) * ((-1.0 * (l * sqrt((2.0 * (U * (n * (U_42_ - U))))))) / Om);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, 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 * 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]}, If[LessEqual[t$95$1, 5e-300], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(U * N[(t + N[(-2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[Sqrt[N[(N[(N[(N[(-2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[n], $MachinePrecision] * N[(N[(-1.0 * N[(l * N[Sqrt[N[(2.0 * N[(U * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.99999999999999996e-300

   1. Initial program 50.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   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 50.2%

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

   3. Applied rewrites 41.6%

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

   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%

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

   if 4.99999999999999996e-300 < (*.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 50.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right)}\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      8. sub-negate-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \color{blue}{\left(U* - U\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} \cdot \left(U* - U\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U* - U\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      11. lift-pow.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}} \cdot n\right) \cdot \left(U* - U\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      12. unpow2 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot n\right) \cdot \left(U* - U\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      13. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \cdot \left(U* - U\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      14. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]

   3. Applied rewrites 51.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)}} \]

   5. Applied rewrites 50.9%

      \[\leadsto \color{blue}{\sqrt{\left(\frac{\mathsf{fma}\left(\ell \cdot \left(U* - U\right), \frac{\ell}{Om} \cdot n, \left(\ell \cdot \ell\right) \cdot -2\right)}{Om} + t\right) \cdot \left(U \cdot \left(n + n\right)\right)}} \]

   6. Taylor expanded in n around 0

      \[\leadsto \sqrt{\left(\frac{\color{blue}{-2 \cdot {\ell}^{2}}}{Om} + t\right) \cdot \left(U \cdot \left(n + n\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{-2 \cdot \color{blue}{{\ell}^{2}}}{Om} + t\right) \cdot \left(U \cdot \left(n + n\right)\right)} \]

      2. lower-pow.f64 43.9%

         \[\leadsto \sqrt{\left(\frac{-2 \cdot {\ell}^{\color{blue}{2}}}{Om} + t\right) \cdot \left(U \cdot \left(n + n\right)\right)} \]

   8. Applied rewrites 43.9%

      \[\leadsto \sqrt{\left(\frac{\color{blue}{-2 \cdot {\ell}^{2}}}{Om} + t\right) \cdot \left(U \cdot \left(n + n\right)\right)} \]

   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 50.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 23.7%

      \[\leadsto \color{blue}{\sqrt{n} \cdot \sqrt{\left(U + U\right) \cdot \mathsf{fma}\left(\left(U* - U\right) \cdot n, \ell \cdot \frac{\ell}{Om \cdot Om}, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)}} \]

   4. Taylor expanded in Om around 0

      \[\leadsto \sqrt{n} \cdot \color{blue}{\frac{\sqrt{2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}}{Om}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \sqrt{n} \cdot \frac{\sqrt{2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}}{\color{blue}{Om}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{n} \cdot \frac{\sqrt{2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}}{Om} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \frac{\sqrt{2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}}{Om} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \frac{\sqrt{2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}}{Om} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \frac{\sqrt{2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}}{Om} \]

      6. lower-pow.f64 N/A

         \[\leadsto \sqrt{n} \cdot \frac{\sqrt{2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}}{Om} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \frac{\sqrt{2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}}{Om} \]

      8. lower--.f64 5.5%

         \[\leadsto \sqrt{n} \cdot \frac{\sqrt{2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}}{Om} \]

   6. Applied rewrites 5.5%

      \[\leadsto \sqrt{n} \cdot \color{blue}{\frac{\sqrt{2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}}{Om}} \]

   7. Taylor expanded in l around -inf

      \[\leadsto \sqrt{n} \cdot \frac{-1 \cdot \left(\ell \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(U* - U\right)\right)\right)}\right)}{Om} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \frac{-1 \cdot \left(\ell \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(U* - U\right)\right)\right)}\right)}{Om} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \frac{-1 \cdot \left(\ell \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(U* - U\right)\right)\right)}\right)}{Om} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{n} \cdot \frac{-1 \cdot \left(\ell \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(U* - U\right)\right)\right)}\right)}{Om} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \frac{-1 \cdot \left(\ell \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(U* - U\right)\right)\right)}\right)}{Om} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \frac{-1 \cdot \left(\ell \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(U* - U\right)\right)\right)}\right)}{Om} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \frac{-1 \cdot \left(\ell \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(U* - U\right)\right)\right)}\right)}{Om} \]

