Toniolo and Linder, Equation (13)

Percentage Accurate: 49.8% → 58.7%

Time: 11.8s

Alternatives: 19

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 49.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 58.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 10^{-91}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(U \cdot \left(\left(\left(U - U*\right) \cdot n\right) \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) - \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot U\right)\right) \cdot n}\\ \mathbf{elif}\;t\_1 \leq 10^{+153}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot n\right)\right) \cdot \mathsf{fma}\left(U - U*, \frac{U}{Om \cdot Om}, \frac{U + U}{Om \cdot n}\right)\right) \cdot -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_1 1e-91)
     (sqrt
      (*
       (*
        -2.0
        (-
         (* U (* (* (- U U*) n) (* l (/ l (* Om Om)))))
         (* (fma -2.0 (* l (/ l Om)) t) U)))
       n))
     (if (<= t_1 1e+153)
       t_1
       (sqrt
        (*
         (*
          (* (* l n) (* l n))
          (fma (- U U*) (/ U (* Om Om)) (/ (+ U U) (* Om n))))
         -2.0))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_1 <= 1e-91) {
		tmp = sqrt(((-2.0 * ((U * (((U - U_42_) * n) * (l * (l / (Om * Om))))) - (fma(-2.0, (l * (l / Om)), t) * U))) * n));
	} else if (t_1 <= 1e+153) {
		tmp = t_1;
	} else {
		tmp = sqrt(((((l * n) * (l * n)) * fma((U - U_42_), (U / (Om * Om)), ((U + U) / (Om * n)))) * -2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = 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_1 <= 1e-91)
		tmp = sqrt(Float64(Float64(-2.0 * Float64(Float64(U * Float64(Float64(Float64(U - U_42_) * n) * Float64(l * Float64(l / Float64(Om * Om))))) - Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) * U))) * n));
	elseif (t_1 <= 1e+153)
		tmp = t_1;
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(l * n) * Float64(l * n)) * fma(Float64(U - U_42_), Float64(U / Float64(Om * Om)), Float64(Float64(U + U) / Float64(Om * n)))) * -2.0));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * 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$1, 1e-91], N[Sqrt[N[(N[(-2.0 * N[(N[(U * N[(N[(N[(U - U$42$), $MachinePrecision] * n), $MachinePrecision] * N[(l * N[(l / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 1e+153], t$95$1, N[Sqrt[N[(N[(N[(N[(l * n), $MachinePrecision] * N[(l * n), $MachinePrecision]), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] * N[(U / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(N[(U + U), $MachinePrecision] / N[(Om * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $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*))))) < 1.00000000000000002e-91

   1. Initial program 39.1%

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

   2. Taylor expanded in n around 0

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

   4. Applied rewrites 48.8%

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

   if 1.00000000000000002e-91 < (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*))))) < 1e153

   1. Initial program 98.2%

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

   if 1e153 < (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 21.2%

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

   2. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 18.5%

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

   5. Taylor expanded in U* around inf

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

      1. count-2-rev N/A

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

      2. associate-/l* N/A

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

      3. associate-/l* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-+.f64 24.6

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

   7. Applied rewrites 24.6%

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

   8. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 35.9%

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

Alternative 2: 57.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 10^{-89}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \left(U \cdot \left(\left(\left(U - U*\right) \cdot n\right) \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) - \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot U\right)\right) \cdot n}\\ \mathbf{elif}\;t\_1 \leq 10^{+153}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \left(t - \frac{\left(n \cdot \frac{\ell}{Om}\right) \cdot \ell}{Om} \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot n\right)\right) \cdot \mathsf{fma}\left(U - U*, \frac{U}{Om \cdot Om}, \frac{U + U}{Om \cdot n}\right)\right) \cdot -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_1 1e-89)
     (sqrt
      (*
       (*
        -2.0
        (-
         (* U (* (* (- U U*) n) (* l (/ l (* Om Om)))))
         (* (fma -2.0 (* l (/ l Om)) t) U)))
       n))
     (if (<= t_1 1e+153)
       (sqrt (* (* (+ n n) U) (- t (* (/ (* (* n (/ l Om)) l) Om) (- U U*)))))
       (sqrt
        (*
         (*
          (* (* l n) (* l n))
          (fma (- U U*) (/ U (* Om Om)) (/ (+ U U) (* Om n))))
         -2.0))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_1 <= 1e-89) {
		tmp = sqrt(((-2.0 * ((U * (((U - U_42_) * n) * (l * (l / (Om * Om))))) - (fma(-2.0, (l * (l / Om)), t) * U))) * n));
	} else if (t_1 <= 1e+153) {
		tmp = sqrt((((n + n) * U) * (t - ((((n * (l / Om)) * l) / Om) * (U - U_42_)))));
	} else {
		tmp = sqrt(((((l * n) * (l * n)) * fma((U - U_42_), (U / (Om * Om)), ((U + U) / (Om * n)))) * -2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = 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_1 <= 1e-89)
		tmp = sqrt(Float64(Float64(-2.0 * Float64(Float64(U * Float64(Float64(Float64(U - U_42_) * n) * Float64(l * Float64(l / Float64(Om * Om))))) - Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) * U))) * n));
	elseif (t_1 <= 1e+153)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * Float64(t - Float64(Float64(Float64(Float64(n * Float64(l / Om)) * l) / Om) * Float64(U - U_42_)))));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(l * n) * Float64(l * n)) * fma(Float64(U - U_42_), Float64(U / Float64(Om * Om)), Float64(Float64(U + U) / Float64(Om * n)))) * -2.0));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * 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$1, 1e-89], N[Sqrt[N[(N[(-2.0 * N[(N[(U * N[(N[(N[(U - U$42$), $MachinePrecision] * n), $MachinePrecision] * N[(l * N[(l / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 1e+153], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * N[(t - N[(N[(N[(N[(n * N[(l / Om), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(l * n), $MachinePrecision] * N[(l * n), $MachinePrecision]), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] * N[(U / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(N[(U + U), $MachinePrecision] / N[(Om * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $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*))))) < 1.00000000000000004e-89

   1. Initial program 39.9%

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

   2. Taylor expanded in n around 0

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

   4. Applied rewrites 48.9%

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

   if 1.00000000000000004e-89 < (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*))))) < 1e153

   1. Initial program 98.2%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 87.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 87.8

