Toniolo and Linder, Equation (13)

Percentage Accurate: 50.6% → 68.4%

Time: 10.4s

Alternatives: 16

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 68.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := \frac{\left|\ell\right|}{Om}\\ t_3 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{\left|\ell\right| \cdot \left|\ell\right|}{Om}\right) - \left(n \cdot {t\_2}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 10^{-286}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(\mathsf{fma}\left(t\_2, \mathsf{fma}\left(U* \cdot n, t\_2, -2 \cdot \left|\ell\right|\right), t\right) \cdot U\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(t\_2, \left(t\_2 \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(t\_2 \cdot \left|\ell\right|, -2, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \frac{2 + \frac{n \cdot \left(U - U*\right)}{Om}}{Om}\right)\right)}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2 (/ (fabs l) Om))
        (t_3
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* (fabs l) (fabs l)) Om)))
           (* (* n (pow t_2 2.0)) (- U U*))))))
   (if (<= t_3 1e-286)
     (sqrt (* (+ n n) (* (fma t_2 (fma (* U* n) t_2 (* -2.0 (fabs l))) t) U)))
     (if (<= t_3 INFINITY)
       (sqrt
        (* t_1 (fma t_2 (* (* t_2 n) (- U* U)) (fma (* t_2 (fabs l)) -2.0 t))))
       (*
        (fabs l)
        (sqrt (* -2.0 (* U (* n (/ (+ 2.0 (/ (* n (- U U*)) Om)) Om))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = fabs(l) / Om;
	double t_3 = t_1 * ((t - (2.0 * ((fabs(l) * fabs(l)) / Om))) - ((n * pow(t_2, 2.0)) * (U - U_42_)));
	double tmp;
	if (t_3 <= 1e-286) {
		tmp = sqrt(((n + n) * (fma(t_2, fma((U_42_ * n), t_2, (-2.0 * fabs(l))), t) * U)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * fma(t_2, ((t_2 * n) * (U_42_ - U)), fma((t_2 * fabs(l)), -2.0, t))));
	} else {
		tmp = fabs(l) * sqrt((-2.0 * (U * (n * ((2.0 + ((n * (U - U_42_)) / Om)) / Om)))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(abs(l) / Om)
	t_3 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(abs(l) * abs(l)) / Om))) - Float64(Float64(n * (t_2 ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_3 <= 1e-286)
		tmp = sqrt(Float64(Float64(n + n) * Float64(fma(t_2, fma(Float64(U_42_ * n), t_2, Float64(-2.0 * abs(l))), t) * U)));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_1 * fma(t_2, Float64(Float64(t_2 * n) * Float64(U_42_ - U)), fma(Float64(t_2 * abs(l)), -2.0, t))));
	else
		tmp = Float64(abs(l) * sqrt(Float64(-2.0 * Float64(U * Float64(n * Float64(Float64(2.0 + Float64(Float64(n * Float64(U - U_42_)) / Om)) / Om))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[l], $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 1e-286], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(N[(t$95$2 * N[(N[(U$42$ * n), $MachinePrecision] * t$95$2 + N[(-2.0 * N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$1 * N[(t$95$2 * N[(N[(t$95$2 * n), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 * N[Abs[l], $MachinePrecision]), $MachinePrecision] * -2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(-2.0 * N[(U * N[(n * N[(N[(2.0 + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.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*)))) < 1.00000000000000005e-286

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 52.2%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 52.2%

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.9%

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

   5. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.9%

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

   7. Applied rewrites 56.2%

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

   8. Taylor expanded in U around 0

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

   10. Applied rewrites 57.3%

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

   if 1.00000000000000005e-286 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 50.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 52.2%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 52.2%