      7. lower--.f64 6.7%

         \[\leadsto \sqrt{n} \cdot \frac{-1 \cdot \left(\ell \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(U* - U\right)\right)\right)}\right)}{Om} \]

   9. Applied rewrites 6.7%

      \[\leadsto \sqrt{n} \cdot \frac{-1 \cdot \left(\ell \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(U* - U\right)\right)\right)}\right)}{Om} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 45.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{\left(U + U\right) \cdot \left(t \cdot n\right)}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{\left(\frac{-2 \cdot {\ell}^{2}}{Om} + t\right) \cdot \left(U \cdot \left(n + n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n} \cdot \frac{-1 \cdot \left(\ell \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(U* - U\right)\right)\right)}\right)}{Om}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* (* 2.0 n) U)
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_1 0.0)
     (sqrt (* (+ U U) (* t n)))
     (if (<= t_1 INFINITY)
       (sqrt (* (+ (/ (* -2.0 (pow l 2.0)) Om) t) (* U (+ n n))))
       (*
        (sqrt n)
        (/ (* -1.0 (* l (sqrt (* 2.0 (* U (* n (- U* U))))))) Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt(((U + U) * (t * n)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt(((((-2.0 * pow(l, 2.0)) / Om) + t) * (U * (n + n))));
	} else {
		tmp = sqrt(n) * ((-1.0 * (l * sqrt((2.0 * (U * (n * (U_42_ - U))))))) / Om);
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.sqrt(((U + U) * (t * n)));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((((-2.0 * Math.pow(l, 2.0)) / Om) + t) * (U * (n + n))));
	} else {
		tmp = Math.sqrt(n) * ((-1.0 * (l * Math.sqrt((2.0 * (U * (n * (U_42_ - U))))))) / Om);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.sqrt(((U + U) * (t * n)))
	elif t_1 <= math.inf:
		tmp = math.sqrt(((((-2.0 * math.pow(l, 2.0)) / Om) + t) * (U * (n + n))))
	else:
		tmp = math.sqrt(n) * ((-1.0 * (l * math.sqrt((2.0 * (U * (n * (U_42_ - U))))))) / Om)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = 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_))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = sqrt(Float64(Float64(U + U) * Float64(t * n)));
	elseif (t_1 <= Inf)
		tmp = sqrt(Float64(Float64(Float64(Float64(-2.0 * (l ^ 2.0)) / Om) + t) * Float64(U * Float64(n + n))));
	else
		tmp = Float64(sqrt(n) * Float64(Float64(-1.0 * Float64(l * sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(U_42_ - U))))))) / Om));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = sqrt(((U + U) * (t * n)));
	elseif (t_1 <= Inf)
		tmp = sqrt(((((-2.0 * (l ^ 2.0)) / Om) + t) * (U * (n + n))));
	else
		tmp = sqrt(n) * ((-1.0 * (l * sqrt((2.0 * (U * (n * (U_42_ - U))))))) / Om);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, 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 * 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]}, If[LessEqual[t$95$1, 0.0], N[Sqrt[N[(N[(U + U), $MachinePrecision] * N[(t * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[Sqrt[N[(N[(N[(N[(-2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[n], $MachinePrecision] * N[(N[(-1.0 * N[(l * N[Sqrt[N[(2.0 * N[(U * N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $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 50.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   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 36.6%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 36.6%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      4. count-2 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      6. lower-*.f64 36.6%

         \[\leadsto \sqrt{\left(U + U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(n \cdot \color{blue}{t}\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(t \cdot \color{blue}{n}\right)} \]

      9. lower-*.f64 36.6%

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(t \cdot \color{blue}{n}\right)} \]

   6. Applied rewrites 36.6%

      \[\leadsto \sqrt{\left(U + U\right) \cdot \color{blue}{\left(t \cdot n\right)}} \]

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 50.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right) \cdot \left(U - U*\right)\right)}} \]

      4. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right) \cdot \left(U - U*\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right)}\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      8. sub-negate-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \color{blue}{\left(U* - U\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} \cdot \left(U* - U\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U* - U\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      11. lift-pow.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}} \cdot n\right) \cdot \left(U* - U\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      12. unpow2 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot n\right) \cdot \left(U* - U\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      13. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \cdot \left(U* - U\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      14. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]