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

   6. Applied rewrites 87.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 87.6

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

   8. Applied rewrites 87.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 87.6

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

   10. Applied rewrites 87.6%

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

   if 1e153 < (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 21.2%

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

   2. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 18.5%

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

   5. Taylor expanded in U* around inf

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

      1. count-2-rev N/A

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

      2. associate-/l* N/A

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

      3. associate-/l* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-+.f64 24.6

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

   7. Applied rewrites 24.6%

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

   8. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 35.9%

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

Alternative 3: 56.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-87}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot \frac{n}{Om \cdot Om}, U - U*, \frac{\ell}{Om} \cdot \left(\ell + \ell\right)\right)\right)\right)}\\ \mathbf{elif}\;t\_1 \leq 10^{+153}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \left(t - \frac{\left(n \cdot \frac{\ell}{Om}\right) \cdot \ell}{Om} \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot n\right)\right) \cdot \mathsf{fma}\left(U - U*, \frac{U}{Om \cdot Om}, \frac{U + U}{Om \cdot n}\right)\right) \cdot -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_1 2e-87)
     (sqrt
      (*
       (+ n n)
       (*
        U
        (-
         t
         (fma (* (* l l) (/ n (* Om Om))) (- U U*) (* (/ l Om) (+ l l)))))))
     (if (<= t_1 1e+153)
       (sqrt (* (* (+ n n) U) (- t (* (/ (* (* n (/ l Om)) l) Om) (- U U*)))))
       (sqrt
        (*
         (*
          (* (* l n) (* l n))
          (fma (- U U*) (/ U (* Om Om)) (/ (+ U U) (* Om n))))
         -2.0))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_1 <= 2e-87) {
		tmp = sqrt(((n + n) * (U * (t - fma(((l * l) * (n / (Om * Om))), (U - U_42_), ((l / Om) * (l + l)))))));
	} else if (t_1 <= 1e+153) {
		tmp = sqrt((((n + n) * U) * (t - ((((n * (l / Om)) * l) / Om) * (U - U_42_)))));
	} else {
		tmp = sqrt(((((l * n) * (l * n)) * fma((U - U_42_), (U / (Om * Om)), ((U + U) / (Om * n)))) * -2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = 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_1 <= 2e-87)
		tmp = sqrt(Float64(Float64(n + n) * Float64(U * Float64(t - fma(Float64(Float64(l * l) * Float64(n / Float64(Om * Om))), Float64(U - U_42_), Float64(Float64(l / Om) * Float64(l + l)))))));
	elseif (t_1 <= 1e+153)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * Float64(t - Float64(Float64(Float64(Float64(n * Float64(l / Om)) * l) / Om) * Float64(U - U_42_)))));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(l * n) * Float64(l * n)) * fma(Float64(U - U_42_), Float64(U / Float64(Om * Om)), Float64(Float64(U + U) / Float64(Om * n)))) * -2.0));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * 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$1, 2e-87], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(U * N[(t - N[(N[(N[(l * l), $MachinePrecision] * N[(n / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + N[(N[(l / Om), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 1e+153], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * N[(t - N[(N[(N[(N[(n * N[(l / Om), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(l * n), $MachinePrecision] * N[(l * n), $MachinePrecision]), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] * N[(U / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(N[(U + U), $MachinePrecision] / N[(Om * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $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*))))) < 2.00000000000000004e-87

   1. Initial program 40.7%

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

   3. Applied rewrites 49.4%

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

   if 2.00000000000000004e-87 < (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*))))) < 1e153

   1. Initial program 98.2%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 87.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 87.9

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

   6. Applied rewrites 87.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 87.6

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

   8. Applied rewrites 87.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 87.6

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

   10. Applied rewrites 87.6%

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

   if 1e153 < (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 21.2%

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

   2. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 18.5%

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

   5. Taylor expanded in U* around inf

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

      1. count-2-rev N/A

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

      2. associate-/l* N/A

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

      3. associate-/l* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-+.f64 24.6

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

   7. Applied rewrites 24.6%

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

   8. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 35.9%

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

Alternative 4: 56.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.95 \cdot 10^{-60}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;n \leq 2.7 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \ell\right), -4, t \cdot \left(\left(U + U\right) \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{U \cdot \left(t - \left(U - U*\right) \cdot \left(\frac{\ell \cdot \ell}{Om \cdot Om} \cdot n\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -1.95e-60)
   (sqrt (* (* (* 2.0 n) U) (- t (* (* n (* (/ l Om) (/ l Om))) (- U U*)))))
   (if (<= n 2.7e-24)
     (sqrt (fma (* U (* (* (/ l Om) n) l)) -4.0 (* t (* (+ U U) n))))
     (*
      (sqrt (+ n n))
      (sqrt (* U (- t (* (- U U*) (* (/ (* l l) (* Om Om)) n)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -1.95e-60) {
		tmp = sqrt((((2.0 * n) * U) * (t - ((n * ((l / Om) * (l / Om))) * (U - U_42_)))));
	} else if (n <= 2.7e-24) {
		tmp = sqrt(fma((U * (((l / Om) * n) * l)), -4.0, (t * ((U + U) * n))));
	} else {
		tmp = sqrt((n + n)) * sqrt((U * (t - ((U - U_42_) * (((l * l) / (Om * Om)) * n)))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -1.95e-60)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t - Float64(Float64(n * Float64(Float64(l / Om) * Float64(l / Om))) * Float64(U - U_42_)))));
	elseif (n <= 2.7e-24)
		tmp = sqrt(fma(Float64(U * Float64(Float64(Float64(l / Om) * n) * l)), -4.0, Float64(t * Float64(Float64(U + U) * n))));
	else
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(Float64(U * Float64(t - Float64(Float64(U - U_42_) * Float64(Float64(Float64(l * l) / Float64(Om * Om)) * n))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -1.95e-60], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t - N[(N[(n * N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 2.7e-24], N[Sqrt[N[(N[(U * N[(N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(t * N[(N[(U + U), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t - N[(N[(U - U$42$), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.9500000000000001e-60

   1. Initial program 55.6%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 59.7%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 59.7

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

   6. Applied rewrites 59.7%

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

   if -1.9500000000000001e-60 < n < 2.70000000000000007e-24

   1. Initial program 44.4%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 41.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 41.2

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

   6. Applied rewrites 41.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 41.1

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

   8. Applied rewrites 41.1%

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

   9. Taylor expanded in Om around inf

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

   11. Applied rewrites 53.5%

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

   if 2.70000000000000007e-24 < n

   1. Initial program 54.3%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 59.7%

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

   6. Applied rewrites 60.3%

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

Alternative 5: 56.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.95 \cdot 10^{-60}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;n \leq 2.25 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \ell\right), -4, t \cdot \left(\left(U + U\right) \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \left(t - \frac{\left(n \cdot \frac{\ell}{Om}\right) \cdot \ell}{Om} \cdot \left(U - U*\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -1.95e-60)
   (sqrt (* (* (* 2.0 n) U) (- t (* (* n (* (/ l Om) (/ l Om))) (- U U*)))))
   (if (<= n 2.25e-32)
     (sqrt (fma (* U (* (* (/ l Om) n) l)) -4.0 (* t (* (+ U U) n))))
     (sqrt (* (* (+ n n) U) (- t (* (/ (* (* n (/ l Om)) l) Om) (- U U*))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -1.95e-60) {
		tmp = sqrt((((2.0 * n) * U) * (t - ((n * ((l / Om) * (l / Om))) * (U - U_42_)))));
	} else if (n <= 2.25e-32) {
		tmp = sqrt(fma((U * (((l / Om) * n) * l)), -4.0, (t * ((U + U) * n))));
	} else {
		tmp = sqrt((((n + n) * U) * (t - ((((n * (l / Om)) * l) / Om) * (U - U_42_)))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -1.95e-60)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t - Float64(Float64(n * Float64(Float64(l / Om) * Float64(l / Om))) * Float64(U - U_42_)))));
	elseif (n <= 2.25e-32)
		tmp = sqrt(fma(Float64(U * Float64(Float64(Float64(l / Om) * n) * l)), -4.0, Float64(t * Float64(Float64(U + U) * n))));
	else
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * Float64(t - Float64(Float64(Float64(Float64(n * Float64(l / Om)) * l) / Om) * Float64(U - U_42_)))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -1.95e-60], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t - N[(N[(n * N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 2.25e-32], N[Sqrt[N[(N[(U * N[(N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(t * N[(N[(U + U), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * N[(t - N[(N[(N[(N[(n * N[(l / Om), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.9500000000000001e-60