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.9%

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

   5. Applied rewrites 55.9%

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

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

   1. Initial program 50.6%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 15.2%

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

   4. Applied rewrites 15.2%

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

   5. Taylor expanded in Om around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 16.9%

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

   7. Applied rewrites 16.9%

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

Alternative 2: 68.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := -2 \cdot \left|\ell\right|\\ t_2 := \frac{\left|\ell\right|}{Om}\\ t_3 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\left|\ell\right| \cdot \left|\ell\right|}{Om}\right) - \left(n \cdot {t\_2}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 10^{-286}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(\mathsf{fma}\left(t\_2, \mathsf{fma}\left(U* \cdot n, t\_2, t\_1\right), t\right) \cdot U\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t\_2 \cdot n, U* - U, t\_1\right), t\_2, t\right) \cdot \left(U \cdot \left(n + n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \frac{2 + \frac{n \cdot \left(U - U*\right)}{Om}}{Om}\right)\right)}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* -2.0 (fabs l)))
        (t_2 (/ (fabs l) Om))
        (t_3
         (*
          (* (* 2.0 n) U)
          (-
           (- t (* 2.0 (/ (* (fabs l) (fabs l)) Om)))
           (* (* n (pow t_2 2.0)) (- U U*))))))
   (if (<= t_3 1e-286)
     (sqrt (* (+ n n) (* (fma t_2 (fma (* U* n) t_2 t_1) t) U)))
     (if (<= t_3 INFINITY)
       (sqrt (* (fma (fma (* t_2 n) (- U* U) t_1) t_2 t) (* U (+ n n))))
       (*
        (fabs l)
        (sqrt (* -2.0 (* U (* n (/ (+ 2.0 (/ (* n (- U U*)) Om)) Om))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = -2.0 * fabs(l);
	double t_2 = fabs(l) / Om;
	double t_3 = ((2.0 * n) * U) * ((t - (2.0 * ((fabs(l) * fabs(l)) / Om))) - ((n * pow(t_2, 2.0)) * (U - U_42_)));
	double tmp;
	if (t_3 <= 1e-286) {
		tmp = sqrt(((n + n) * (fma(t_2, fma((U_42_ * n), t_2, t_1), t) * U)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((fma(fma((t_2 * n), (U_42_ - U), t_1), t_2, t) * (U * (n + n))));
	} else {
		tmp = fabs(l) * sqrt((-2.0 * (U * (n * ((2.0 + ((n * (U - U_42_)) / Om)) / Om)))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(-2.0 * abs(l))
	t_2 = Float64(abs(l) / Om)
	t_3 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(abs(l) * abs(l)) / Om))) - Float64(Float64(n * (t_2 ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_3 <= 1e-286)
		tmp = sqrt(Float64(Float64(n + n) * Float64(fma(t_2, fma(Float64(U_42_ * n), t_2, t_1), t) * U)));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(fma(fma(Float64(t_2 * n), Float64(U_42_ - U), t_1), t_2, t) * Float64(U * Float64(n + n))));
	else
		tmp = Float64(abs(l) * sqrt(Float64(-2.0 * Float64(U * Float64(n * Float64(Float64(2.0 + Float64(Float64(n * Float64(U - U_42_)) / Om)) / Om))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(-2.0 * N[Abs[l], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[l], $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 1e-286], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(N[(t$95$2 * N[(N[(U$42$ * n), $MachinePrecision] * t$95$2 + t$95$1), $MachinePrecision] + t), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(N[(N[(N[(t$95$2 * n), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision] + t$95$1), $MachinePrecision] * t$95$2 + t), $MachinePrecision] * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(-2.0 * N[(U * N[(n * N[(N[(2.0 + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.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*)))) < 1.00000000000000005e-286

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 52.2%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 52.2%

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.9%

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

   5. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.9%

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

   7. Applied rewrites 56.2%

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

   8. Taylor expanded in U around 0

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

   10. Applied rewrites 57.3%

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

   if 1.00000000000000005e-286 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 50.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 52.2%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 52.2%

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.9%

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

   5. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.9%

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

   7. Applied rewrites 56.2%

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

   9. Applied rewrites 60.0%

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

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

   1. Initial program 50.6%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 15.2%

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

   4. Applied rewrites 15.2%

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

   5. Taylor expanded in Om around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 16.9%

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

   7. Applied rewrites 16.9%

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

Alternative 3: 61.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 50.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 52.2%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 52.2%