   3. Applied rewrites 51.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)}} \]

   5. Applied rewrites 50.9%

      \[\leadsto \color{blue}{\sqrt{\left(\frac{\mathsf{fma}\left(\ell \cdot \left(U* - U\right), \frac{\ell}{Om} \cdot n, \left(\ell \cdot \ell\right) \cdot -2\right)}{Om} + t\right) \cdot \left(U \cdot \left(n + n\right)\right)}} \]

   6. Taylor expanded in n around 0

      \[\leadsto \sqrt{\left(\frac{\color{blue}{-2 \cdot {\ell}^{2}}}{Om} + t\right) \cdot \left(U \cdot \left(n + n\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{-2 \cdot \color{blue}{{\ell}^{2}}}{Om} + t\right) \cdot \left(U \cdot \left(n + n\right)\right)} \]

      2. lower-pow.f64 43.9%

         \[\leadsto \sqrt{\left(\frac{-2 \cdot {\ell}^{\color{blue}{2}}}{Om} + t\right) \cdot \left(U \cdot \left(n + n\right)\right)} \]

   8. Applied rewrites 43.9%

      \[\leadsto \sqrt{\left(\frac{\color{blue}{-2 \cdot {\ell}^{2}}}{Om} + t\right) \cdot \left(U \cdot \left(n + n\right)\right)} \]

   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 50.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 23.7%

      \[\leadsto \color{blue}{\sqrt{n} \cdot \sqrt{\left(U + U\right) \cdot \mathsf{fma}\left(\left(U* - U\right) \cdot n, \ell \cdot \frac{\ell}{Om \cdot Om}, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)}} \]

   4. Taylor expanded in Om around 0

      \[\leadsto \sqrt{n} \cdot \color{blue}{\frac{\sqrt{2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}}{Om}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \sqrt{n} \cdot \frac{\sqrt{2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}}{\color{blue}{Om}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{n} \cdot \frac{\sqrt{2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}}{Om} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \frac{\sqrt{2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}}{Om} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \frac{\sqrt{2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}}{Om} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \frac{\sqrt{2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}}{Om} \]

      6. lower-pow.f64 N/A

         \[\leadsto \sqrt{n} \cdot \frac{\sqrt{2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}}{Om} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \frac{\sqrt{2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}}{Om} \]

      8. lower--.f64 5.5%

         \[\leadsto \sqrt{n} \cdot \frac{\sqrt{2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}}{Om} \]

   6. Applied rewrites 5.5%

      \[\leadsto \sqrt{n} \cdot \color{blue}{\frac{\sqrt{2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right)}}{Om}} \]

   7. Taylor expanded in l around -inf

      \[\leadsto \sqrt{n} \cdot \frac{-1 \cdot \left(\ell \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(U* - U\right)\right)\right)}\right)}{Om} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \frac{-1 \cdot \left(\ell \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(U* - U\right)\right)\right)}\right)}{Om} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \frac{-1 \cdot \left(\ell \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(U* - U\right)\right)\right)}\right)}{Om} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{n} \cdot \frac{-1 \cdot \left(\ell \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(U* - U\right)\right)\right)}\right)}{Om} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \frac{-1 \cdot \left(\ell \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(U* - U\right)\right)\right)}\right)}{Om} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \frac{-1 \cdot \left(\ell \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(U* - U\right)\right)\right)}\right)}{Om} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \frac{-1 \cdot \left(\ell \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(U* - U\right)\right)\right)}\right)}{Om} \]

      7. lower--.f64 6.7%

         \[\leadsto \sqrt{n} \cdot \frac{-1 \cdot \left(\ell \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(U* - U\right)\right)\right)}\right)}{Om} \]