   1. Initial program 55.6%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 59.7%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 59.7

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

   6. Applied rewrites 59.7%

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

   if -1.9500000000000001e-60 < n < 2.25000000000000002e-32

   1. Initial program 44.3%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 41.0%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 41.0

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

   6. Applied rewrites 41.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 40.9

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

   8. Applied rewrites 40.9%

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

   9. Taylor expanded in Om around inf

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

   11. Applied rewrites 53.5%

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

   if 2.25000000000000002e-32 < n

   1. Initial program 54.5%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 59.8

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

   6. Applied rewrites 59.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 60.5

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

   8. Applied rewrites 60.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 60.5

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

   10. Applied rewrites 60.5%

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

Alternative 6: 56.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \left(t - \frac{\left(n \cdot \frac{\ell}{Om}\right) \cdot \ell}{Om} \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;n \leq -1.95 \cdot 10^{-60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 2.25 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \ell\right), -4, t \cdot \left(\left(U + U\right) \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (* (* (+ n n) U) (- t (* (/ (* (* n (/ l Om)) l) Om) (- U U*)))))))
   (if (<= n -1.95e-60)
     t_1
     (if (<= n 2.25e-32)
       (sqrt (fma (* U (* (* (/ l Om) n) l)) -4.0 (* t (* (+ U U) n))))
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((((n + n) * U) * (t - ((((n * (l / Om)) * l) / Om) * (U - U_42_)))));
	double tmp;
	if (n <= -1.95e-60) {
		tmp = t_1;
	} else if (n <= 2.25e-32) {
		tmp = sqrt(fma((U * (((l / Om) * n) * l)), -4.0, (t * ((U + U) * n))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(n + n) * U) * Float64(t - Float64(Float64(Float64(Float64(n * Float64(l / Om)) * l) / Om) * Float64(U - U_42_)))))
	tmp = 0.0
	if (n <= -1.95e-60)
		tmp = t_1;
	elseif (n <= 2.25e-32)
		tmp = sqrt(fma(Float64(U * Float64(Float64(Float64(l / Om) * n) * l)), -4.0, Float64(t * Float64(Float64(U + U) * n))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * N[(t - N[(N[(N[(N[(n * N[(l / Om), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -1.95e-60], t$95$1, If[LessEqual[n, 2.25e-32], N[Sqrt[N[(N[(U * N[(N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(t * N[(N[(U + U), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.9500000000000001e-60 or 2.25000000000000002e-32 < n

   1. Initial program 55.0%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 59.8

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

   6. Applied rewrites 59.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 60.4

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

   8. Applied rewrites 60.4%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 60.4

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

   10. Applied rewrites 60.4%

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

   if -1.9500000000000001e-60 < n < 2.25000000000000002e-32

   1. Initial program 44.3%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 41.0%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 41.0

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

   6. Applied rewrites 41.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 40.9

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

   8. Applied rewrites 40.9%

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

   9. Taylor expanded in Om around inf

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

   11. Applied rewrites 53.5%

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

Alternative 7: 55.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.95 \cdot 10^{-60}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \left(t - \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right) \cdot \left(-U*\right)\right)}\\ \mathbf{elif}\;n \leq 2.7 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \ell\right), -4, t \cdot \left(\left(U + U\right) \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{U \cdot \left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{n}{Om \cdot Om}\right) \cdot \left(-U*\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n -1.95e-60)
   (sqrt (* (* (+ n n) U) (- t (* (* n (* (/ l Om) (/ l Om))) (- U*)))))
   (if (<= n 2.7e-24)
     (sqrt (fma (* U (* (* (/ l Om) n) l)) -4.0 (* t (* (+ U U) n))))
     (*
      (sqrt (+ n n))
      (sqrt (* U (- t (* (* (* l l) (/ n (* Om Om))) (- U*)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= -1.95e-60) {
		tmp = sqrt((((n + n) * U) * (t - ((n * ((l / Om) * (l / Om))) * -U_42_))));
	} else if (n <= 2.7e-24) {
		tmp = sqrt(fma((U * (((l / Om) * n) * l)), -4.0, (t * ((U + U) * n))));
	} else {
		tmp = sqrt((n + n)) * sqrt((U * (t - (((l * l) * (n / (Om * Om))) * -U_42_))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= -1.95e-60)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * Float64(t - Float64(Float64(n * Float64(Float64(l / Om) * Float64(l / Om))) * Float64(-U_42_)))));
	elseif (n <= 2.7e-24)
		tmp = sqrt(fma(Float64(U * Float64(Float64(Float64(l / Om) * n) * l)), -4.0, Float64(t * Float64(Float64(U + U) * n))));
	else
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(Float64(U * Float64(t - Float64(Float64(Float64(l * l) * Float64(n / Float64(Om * Om))) * Float64(-U_42_))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, -1.95e-60], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * N[(t - N[(N[(n * N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-U$42$)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 2.7e-24], N[Sqrt[N[(N[(U * N[(N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(t * N[(N[(U + U), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t - N[(N[(N[(l * l), $MachinePrecision] * N[(n / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-U$42$)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.9500000000000001e-60

   1. Initial program 55.6%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 59.7%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 59.7

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

   6. Applied rewrites 59.7%

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

   7. Taylor expanded in U around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 59.7

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

   9. Applied rewrites 59.7%

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

       1. lift-*.f64 N/A

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

       2. count-2-rev N/A

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

       3. lower-+.f64 59.7

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

   11. Applied rewrites 59.7%

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

   if -1.9500000000000001e-60 < n < 2.70000000000000007e-24

   1. Initial program 44.4%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 41.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 41.2

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

   6. Applied rewrites 41.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 41.1

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

   8. Applied rewrites 41.1%

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

   9. Taylor expanded in Om around inf

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

   11. Applied rewrites 53.5%

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

   if 2.70000000000000007e-24 < n

   1. Initial program 54.3%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 59.7%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 59.7

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

   6. Applied rewrites 59.7%

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

   7. Taylor expanded in U around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 59.7

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

   9. Applied rewrites 59.7%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l* N/A

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

       6. sqrt-prod N/A

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

       7. lower-*.f64 N/A

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

   11. Applied rewrites 58.6%

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

Alternative 8: 55.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \left(t - \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right) \cdot \left(-U*\right)\right)}\\ \mathbf{if}\;n \leq -1.95 \cdot 10^{-60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 2.25 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \ell\right), -4, t \cdot \left(\left(U + U\right) \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (* (* (+ n n) U) (- t (* (* n (* (/ l Om) (/ l Om))) (- U*)))))))
   (if (<= n -1.95e-60)
     t_1
     (if (<= n 2.25e-32)
       (sqrt (fma (* U (* (* (/ l Om) n) l)) -4.0 (* t (* (+ U U) n))))
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((((n + n) * U) * (t - ((n * ((l / Om) * (l / Om))) * -U_42_))));
	double tmp;
	if (n <= -1.95e-60) {
		tmp = t_1;
	} else if (n <= 2.25e-32) {
		tmp = sqrt(fma((U * (((l / Om) * n) * l)), -4.0, (t * ((U + U) * n))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(n + n) * U) * Float64(t - Float64(Float64(n * Float64(Float64(l / Om) * Float64(l / Om))) * Float64(-U_42_)))))
	tmp = 0.0
	if (n <= -1.95e-60)
		tmp = t_1;
	elseif (n <= 2.25e-32)
		tmp = sqrt(fma(Float64(U * Float64(Float64(Float64(l / Om) * n) * l)), -4.0, Float64(t * Float64(Float64(U + U) * n))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision] * N[(t - N[(N[(n * N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-U$42$)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -1.95e-60], t$95$1, If[LessEqual[n, 2.25e-32], N[Sqrt[N[(N[(U * N[(N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(t * N[(N[(U + U), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.9500000000000001e-60 or 2.25000000000000002e-32 < n