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.9%

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

   5. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.9%

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

   7. Applied rewrites 56.2%

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

   8. Taylor expanded in U around 0

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

   10. Applied rewrites 57.3%

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

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

   1. Initial program 50.6%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 15.2%

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

   4. Applied rewrites 15.2%

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

   5. Taylor expanded in Om around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 16.9%

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

   7. Applied rewrites 16.9%

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

Alternative 4: 60.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|\ell\right|}{Om}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\left|\ell\right| \cdot \left|\ell\right|}{Om}\right) - \left(n \cdot {t\_1}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_2 \leq 10^{+20}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(\mathsf{fma}\left(t\_1, -2 \cdot \left|\ell\right|, t\right) \cdot U\right)}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(\frac{U - U*}{Om}, \frac{n}{Om}, \frac{2}{Om}\right)\right)\right)}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (fabs l) Om))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* (fabs l) (fabs l)) Om)))
            (* (* n (pow t_1 2.0)) (- U U*)))))))
   (if (<= t_2 1e+20)
     (sqrt (* (+ n n) (* (fma t_1 (* -2.0 (fabs l)) t) U)))
     (if (<= t_2 2e+148)
       (sqrt (* (* U (+ n n)) t))
       (*
        (fabs l)
        (sqrt
         (* -2.0 (* U (* n (fma (/ (- U U*) Om) (/ n Om) (/ 2.0 Om)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = fabs(l) / Om;
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((fabs(l) * fabs(l)) / Om))) - ((n * pow(t_1, 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 1e+20) {
		tmp = sqrt(((n + n) * (fma(t_1, (-2.0 * fabs(l)), t) * U)));
	} else if (t_2 <= 2e+148) {
		tmp = sqrt(((U * (n + n)) * t));
	} else {
		tmp = fabs(l) * sqrt((-2.0 * (U * (n * fma(((U - U_42_) / Om), (n / Om), (2.0 / Om))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(abs(l) / Om)
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(abs(l) * abs(l)) / Om))) - Float64(Float64(n * (t_1 ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_2 <= 1e+20)
		tmp = sqrt(Float64(Float64(n + n) * Float64(fma(t_1, Float64(-2.0 * abs(l)), t) * U)));
	elseif (t_2 <= 2e+148)
		tmp = sqrt(Float64(Float64(U * Float64(n + n)) * t));
	else
		tmp = Float64(abs(l) * sqrt(Float64(-2.0 * Float64(U * Float64(n * fma(Float64(Float64(U - U_42_) / Om), Float64(n / Om), Float64(2.0 / Om)))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[Abs[l], $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 1e+20], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(N[(t$95$1 * N[(-2.0 * N[Abs[l], $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 2e+148], N[Sqrt[N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(-2.0 * N[(U * N[(n * N[(N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] * N[(n / Om), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1e20

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 52.2%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 52.2%

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.9%

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

   5. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.9%

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

   7. Applied rewrites 56.2%

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

   8. Taylor expanded in n around 0

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

      1. lower-*.f64 47.1%

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

   10. Applied rewrites 47.1%

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

   if 1e20 < (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*))))) < 2.0000000000000001e148

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 36.3%

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

   4. Applied rewrites 36.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. count-2 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-+.f64 N/A

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

      9. count-2 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 36.1%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 36.1%

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

      19. lift-*.f64 N/A

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

      20. count-2-rev N/A

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

      21. lift-+.f64 36.1%

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

   6. Applied rewrites 36.1%

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

   if 2.0000000000000001e148 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 50.6%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 15.2%

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

   4. Applied rewrites 15.2%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. mult-flip-rev N/A

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

      14. lower-/.f64 17.0%

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

   6. Applied rewrites 17.0%

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

Alternative 5: 60.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\frac{\ell}{Om}, \frac{U* \cdot \left(\ell \cdot n\right)}{Om}, t\right) \cdot U\\ \mathbf{if}\;n \leq -1.75 \cdot 10^{-231}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot t\_1}\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{+20}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-4, \frac{\left(\ell \cdot n\right) \cdot \left(\ell \cdot U\right)}{Om}, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{t\_1}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (fma (/ l Om) (/ (* U* (* l n)) Om) t) U)))
   (if (<= n -1.75e-231)
     (sqrt (* (+ n n) t_1))
     (if (<= n 1.1e+20)
       (sqrt (fma -4.0 (/ (* (* l n) (* l U)) Om) (* 2.0 (* U (* n t)))))
       (* (sqrt (+ n n)) (sqrt t_1))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = fma((l / Om), ((U_42_ * (l * n)) / Om), t) * U;
	double tmp;
	if (n <= -1.75e-231) {
		tmp = sqrt(((n + n) * t_1));
	} else if (n <= 1.1e+20) {
		tmp = sqrt(fma(-4.0, (((l * n) * (l * U)) / Om), (2.0 * (U * (n * t)))));
	} else {
		tmp = sqrt((n + n)) * sqrt(t_1);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(fma(Float64(l / Om), Float64(Float64(U_42_ * Float64(l * n)) / Om), t) * U)
	tmp = 0.0
	if (n <= -1.75e-231)
		tmp = sqrt(Float64(Float64(n + n) * t_1));
	elseif (n <= 1.1e+20)
		tmp = sqrt(fma(-4.0, Float64(Float64(Float64(l * n) * Float64(l * U)) / Om), Float64(2.0 * Float64(U * Float64(n * t)))));
	else
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(t_1));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(l / Om), $MachinePrecision] * N[(N[(U$42$ * N[(l * n), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * U), $MachinePrecision]}, If[LessEqual[n, -1.75e-231], N[Sqrt[N[(N[(n + n), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 1.1e+20], N[Sqrt[N[(-4.0 * N[(N[(N[(l * n), $MachinePrecision] * N[(l * U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.7500000000000001e-231