   9. Applied rewrites 6.7%

      \[\leadsto \sqrt{n} \cdot \frac{-1 \cdot \left(\ell \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(U* - U\right)\right)\right)}\right)}{Om} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 41.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ t_2 := \left(\left(U + U\right) \cdot t\right) \cdot n\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{\left(U + U\right) \cdot \left(t \cdot n\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\sqrt{\left|t \cdot \left(U \cdot \left(n + n\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{t\_2 \cdot t\_2}}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* (* 2.0 n) U)
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*)))))
        (t_2 (* (* (+ U U) t) n)))
   (if (<= t_1 0.0)
     (sqrt (* (+ U U) (* t n)))
     (if (<= t_1 2e+289)
       (sqrt (fabs (* t (* U (+ n n)))))
       (sqrt (sqrt (* t_2 t_2)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double t_2 = ((U + U) * t) * n;
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt(((U + U) * (t * n)));
	} else if (t_1 <= 2e+289) {
		tmp = sqrt(fabs((t * (U * (n + n)))));
	} else {
		tmp = sqrt(sqrt((t_2 * t_2)));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))
    t_2 = ((u + u) * t) * n
    if (t_1 <= 0.0d0) then
        tmp = sqrt(((u + u) * (t * n)))
    else if (t_1 <= 2d+289) then
        tmp = sqrt(abs((t * (u * (n + n)))))
    else
        tmp = sqrt(sqrt((t_2 * t_2)))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)));
	double t_2 = ((U + U) * t) * n;
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.sqrt(((U + U) * (t * n)));
	} else if (t_1 <= 2e+289) {
		tmp = Math.sqrt(Math.abs((t * (U * (n + n)))));
	} else {
		tmp = Math.sqrt(Math.sqrt((t_2 * t_2)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))
	t_2 = ((U + U) * t) * n
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.sqrt(((U + U) * (t * n)))
	elif t_1 <= 2e+289:
		tmp = math.sqrt(math.fabs((t * (U * (n + n)))))
	else:
		tmp = math.sqrt(math.sqrt((t_2 * t_2)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = 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_))))
	t_2 = Float64(Float64(Float64(U + U) * t) * n)
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = sqrt(Float64(Float64(U + U) * Float64(t * n)));
	elseif (t_1 <= 2e+289)
		tmp = sqrt(abs(Float64(t * Float64(U * Float64(n + n)))));
	else
		tmp = sqrt(sqrt(Float64(t_2 * t_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)));
	t_2 = ((U + U) * t) * n;
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = sqrt(((U + U) * (t * n)));
	elseif (t_1 <= 2e+289)
		tmp = sqrt(abs((t * (U * (n + n)))));
	else
		tmp = sqrt(sqrt((t_2 * t_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, 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 * 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]}, Block[{t$95$2 = N[(N[(N[(U + U), $MachinePrecision] * t), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[Sqrt[N[(N[(U + U), $MachinePrecision] * N[(t * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 2e+289], N[Sqrt[N[Abs[N[(t * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Sqrt[N[(t$95$2 * t$95$2), $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 50.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   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 36.6%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 36.6%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      4. count-2 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      6. lower-*.f64 36.6%

         \[\leadsto \sqrt{\left(U + U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(n \cdot \color{blue}{t}\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(t \cdot \color{blue}{n}\right)} \]

      9. lower-*.f64 36.6%

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(t \cdot \color{blue}{n}\right)} \]

   6. Applied rewrites 36.6%

      \[\leadsto \sqrt{\left(U + U\right) \cdot \color{blue}{\left(t \cdot n\right)}} \]

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 2.0000000000000001e289

   1. Initial program 50.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   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 36.6%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 36.6%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      4. sqr-abs-rev N/A

         \[\leadsto \sqrt{\color{blue}{\left|\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right| \cdot \left|\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right|}} \]

   6. Applied rewrites 38.6%

      \[\leadsto \sqrt{\color{blue}{\left|t \cdot \left(U \cdot \left(n + n\right)\right)\right|}} \]

   if 2.0000000000000001e289 < (*.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 50.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   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 36.6%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 36.6%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      4. count-2 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      6. lower-*.f64 36.6%

         \[\leadsto \sqrt{\left(U + U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(n \cdot \color{blue}{t}\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(t \cdot \color{blue}{n}\right)} \]

      9. lower-*.f64 36.6%

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(t \cdot \color{blue}{n}\right)} \]

   6. Applied rewrites 36.6%

      \[\leadsto \sqrt{\left(U + U\right) \cdot \color{blue}{\left(t \cdot n\right)}} \]
   7. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{\left(U + U\right) \cdot \left(t \cdot n\right)} \cdot \sqrt{\left(U + U\right) \cdot \left(t \cdot n\right)}}} \]