   1. Initial program 55.0%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 59.8

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

   6. Applied rewrites 59.8%

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

   7. Taylor expanded in U around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 59.7

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

   9. Applied rewrites 59.7%

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

       1. lift-*.f64 N/A

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

       2. count-2-rev N/A

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

       3. lower-+.f64 59.7

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

   11. Applied rewrites 59.7%

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

   if -1.9500000000000001e-60 < n < 2.25000000000000002e-32

   1. Initial program 44.3%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 41.0%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 41.0

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

   6. Applied rewrites 41.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 40.9

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

   8. Applied rewrites 40.9%

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

   9. Taylor expanded in Om around inf

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

   11. Applied rewrites 53.5%

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

Alternative 9: 54.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(n + n\right) \cdot \left(U \cdot \left(t - \left(\left(\ell \cdot \ell\right) \cdot \frac{n}{Om \cdot Om}\right) \cdot \left(-U*\right)\right)\right)}\\ \mathbf{if}\;n \leq -1.5 \cdot 10^{-59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-21}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \ell\right), -4, t \cdot \left(\left(U + U\right) \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (* (+ n n) (* U (- t (* (* (* l l) (/ n (* Om Om))) (- U*))))))))
   (if (<= n -1.5e-59)
     t_1
     (if (<= n 1.75e-21)
       (sqrt (fma (* U (* (* (/ l Om) n) l)) -4.0 (* t (* (+ U U) n))))
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((n + n) * (U * (t - (((l * l) * (n / (Om * Om))) * -U_42_)))));
	double tmp;
	if (n <= -1.5e-59) {
		tmp = t_1;
	} else if (n <= 1.75e-21) {
		tmp = sqrt(fma((U * (((l / Om) * n) * l)), -4.0, (t * ((U + U) * n))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(n + n) * Float64(U * Float64(t - Float64(Float64(Float64(l * l) * Float64(n / Float64(Om * Om))) * Float64(-U_42_))))))
	tmp = 0.0
	if (n <= -1.5e-59)
		tmp = t_1;
	elseif (n <= 1.75e-21)
		tmp = sqrt(fma(Float64(U * Float64(Float64(Float64(l / Om) * n) * l)), -4.0, Float64(t * Float64(Float64(U + U) * n))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(U * N[(t - N[(N[(N[(l * l), $MachinePrecision] * N[(n / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-U$42$)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -1.5e-59], t$95$1, If[LessEqual[n, 1.75e-21], N[Sqrt[N[(N[(U * N[(N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(t * N[(N[(U + U), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.5e-59 or 1.7500000000000002e-21 < n

   1. Initial program 54.9%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 59.7%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 59.7

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

   6. Applied rewrites 59.7%

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

   7. Taylor expanded in U around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 59.7

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

   9. Applied rewrites 59.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. count-2-rev N/A

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

       7. lower-+.f64 N/A

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

       8. lower-*.f64 58.7

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

   11. Applied rewrites 51.1%

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

   if -1.5e-59 < n < 1.7500000000000002e-21

   1. Initial program 44.6%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 41.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 41.3

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

   6. Applied rewrites 41.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 41.2

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

   8. Applied rewrites 41.2%

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

   9. Taylor expanded in Om around inf

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

   11. Applied rewrites 53.6%

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

Alternative 10: 54.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\frac{\ell}{Om} \cdot n\right) \cdot \ell\\ t_2 := \left(U + U\right) \cdot n\\ \mathbf{if}\;n \leq -1.45 \cdot 10^{+177}:\\ \;\;\;\;{\left(\left(U + U\right) \cdot \left(t \cdot n\right)\right)}^{0.5}\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{+159}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U \cdot t\_1, -4, t \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{U*}{Om} \cdot t\_1\right) \cdot t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* (/ l Om) n) l)) (t_2 (* (+ U U) n)))
   (if (<= n -1.45e+177)
     (pow (* (+ U U) (* t n)) 0.5)
     (if (<= n 3.8e+159)
       (sqrt (fma (* U t_1) -4.0 (* t t_2)))
       (sqrt (* (* (/ U* Om) t_1) t_2))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = ((l / Om) * n) * l;
	double t_2 = (U + U) * n;
	double tmp;
	if (n <= -1.45e+177) {
		tmp = pow(((U + U) * (t * n)), 0.5);
	} else if (n <= 3.8e+159) {
		tmp = sqrt(fma((U * t_1), -4.0, (t * t_2)));
	} else {
		tmp = sqrt((((U_42_ / Om) * t_1) * t_2));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(Float64(l / Om) * n) * l)
	t_2 = Float64(Float64(U + U) * n)
	tmp = 0.0
	if (n <= -1.45e+177)
		tmp = Float64(Float64(U + U) * Float64(t * n)) ^ 0.5;
	elseif (n <= 3.8e+159)
		tmp = sqrt(fma(Float64(U * t_1), -4.0, Float64(t * t_2)));
	else
		tmp = sqrt(Float64(Float64(Float64(U_42_ / Om) * t_1) * t_2));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision] * l), $MachinePrecision]}, Block[{t$95$2 = N[(N[(U + U), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -1.45e+177], N[Power[N[(N[(U + U), $MachinePrecision] * N[(t * n), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[n, 3.8e+159], N[Sqrt[N[(N[(U * t$95$1), $MachinePrecision] * -4.0 + N[(t * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(U$42$ / Om), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.45000000000000007e177

   1. Initial program 54.5%

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

   2. Taylor expanded in t around inf

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

      1. count-2-rev N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. count-2-rev N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 33.3

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

   4. Applied rewrites 33.3%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(t \cdot \left(\left(n + n\right) \cdot U\right)\right)}^{\frac{1}{2}}} \]

      3. lower-pow.f64 40.9

         \[\leadsto \color{blue}{{\left(t \cdot \left(\left(n + n\right) \cdot U\right)\right)}^{0.5}} \]

   6. Applied rewrites 35.9%

      \[\leadsto \color{blue}{{\left(\left(U + U\right) \cdot \left(t \cdot n\right)\right)}^{0.5}} \]

   if -1.45000000000000007e177 < n < 3.79999999999999965e159

   1. Initial program 48.8%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 47.7%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 47.7

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

   6. Applied rewrites 47.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 47.6

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

   8. Applied rewrites 47.6%

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

   9. Taylor expanded in Om around inf

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

   11. Applied rewrites 51.7%

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

   if 3.79999999999999965e159 < n

   1. Initial program 54.0%

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

   2. Taylor expanded in U* around inf

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 32.4

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

   4. Applied rewrites 32.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 32.4

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 32.4

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

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(\left(\ell \cdot \ell\right) \cdot U*\right) \cdot \frac{n}{Om \cdot Om}\right) \cdot \mathsf{Rewrite=>}\left(lift-*.f64, \left(\left(2 \cdot n\right) \cdot U\right)\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(\left(\ell \cdot \ell\right) \cdot U*\right) \cdot \frac{n}{Om \cdot Om}\right) \cdot \left(\mathsf{Rewrite=>}\left(lift-*.f64, \left(2 \cdot n\right)\right) \cdot U\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(\left(\ell \cdot \ell\right) \cdot U*\right) \cdot \frac{n}{Om \cdot Om}\right) \cdot \mathsf{Rewrite=>}\left(associate-*l*, \left(2 \cdot \left(n \cdot U\right)\right)\right)} \]

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(\left(\ell \cdot \ell\right) \cdot U*\right) \cdot \frac{n}{Om \cdot Om}\right) \cdot \mathsf{Rewrite=>}\left(associate-*r*, \left(\left(2 \cdot U\right) \cdot n\right)\right)} \]

   6. Applied rewrites 32.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. pow2 N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. pow2 N/A

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

      18. associate-*l/ N/A

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

      19. lift-/.f64 N/A

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

      20. associate-*l* N/A

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

      21. lift-*.f64 N/A

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

      22. lift-*.f64 41.3

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 41.3

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

   8. Applied rewrites 41.3%

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

Alternative 11: 54.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = fma(-2.0, Float64(l * Float64(l / Om)), t)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = sqrt(Float64(t_2 * 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_3 <= 0.0)
		tmp = sqrt(Float64(Float64(U + U) * Float64(t_1 * n)));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * t_1));
	else
		tmp = sqrt(Float64(Float64(Float64(U_42_ / Om) * Float64(Float64(Float64(l / Om) * n) * l)) * Float64(Float64(U + U) * n)));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 12.0%

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

   2. Taylor expanded in 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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 37.5

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

   4. Applied rewrites 37.5%

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

   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*))))) < +inf.0

   1. Initial program 68.8%

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

   2. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 63.1

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

   4. Applied rewrites 63.1%

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

   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 0.0%

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

   2. Taylor expanded in U* around inf

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 26.7

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

   4. Applied rewrites 26.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 26.7

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 26.7

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

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(\left(\ell \cdot \ell\right) \cdot U*\right) \cdot \frac{n}{Om \cdot Om}\right) \cdot \mathsf{Rewrite=>}\left(lift-*.f64, \left(\left(2 \cdot n\right) \cdot U\right)\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(\left(\ell \cdot \ell\right) \cdot U*\right) \cdot \frac{n}{Om \cdot Om}\right) \cdot \left(\mathsf{Rewrite=>}\left(lift-*.f64, \left(2 \cdot n\right)\right) \cdot U\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(\left(\ell \cdot \ell\right) \cdot U*\right) \cdot \frac{n}{Om \cdot Om}\right) \cdot \mathsf{Rewrite=>}\left(associate-*l*, \left(2 \cdot \left(n \cdot U\right)\right)\right)} \]

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(\left(\ell \cdot \ell\right) \cdot U*\right) \cdot \frac{n}{Om \cdot Om}\right) \cdot \mathsf{Rewrite=>}\left(associate-*r*, \left(\left(2 \cdot U\right) \cdot n\right)\right)} \]

   6. Applied rewrites 26.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. pow2 N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. pow2 N/A

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

      18. associate-*l/ N/A

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

      19. lift-/.f64 N/A

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

      20. associate-*l* N/A

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

      21. lift-*.f64 N/A

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

      22. lift-*.f64 30.2

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 30.2

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

   8. Applied rewrites 30.2%

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

Alternative 12: 52.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = fma(-2.0, Float64(l * Float64(l / Om)), t)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * 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_3 <= 0.0)
		tmp = sqrt(Float64(Float64(U + U) * Float64(t_1 * n)));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * t_1));
	else
		tmp = sqrt(Float64(Float64(n + n) * Float64(U * Float64(Float64(Float64(l * l) * U_42_) * Float64(n / Float64(Om * Om))))));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 10.5%

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

   2. Taylor expanded in n around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 37.2

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

   4. Applied rewrites 37.2%

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

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

   1. Initial program 68.8%

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

   2. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 63.1

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

   4. Applied rewrites 63.1%

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

   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 0.0%

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

   2. Taylor expanded in U* around inf

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 29.1

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

   4. Applied rewrites 29.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. lower-*.f64 30.4

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

   6. Applied rewrites 30.4%

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

Alternative 13: 49.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = fma(-2.0, Float64(l * Float64(l / Om)), t)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * 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_3 <= 0.0)
		tmp = sqrt(Float64(Float64(U + U) * Float64(t_1 * n)));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * t_1));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(l * l) * Float64(U_42_ / Float64(Om * Om))) * Float64(U + U)) * n) * n));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * 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$3, 0.0], N[Sqrt[N[(N[(U + U), $MachinePrecision] * N[(t$95$1 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * t$95$1), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(l * l), $MachinePrecision] * N[(U$42$ / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U + U), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * n), $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 10.5%

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

   2. Taylor expanded in n around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 37.2

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

   4. Applied rewrites 37.2%

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

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

   1. Initial program 68.8%

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

   2. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 63.1

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

   4. Applied rewrites 63.1%

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

   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 0.0%

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

   2. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 0.6%

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

   5. Taylor expanded in U* around inf

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

      1. count-2-rev N/A

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

      2. associate-/l* N/A

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

      3. associate-/l* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-+.f64 28.0

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

   7. Applied rewrites 28.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

   9. Applied rewrites 30.3%

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

Alternative 14: 48.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\\ \mathbf{if}\;U \leq -3.7 \cdot 10^{-101}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U + U\right) \cdot \left(t\_1 \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (fma -2.0 (* l (/ l Om)) t)))
   (if (<= U -3.7e-101)
     (sqrt (* (* (* 2.0 n) U) t_1))
     (sqrt (* (+ U U) (* t_1 n))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = fma(-2.0, (l * (l / Om)), t);
	double tmp;
	if (U <= -3.7e-101) {
		tmp = sqrt((((2.0 * n) * U) * t_1));
	} else {
		tmp = sqrt(((U + U) * (t_1 * n)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = fma(-2.0, Float64(l * Float64(l / Om)), t)
	tmp = 0.0
	if (U <= -3.7e-101)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t_1));
	else
		tmp = sqrt(Float64(Float64(U + U) * Float64(t_1 * n)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[U, -3.7e-101], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(U + U), $MachinePrecision] * N[(t$95$1 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if U < -3.70000000000000005e-101

   1. Initial program 59.3%

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

   2. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 55.6

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

   4. Applied rewrites 55.6%

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

   if -3.70000000000000005e-101 < U

   1. Initial program 45.9%

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

   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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 45.5

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

   4. Applied rewrites 45.5%

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

Alternative 15: 47.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq -1.28 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U + U\right) \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U -1.28e+147)
   (sqrt (* t (* (+ n n) U)))
   (sqrt (* (+ U U) (* (fma -2.0 (* l (/ l Om)) t) n)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -1.28e+147) {
		tmp = sqrt((t * ((n + n) * U)));
	} else {
		tmp = sqrt(((U + U) * (fma(-2.0, (l * (l / Om)), t) * n)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= -1.28e+147)
		tmp = sqrt(Float64(t * Float64(Float64(n + n) * U)));
	else
		tmp = sqrt(Float64(Float64(U + U) * Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) * n)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, -1.28e+147], N[Sqrt[N[(t * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(U + U), $MachinePrecision] * N[(N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -1.28e147