      2. sqrt-unprod N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{\left(\left(U + U\right) \cdot \left(t \cdot n\right)\right) \cdot \left(\left(U + U\right) \cdot \left(t \cdot n\right)\right)}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{\left(\left(U + U\right) \cdot \left(t \cdot n\right)\right) \cdot \left(\left(U + U\right) \cdot \left(t \cdot n\right)\right)}}} \]

      4. lower-*.f64 28.6%

         \[\leadsto \sqrt{\sqrt{\color{blue}{\left(\left(U + U\right) \cdot \left(t \cdot n\right)\right) \cdot \left(\left(U + U\right) \cdot \left(t \cdot n\right)\right)}}} \]

   8. Applied rewrites 28.1%

      \[\leadsto \sqrt{\color{blue}{\sqrt{\left(\left(\left(U + U\right) \cdot t\right) \cdot n\right) \cdot \left(\left(\left(U + U\right) \cdot t\right) \cdot n\right)}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 41.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1 \cdot \left(t \cdot \sqrt{2 \cdot \frac{U \cdot n}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t} \cdot \sqrt{U \cdot \left(n + n\right)}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t -5e-310)
   (* -1.0 (* t (sqrt (* 2.0 (/ (* U n) t)))))
   (* (sqrt t) (sqrt (* U (+ n n))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -5e-310) {
		tmp = -1.0 * (t * sqrt((2.0 * ((U * n) / t))));
	} else {
		tmp = sqrt(t) * sqrt((U * (n + n)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (t <= (-5d-310)) then
        tmp = (-1.0d0) * (t * sqrt((2.0d0 * ((u * n) / t))))
    else
        tmp = sqrt(t) * sqrt((u * (n + n)))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -5e-310) {
		tmp = -1.0 * (t * Math.sqrt((2.0 * ((U * n) / t))));
	} else {
		tmp = Math.sqrt(t) * Math.sqrt((U * (n + n)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= -5e-310:
		tmp = -1.0 * (t * math.sqrt((2.0 * ((U * n) / t))))
	else:
		tmp = math.sqrt(t) * math.sqrt((U * (n + n)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= -5e-310)
		tmp = Float64(-1.0 * Float64(t * sqrt(Float64(2.0 * Float64(Float64(U * n) / t)))));
	else
		tmp = Float64(sqrt(t) * sqrt(Float64(U * Float64(n + n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= -5e-310)
		tmp = -1.0 * (t * sqrt((2.0 * ((U * n) / t))));
	else
		tmp = sqrt(t) * sqrt((U * (n + n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, -5e-310], N[(-1.0 * N[(t * N[Sqrt[N[(2.0 * N[(N[(U * n), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[t], $MachinePrecision] * N[Sqrt[N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 50.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(t \cdot \sqrt{2 \cdot \frac{U \cdot n}{t}}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(t \cdot \sqrt{2 \cdot \frac{U \cdot n}{t}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(t \cdot \color{blue}{\sqrt{2 \cdot \frac{U \cdot n}{t}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(t \cdot \sqrt{2 \cdot \frac{U \cdot n}{t}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(t \cdot \sqrt{2 \cdot \frac{U \cdot n}{t}}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(t \cdot \sqrt{2 \cdot \frac{U \cdot n}{t}}\right) \]

      6. lower-*.f64 19.0%

         \[\leadsto -1 \cdot \left(t \cdot \sqrt{2 \cdot \frac{U \cdot n}{t}}\right) \]

   4. Applied rewrites 19.0%

      \[\leadsto \color{blue}{-1 \cdot \left(t \cdot \sqrt{2 \cdot \frac{U \cdot n}{t}}\right)} \]

   if -4.999999999999985e-310 < t

   1. Initial program 50.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      4. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}} \]

      5. lower-unsound-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}} \]

   3. Applied rewrites 25.2%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\left(U* - U\right) \cdot n, \ell \cdot \frac{\ell}{Om \cdot Om}, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)} \cdot \sqrt{U \cdot \left(n + n\right)}} \]

   4. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\sqrt{t}} \cdot \sqrt{U \cdot \left(n + n\right)} \]
   5. Step-by-step derivation

      1. lower-sqrt.f64 21.8%

         \[\leadsto \sqrt{t} \cdot \sqrt{U \cdot \left(n + n\right)} \]

   6. Applied rewrites 21.8%

      \[\leadsto \color{blue}{\sqrt{t}} \cdot \sqrt{U \cdot \left(n + n\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 41.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|t \cdot \left(U \cdot \left(n + n\right)\right)\right|}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<=
      (sqrt
       (*
        (* (* 2.0 n) U)
        (-
         (- t (* 2.0 (/ (* l l) Om)))
         (* (* n (pow (/ l Om) 2.0)) (- U U*)))))
      0.0)
   (* (sqrt n) (sqrt (* 2.0 (* U t))))
   (sqrt (fabs (* t (* U (+ n n)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_))))) <= 0.0) {
		tmp = sqrt(n) * sqrt((2.0 * (U * t)));
	} else {
		tmp = sqrt(fabs((t * (U * (n + n)))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42))))) <= 0.0d0) then
        tmp = sqrt(n) * sqrt((2.0d0 * (u * t)))
    else
        tmp = sqrt(abs((t * (u * (n + n)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_))))) <= 0.0) {
		tmp = Math.sqrt(n) * Math.sqrt((2.0 * (U * t)));
	} else {
		tmp = Math.sqrt(Math.abs((t * (U * (n + n)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_))))) <= 0.0:
		tmp = math.sqrt(n) * math.sqrt((2.0 * (U * t)))
	else:
		tmp = math.sqrt(math.fabs((t * (U * (n + n)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (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_))))) <= 0.0)
		tmp = Float64(sqrt(n) * sqrt(Float64(2.0 * Float64(U * t))));
	else
		tmp = sqrt(abs(Float64(t * Float64(U * Float64(n + n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_))))) <= 0.0)
		tmp = sqrt(n) * sqrt((2.0 * (U * t)));
	else
		tmp = sqrt(abs((t * (U * (n + n)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * 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], 0.0], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[Abs[N[(t * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 0.0

   1. Initial program 50.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 23.7%

      \[\leadsto \color{blue}{\sqrt{n} \cdot \sqrt{\left(U + U\right) \cdot \mathsf{fma}\left(\left(U* - U\right) \cdot n, \ell \cdot \frac{\ell}{Om \cdot Om}, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)}} \]

   4. Taylor expanded in t around inf

      \[\leadsto \sqrt{n} \cdot \sqrt{\color{blue}{2 \cdot \left(U \cdot t\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{2 \cdot \color{blue}{\left(U \cdot t\right)}} \]

      2. lower-*.f64 20.8%

         \[\leadsto \sqrt{n} \cdot \sqrt{2 \cdot \left(U \cdot \color{blue}{t}\right)} \]

   6. Applied rewrites 20.8%

      \[\leadsto \sqrt{n} \cdot \sqrt{\color{blue}{2 \cdot \left(U \cdot t\right)}} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 50.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   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 36.6%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 36.6%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      4. sqr-abs-rev N/A

         \[\leadsto \sqrt{\color{blue}{\left|\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right| \cdot \left|\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right|}} \]

   6. Applied rewrites 38.6%

      \[\leadsto \sqrt{\color{blue}{\left|t \cdot \left(U \cdot \left(n + n\right)\right)\right|}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 39.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(U + U\right) \cdot \left(t \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|t \cdot \left(U \cdot \left(n + n\right)\right)\right|}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<=
      (sqrt
       (*
        (* (* 2.0 n) U)
        (-
         (- t (* 2.0 (/ (* l l) Om)))
         (* (* n (pow (/ l Om) 2.0)) (- U U*)))))
      0.0)
   (sqrt (* (+ U U) (* t n)))
   (sqrt (fabs (* t (* U (+ n n)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_))))) <= 0.0) {
		tmp = sqrt(((U + U) * (t * n)));
	} else {
		tmp = sqrt(fabs((t * (U * (n + n)))));
	}
	return tmp;
}

Fortran

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
    real(8) :: tmp
    if (sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42))))) <= 0.0d0) then
        tmp = sqrt(((u + u) * (t * n)))
    else
        tmp = sqrt(abs((t * (u * (n + n)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_))))) <= 0.0) {
		tmp = Math.sqrt(((U + U) * (t * n)));
	} else {
		tmp = Math.sqrt(Math.abs((t * (U * (n + n)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_))))) <= 0.0:
		tmp = math.sqrt(((U + U) * (t * n)))
	else:
		tmp = math.sqrt(math.fabs((t * (U * (n + n)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (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_))))) <= 0.0)
		tmp = sqrt(Float64(Float64(U + U) * Float64(t * n)));
	else
		tmp = sqrt(abs(Float64(t * Float64(U * Float64(n + n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_))))) <= 0.0)
		tmp = sqrt(((U + U) * (t * n)));
	else
		tmp = sqrt(abs((t * (U * (n + n)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * 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], 0.0], N[Sqrt[N[(N[(U + U), $MachinePrecision] * N[(t * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(t * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 0.0