   1. Initial program 62.5%

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

   2. Taylor expanded in t around inf

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

      1. count-2-rev N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. count-2-rev N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 49.1

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

   4. Applied rewrites 49.1%

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

   if -1.28e147 < U

   1. Initial program 48.6%

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

   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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 47.3

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

   4. Applied rewrites 47.3%

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

Alternative 16: 37.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 5 \cdot 10^{-90}:\\ \;\;\;\;\sqrt{\left(\left(U + U\right) \cdot t\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U + U\right) \cdot \left(t \cdot n\right)}\\ \end{array} \end{array} \]

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*))))
      5e-90)
   (sqrt (* (* (+ U U) t) n))
   (sqrt (* (+ U U) (* t n)))))

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_)))) <= 5e-90) {
		tmp = sqrt((((U + U) * t) * n));
	} else {
		tmp = sqrt(((U + U) * (t * 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 ((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))) <= 5d-90) then
        tmp = sqrt((((u + u) * t) * n))
    else
        tmp = sqrt(((u + u) * (t * 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 ((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))) <= 5e-90) {
		tmp = Math.sqrt((((U + U) * t) * n));
	} else {
		tmp = Math.sqrt(((U + U) * (t * n)));
	}
	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_)))) <= 5e-90:
		tmp = math.sqrt((((U + U) * t) * n))
	else:
		tmp = math.sqrt(((U + U) * (t * n)))
	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_)))) <= 5e-90)
		tmp = sqrt(Float64(Float64(Float64(U + U) * t) * n));
	else
		tmp = sqrt(Float64(Float64(U + U) * Float64(t * n)));
	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_)))) <= 5e-90)
		tmp = sqrt((((U + U) * t) * n));
	else
		tmp = sqrt(((U + U) * (t * n)));
	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], 5e-90], N[Sqrt[N[(N[(N[(U + U), $MachinePrecision] * t), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(U + U), $MachinePrecision] * N[(t * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.00000000000000019e-90

   1. Initial program 48.1%

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

   2. Taylor expanded in t around inf

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

      1. count-2-rev N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. count-2-rev N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

         \[\leadsto \sqrt{t \cdot \left(\left(2 \cdot n\right) \cdot \color{blue}{U}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{t \cdot \left(\left(2 \cdot n\right) \cdot \color{blue}{U}\right)} \]

      10. count-2-rev N/A

          \[\leadsto \sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)} \]

      11. lower-+.f64 39.9

          \[\leadsto \sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)} \]

   4. Applied rewrites 39.9%

      \[\leadsto \sqrt{\color{blue}{t \cdot \left(\left(n + n\right) \cdot U\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{t \cdot \color{blue}{\left(\left(n + n\right) \cdot U\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{t \cdot \left(\left(n + n\right) \cdot \color{blue}{U}\right)} \]

      3. lift-+.f64 N/A

         \[\leadsto \sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)} \]

      4. count-2-rev N/A

         \[\leadsto \sqrt{t \cdot \left(\left(2 \cdot n\right) \cdot U\right)} \]

      5. associate-*l* N/A

         \[\leadsto \sqrt{t \cdot \left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right)} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{t \cdot \left(2 \cdot \left(U \cdot \color{blue}{n}\right)\right)} \]

      7. count-2-rev N/A

         \[\leadsto \sqrt{t \cdot \left(U \cdot n + \color{blue}{U \cdot n}\right)} \]

      8. distribute-rgt-out N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot t + \color{blue}{\left(U \cdot n\right) \cdot t}} \]

      9. associate-*r* N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot t\right) + \color{blue}{\left(U \cdot n\right)} \cdot t} \]

      10. associate-*r* N/A

          \[\leadsto \sqrt{U \cdot \left(n \cdot t\right) + U \cdot \color{blue}{\left(n \cdot t\right)}} \]

      11. count-2-rev N/A

          \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      12. associate-*r* N/A

          \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      13. count-2-rev N/A

          \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(U + U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      15. lower-+.f64 N/A

          \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\left(U + U\right) \cdot \left(t \cdot \color{blue}{n}\right)} \]

      17. lower-*.f64 44.9

          \[\leadsto \sqrt{\left(U + U\right) \cdot \left(t \cdot \color{blue}{n}\right)} \]

   6. Applied rewrites 44.9%

      \[\leadsto \sqrt{\left(U + U\right) \cdot \color{blue}{\left(t \cdot n\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(t \cdot \color{blue}{n}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \color{blue}{\left(t \cdot n\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(U + U\right) \cdot t\right) \cdot \color{blue}{n}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(U + U\right) \cdot t\right) \cdot \color{blue}{n}} \]

      5. lower-*.f64 47.0

         \[\leadsto \sqrt{\left(\left(U + U\right) \cdot t\right) \cdot n} \]

   8. Applied rewrites 47.0%

      \[\leadsto \sqrt{\left(\left(U + U\right) \cdot t\right) \cdot \color{blue}{n}} \]

   if 5.00000000000000019e-90 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot t\right) + \color{blue}{U \cdot \left(n \cdot t\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot t + \color{blue}{U} \cdot \left(n \cdot t\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot t + \left(U \cdot n\right) \cdot \color{blue}{t}} \]

      4. distribute-rgt-out N/A

         \[\leadsto \sqrt{t \cdot \color{blue}{\left(U \cdot n + U \cdot n\right)}} \]

      5. count-2-rev N/A

         \[\leadsto \sqrt{t \cdot \left(2 \cdot \color{blue}{\left(U \cdot n\right)}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{t \cdot \color{blue}{\left(2 \cdot \left(U \cdot n\right)\right)}} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{t \cdot \left(2 \cdot \left(n \cdot \color{blue}{U}\right)\right)} \]

      8. associate-*l* N/A

         \[\leadsto \sqrt{t \cdot \left(\left(2 \cdot n\right) \cdot \color{blue}{U}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{t \cdot \left(\left(2 \cdot n\right) \cdot \color{blue}{U}\right)} \]

      10. count-2-rev N/A

          \[\leadsto \sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)} \]

      11. lower-+.f64 34.3

          \[\leadsto \sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)} \]

   4. Applied rewrites 34.3%

      \[\leadsto \sqrt{\color{blue}{t \cdot \left(\left(n + n\right) \cdot U\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{t \cdot \color{blue}{\left(\left(n + n\right) \cdot U\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{t \cdot \left(\left(n + n\right) \cdot \color{blue}{U}\right)} \]

      3. lift-+.f64 N/A

         \[\leadsto \sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)} \]

      4. count-2-rev N/A

         \[\leadsto \sqrt{t \cdot \left(\left(2 \cdot n\right) \cdot U\right)} \]

      5. associate-*l* N/A

         \[\leadsto \sqrt{t \cdot \left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right)} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{t \cdot \left(2 \cdot \left(U \cdot \color{blue}{n}\right)\right)} \]

      7. count-2-rev N/A

         \[\leadsto \sqrt{t \cdot \left(U \cdot n + \color{blue}{U \cdot n}\right)} \]

      8. distribute-rgt-out N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot t + \color{blue}{\left(U \cdot n\right) \cdot t}} \]

      9. associate-*r* N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot t\right) + \color{blue}{\left(U \cdot n\right)} \cdot t} \]

      10. associate-*r* N/A

          \[\leadsto \sqrt{U \cdot \left(n \cdot t\right) + U \cdot \color{blue}{\left(n \cdot t\right)}} \]

      11. count-2-rev N/A

          \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      12. associate-*r* N/A

          \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      13. count-2-rev N/A

          \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(U + U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      15. lower-+.f64 N/A

          \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\left(U + U\right) \cdot \left(t \cdot \color{blue}{n}\right)} \]

      17. lower-*.f64 32.7

          \[\leadsto \sqrt{\left(U + U\right) \cdot \left(t \cdot \color{blue}{n}\right)} \]

   6. Applied rewrites 32.7%

      \[\leadsto \sqrt{\left(U + U\right) \cdot \color{blue}{\left(t \cdot n\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 36.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 10^{-93}:\\ \;\;\;\;\sqrt{\left(\left(U + U\right) \cdot t\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot n\right) \cdot \left(t + t\right)}\\ \end{array} \end{array} \]

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*))))
      1e-93)
   (sqrt (* (* (+ U U) t) n))
   (sqrt (* (* U n) (+ t t)))))

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_)))) <= 1e-93) {
		tmp = sqrt((((U + U) * t) * n));
	} else {
		tmp = sqrt(((U * n) * (t + t)));
	}
	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 ((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))) <= 1d-93) then
        tmp = sqrt((((u + u) * t) * n))
    else
        tmp = sqrt(((u * n) * (t + t)))
    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 ((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))) <= 1e-93) {
		tmp = Math.sqrt((((U + U) * t) * n));
	} else {
		tmp = Math.sqrt(((U * n) * (t + t)));
	}
	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_)))) <= 1e-93:
		tmp = math.sqrt((((U + U) * t) * n))
	else:
		tmp = math.sqrt(((U * n) * (t + t)))
	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_)))) <= 1e-93)
		tmp = sqrt(Float64(Float64(Float64(U + U) * t) * n));
	else
		tmp = sqrt(Float64(Float64(U * n) * Float64(t + t)));
	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_)))) <= 1e-93)
		tmp = sqrt((((U + U) * t) * n));
	else
		tmp = sqrt(((U * n) * (t + t)));
	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], 1e-93], N[Sqrt[N[(N[(N[(U + U), $MachinePrecision] * t), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(U * n), $MachinePrecision] * N[(t + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.999999999999999e-94

   1. Initial program 47.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   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. count-2-rev N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot t\right) + \color{blue}{U \cdot \left(n \cdot t\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot t + \color{blue}{U} \cdot \left(n \cdot t\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot t + \left(U \cdot n\right) \cdot \color{blue}{t}} \]

      4. distribute-rgt-out N/A

         \[\leadsto \sqrt{t \cdot \color{blue}{\left(U \cdot n + U \cdot n\right)}} \]

      5. count-2-rev N/A

         \[\leadsto \sqrt{t \cdot \left(2 \cdot \color{blue}{\left(U \cdot n\right)}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{t \cdot \color{blue}{\left(2 \cdot \left(U \cdot n\right)\right)}} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{t \cdot \left(2 \cdot \left(n \cdot \color{blue}{U}\right)\right)} \]

      8. associate-*l* N/A

         \[\leadsto \sqrt{t \cdot \left(\left(2 \cdot n\right) \cdot \color{blue}{U}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{t \cdot \left(\left(2 \cdot n\right) \cdot \color{blue}{U}\right)} \]

      10. count-2-rev N/A

          \[\leadsto \sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)} \]

      11. lower-+.f64 39.6

          \[\leadsto \sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)} \]

   4. Applied rewrites 39.6%

      \[\leadsto \sqrt{\color{blue}{t \cdot \left(\left(n + n\right) \cdot U\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{t \cdot \color{blue}{\left(\left(n + n\right) \cdot U\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{t \cdot \left(\left(n + n\right) \cdot \color{blue}{U}\right)} \]

      3. lift-+.f64 N/A

         \[\leadsto \sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)} \]

      4. count-2-rev N/A

         \[\leadsto \sqrt{t \cdot \left(\left(2 \cdot n\right) \cdot U\right)} \]

      5. associate-*l* N/A

         \[\leadsto \sqrt{t \cdot \left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right)} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{t \cdot \left(2 \cdot \left(U \cdot \color{blue}{n}\right)\right)} \]

      7. count-2-rev N/A

         \[\leadsto \sqrt{t \cdot \left(U \cdot n + \color{blue}{U \cdot n}\right)} \]

      8. distribute-rgt-out N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot t + \color{blue}{\left(U \cdot n\right) \cdot t}} \]

      9. associate-*r* N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot t\right) + \color{blue}{\left(U \cdot n\right)} \cdot t} \]

      10. associate-*r* N/A

          \[\leadsto \sqrt{U \cdot \left(n \cdot t\right) + U \cdot \color{blue}{\left(n \cdot t\right)}} \]

      11. count-2-rev N/A

          \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      12. associate-*r* N/A

          \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      13. count-2-rev N/A

          \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(U + U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      15. lower-+.f64 N/A

          \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\left(U + U\right) \cdot \left(t \cdot \color{blue}{n}\right)} \]

      17. lower-*.f64 44.7

          \[\leadsto \sqrt{\left(U + U\right) \cdot \left(t \cdot \color{blue}{n}\right)} \]

   6. Applied rewrites 44.7%

      \[\leadsto \sqrt{\left(U + U\right) \cdot \color{blue}{\left(t \cdot n\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(t \cdot \color{blue}{n}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \color{blue}{\left(t \cdot n\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(U + U\right) \cdot t\right) \cdot \color{blue}{n}} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(U + U\right) \cdot t\right) \cdot \color{blue}{n}} \]

      5. lower-*.f64 46.8

         \[\leadsto \sqrt{\left(\left(U + U\right) \cdot t\right) \cdot n} \]

   8. Applied rewrites 46.8%

      \[\leadsto \sqrt{\left(\left(U + U\right) \cdot t\right) \cdot \color{blue}{n}} \]

   if 9.999999999999999e-94 < (*.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 50.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   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. count-2-rev N/A

         \[\leadsto \sqrt{U \cdot \left(n \cdot t\right) + \color{blue}{U \cdot \left(n \cdot t\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot t + \color{blue}{U} \cdot \left(n \cdot t\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot t + \left(U \cdot n\right) \cdot \color{blue}{t}} \]

      4. distribute-rgt-out N/A

         \[\leadsto \sqrt{t \cdot \color{blue}{\left(U \cdot n + U \cdot n\right)}} \]

      5. count-2-rev N/A

         \[\leadsto \sqrt{t \cdot \left(2 \cdot \color{blue}{\left(U \cdot n\right)}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{t \cdot \color{blue}{\left(2 \cdot \left(U \cdot n\right)\right)}} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{t \cdot \left(2 \cdot \left(n \cdot \color{blue}{U}\right)\right)} \]