   1. Initial program 50.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   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 36.6%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 36.6%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      4. count-2 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      6. lower-*.f64 36.6%

         \[\leadsto \sqrt{\left(U + U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(n \cdot \color{blue}{t}\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(t \cdot \color{blue}{n}\right)} \]

      9. lower-*.f64 36.6%

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(t \cdot \color{blue}{n}\right)} \]

   6. Applied rewrites 36.6%

      \[\leadsto \sqrt{\left(U + U\right) \cdot \color{blue}{\left(t \cdot n\right)}} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 50.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   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 36.6%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 36.6%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      4. sqr-abs-rev N/A

         \[\leadsto \sqrt{\color{blue}{\left|\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right| \cdot \left|\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right|}} \]

   6. Applied rewrites 38.6%

      \[\leadsto \sqrt{\color{blue}{\left|t \cdot \left(U \cdot \left(n + n\right)\right)\right|}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 36.6% accurate, 4.6× speedup

Math

\[\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (sqrt (* 2.0 (* U (* n t)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((2.0 * (U * (n * t))));
}

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 * (u * (n * t))))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((2.0 * (U * (n * t))));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((2.0 * (U * (n * t))))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(2.0 * Float64(U * Float64(n * t))))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((2.0 * (U * (n * t))));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.2%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

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 36.6%

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

4. Applied rewrites 36.6%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
5. Add Preprocessing

Alternative 21: 36.6% accurate, 4.7× speedup

Math

\[\sqrt{\left(U + U\right) \cdot \left(t \cdot n\right)} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (sqrt (* (+ U U) (* t n))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt(((U + U) * (t * n)));
}

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(((u + u) * (t * n)))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt(((U + U) * (t * n)));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt(((U + U) * (t * n)))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(U + U) * Float64(t * n)))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt(((U + U) * (t * n)));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(U + U), $MachinePrecision] * N[(t * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.2%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

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 36.6%

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

4. Applied rewrites 36.6%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

   3. associate-*r* N/A

      \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   4. count-2 N/A

      \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

   5. lift-+.f64 N/A

      \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

   6. lower-*.f64 36.6%

      \[\leadsto \sqrt{\left(U + U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   7. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(U + U\right) \cdot \left(n \cdot \color{blue}{t}\right)} \]

   8. *-commutative N/A

      \[\leadsto \sqrt{\left(U + U\right) \cdot \left(t \cdot \color{blue}{n}\right)} \]

   9. lower-*.f64 36.6%

      \[\leadsto \sqrt{\left(U + U\right) \cdot \left(t \cdot \color{blue}{n}\right)} \]

6. Applied rewrites 36.6%

   \[\leadsto \sqrt{\left(U + U\right) \cdot \color{blue}{\left(t \cdot n\right)}} \]
7. Add Preprocessing

Alternative 22: 36.2% accurate, 4.7× speedup

Math

\[\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (sqrt (* (* U (+ n n)) t)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt(((U * (n + n)) * t));
}

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(((u * (n + n)) * t))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt(((U * (n + n)) * t));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt(((U * (n + n)) * t))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(U * Float64(n + n)) * t))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt(((U * (n + n)) * t));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.2%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

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 36.6%

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

4. Applied rewrites 36.6%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. associate-*r* N/A

      \[\leadsto \sqrt{2 \cdot \left(\left(U \cdot n\right) \cdot \color{blue}{t}\right)} \]

   5. associate-*r* N/A

      \[\leadsto \sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \color{blue}{t}} \]

   6. *-commutative N/A

      \[\leadsto \sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t} \]

   7. associate-*l* N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

   8. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

   9. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

   10. lower-*.f64 36.2%

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

   11. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

   12. *-commutative N/A

       \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

   13. lower-*.f64 36.2%

       \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

   14. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

   15. count-2-rev N/A

       \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   16. lift-+.f64 36.2%

       \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

6. Applied rewrites 36.2%

   \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot \color{blue}{t}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025183 
(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*))))))