      8. associate-*l* N/A

         \[\leadsto \sqrt{t \cdot \left(\left(2 \cdot n\right) \cdot \color{blue}{U}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{t \cdot \left(\left(2 \cdot n\right) \cdot \color{blue}{U}\right)} \]

      10. count-2-rev N/A

          \[\leadsto \sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)} \]

      11. lower-+.f64 34.4

          \[\leadsto \sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)} \]

   4. Applied rewrites 34.4%

      \[\leadsto \sqrt{\color{blue}{t \cdot \left(\left(n + n\right) \cdot U\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{t \cdot \color{blue}{\left(\left(n + n\right) \cdot U\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{t \cdot \left(\left(n + n\right) \cdot \color{blue}{U}\right)} \]

      3. lift-+.f64 N/A

         \[\leadsto \sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)} \]

      4. count-2-rev N/A

         \[\leadsto \sqrt{t \cdot \left(\left(2 \cdot n\right) \cdot U\right)} \]

      5. associate-*l* N/A

         \[\leadsto \sqrt{t \cdot \left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right)} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{t \cdot \left(2 \cdot \left(U \cdot \color{blue}{n}\right)\right)} \]

      7. count-2-rev N/A

         \[\leadsto \sqrt{t \cdot \left(U \cdot n + \color{blue}{U \cdot n}\right)} \]

      8. distribute-rgt-out N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot t + \color{blue}{\left(U \cdot n\right) \cdot t}} \]

      9. distribute-lft-out N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \color{blue}{\left(t + t\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \color{blue}{\left(t + t\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(\color{blue}{t} + t\right)} \]

      12. lower-+.f64 34.5

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(t + \color{blue}{t}\right)} \]

   6. Applied rewrites 34.5%

      \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \color{blue}{\left(t + t\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 35.8% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(U \cdot n\right) \cdot \left(t + t\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (sqrt (* (* U n) (+ t t))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt(((U * n) * (t + 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) * (t + 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) * (t + t)));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt(((U * n) * (t + t)))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(U * n) * Float64(t + t)))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt(((U * n) * (t + t)));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(U * n), $MachinePrecision] * N[(t + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 49.8%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

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. count-2-rev N/A

      \[\leadsto \sqrt{U \cdot \left(n \cdot t\right) + \color{blue}{U \cdot \left(n \cdot t\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \sqrt{\left(U \cdot n\right) \cdot t + \color{blue}{U} \cdot \left(n \cdot t\right)} \]

   3. associate-*r* N/A

      \[\leadsto \sqrt{\left(U \cdot n\right) \cdot t + \left(U \cdot n\right) \cdot \color{blue}{t}} \]

   4. distribute-rgt-out N/A

      \[\leadsto \sqrt{t \cdot \color{blue}{\left(U \cdot n + U \cdot n\right)}} \]

   5. count-2-rev N/A

      \[\leadsto \sqrt{t \cdot \left(2 \cdot \color{blue}{\left(U \cdot n\right)}\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \sqrt{t \cdot \color{blue}{\left(2 \cdot \left(U \cdot n\right)\right)}} \]

   7. *-commutative N/A

      \[\leadsto \sqrt{t \cdot \left(2 \cdot \left(n \cdot \color{blue}{U}\right)\right)} \]

   8. associate-*l* N/A

      \[\leadsto \sqrt{t \cdot \left(\left(2 \cdot n\right) \cdot \color{blue}{U}\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \sqrt{t \cdot \left(\left(2 \cdot n\right) \cdot \color{blue}{U}\right)} \]

   10. count-2-rev N/A

       \[\leadsto \sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)} \]

   11. lower-+.f64 35.8

       \[\leadsto \sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)} \]

4. Applied rewrites 35.8%

   \[\leadsto \sqrt{\color{blue}{t \cdot \left(\left(n + n\right) \cdot U\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{t \cdot \color{blue}{\left(\left(n + n\right) \cdot U\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \sqrt{t \cdot \left(\left(n + n\right) \cdot \color{blue}{U}\right)} \]

   3. lift-+.f64 N/A

      \[\leadsto \sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)} \]

   4. count-2-rev N/A

      \[\leadsto \sqrt{t \cdot \left(\left(2 \cdot n\right) \cdot U\right)} \]

   5. associate-*l* N/A

      \[\leadsto \sqrt{t \cdot \left(2 \cdot \color{blue}{\left(n \cdot U\right)}\right)} \]

   6. *-commutative N/A

      \[\leadsto \sqrt{t \cdot \left(2 \cdot \left(U \cdot \color{blue}{n}\right)\right)} \]

   7. count-2-rev N/A

      \[\leadsto \sqrt{t \cdot \left(U \cdot n + \color{blue}{U \cdot n}\right)} \]

   8. distribute-rgt-out N/A

      \[\leadsto \sqrt{\left(U \cdot n\right) \cdot t + \color{blue}{\left(U \cdot n\right) \cdot t}} \]

   9. distribute-lft-out N/A

      \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \color{blue}{\left(t + t\right)}} \]

   10. lower-*.f64 N/A

       \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \color{blue}{\left(t + t\right)}} \]

   11. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(\color{blue}{t} + t\right)} \]

   12. lower-+.f64 35.8

       \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(t + \color{blue}{t}\right)} \]

6. Applied rewrites 35.8%

   \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \color{blue}{\left(t + t\right)}} \]
7. Add Preprocessing

Alternative 19: 35.8% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (sqrt (* t (* (+ n n) U))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((t * ((n + n) * U)));
}

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((t * ((n + n) * u)))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((t * ((n + n) * U)));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((t * ((n + n) * U)))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(t * Float64(Float64(n + n) * U)))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((t * ((n + n) * U)));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(t * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 49.8%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

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. count-2-rev N/A

      \[\leadsto \sqrt{U \cdot \left(n \cdot t\right) + \color{blue}{U \cdot \left(n \cdot t\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \sqrt{\left(U \cdot n\right) \cdot t + \color{blue}{U} \cdot \left(n \cdot t\right)} \]

   3. associate-*r* N/A

      \[\leadsto \sqrt{\left(U \cdot n\right) \cdot t + \left(U \cdot n\right) \cdot \color{blue}{t}} \]

   4. distribute-rgt-out N/A

      \[\leadsto \sqrt{t \cdot \color{blue}{\left(U \cdot n + U \cdot n\right)}} \]

   5. count-2-rev N/A

      \[\leadsto \sqrt{t \cdot \left(2 \cdot \color{blue}{\left(U \cdot n\right)}\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \sqrt{t \cdot \color{blue}{\left(2 \cdot \left(U \cdot n\right)\right)}} \]

   7. *-commutative N/A

      \[\leadsto \sqrt{t \cdot \left(2 \cdot \left(n \cdot \color{blue}{U}\right)\right)} \]

   8. associate-*l* N/A

      \[\leadsto \sqrt{t \cdot \left(\left(2 \cdot n\right) \cdot \color{blue}{U}\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \sqrt{t \cdot \left(\left(2 \cdot n\right) \cdot \color{blue}{U}\right)} \]

   10. count-2-rev N/A

       \[\leadsto \sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)} \]

   11. lower-+.f64 35.8

       \[\leadsto \sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)} \]

4. Applied rewrites 35.8%

   \[\leadsto \sqrt{\color{blue}{t \cdot \left(\left(n + n\right) \cdot U\right)}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025130 
(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*))))))