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 52.2%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 52.2%

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.9%

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

   5. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.9%

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

   7. Applied rewrites 56.2%

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

   8. Taylor expanded in U* around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 50.3%

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

   10. Applied rewrites 50.3%

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

   if -1.7500000000000001e-231 < n < 1.1e20

   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 Om around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 43.3%

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

   4. Applied rewrites 43.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 47.0%

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

   6. Applied rewrites 47.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 48.5%

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

   8. Applied rewrites 48.5%

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

   if 1.1e20 < n

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 52.2%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 52.2%

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.9%

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

   5. Applied rewrites 55.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. sqrt-prod N/A

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lift-+.f64 N/A

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

      11. lower-unsound-sqrt.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 33.6%

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

   7. Applied rewrites 32.7%

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

   8. Taylor expanded in U* around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 29.0%

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

   10. Applied rewrites 29.0%

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

Alternative 6: 57.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|\ell\right|}{Om}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\left|\ell\right| \cdot \left|\ell\right|}{Om}\right) - \left(n \cdot {t\_1}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_2 \leq 10^{+20}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(\mathsf{fma}\left(t\_1, -2 \cdot \left|\ell\right|, t\right) \cdot U\right)}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \frac{2 + \frac{n \cdot \left(U - U*\right)}{Om}}{Om}\right)\right)}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (fabs l) Om))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* (fabs l) (fabs l)) Om)))
            (* (* n (pow t_1 2.0)) (- U U*)))))))
   (if (<= t_2 1e+20)
     (sqrt (* (+ n n) (* (fma t_1 (* -2.0 (fabs l)) t) U)))
     (if (<= t_2 2e+148)
       (sqrt (* (* U (+ n n)) t))
       (*
        (fabs l)
        (sqrt (* -2.0 (* U (* n (/ (+ 2.0 (/ (* n (- U U*)) Om)) Om))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = fabs(l) / Om;
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((fabs(l) * fabs(l)) / Om))) - ((n * pow(t_1, 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 1e+20) {
		tmp = sqrt(((n + n) * (fma(t_1, (-2.0 * fabs(l)), t) * U)));
	} else if (t_2 <= 2e+148) {
		tmp = sqrt(((U * (n + n)) * t));
	} else {
		tmp = fabs(l) * sqrt((-2.0 * (U * (n * ((2.0 + ((n * (U - U_42_)) / Om)) / Om)))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(abs(l) / Om)
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(abs(l) * abs(l)) / Om))) - Float64(Float64(n * (t_1 ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_2 <= 1e+20)
		tmp = sqrt(Float64(Float64(n + n) * Float64(fma(t_1, Float64(-2.0 * abs(l)), t) * U)));
	elseif (t_2 <= 2e+148)
		tmp = sqrt(Float64(Float64(U * Float64(n + n)) * t));
	else
		tmp = Float64(abs(l) * sqrt(Float64(-2.0 * Float64(U * Float64(n * Float64(Float64(2.0 + Float64(Float64(n * Float64(U - U_42_)) / Om)) / Om))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[Abs[l], $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 1e+20], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(N[(t$95$1 * N[(-2.0 * N[Abs[l], $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 2e+148], N[Sqrt[N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(-2.0 * N[(U * N[(n * N[(N[(2.0 + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1e20

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 52.2%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 52.2%

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.9%

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

   5. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.9%

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

   7. Applied rewrites 56.2%

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

   8. Taylor expanded in n around 0

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

      1. lower-*.f64 47.1%

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

   10. Applied rewrites 47.1%

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

   if 1e20 < (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*))))) < 2.0000000000000001e148

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 36.3%

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

   4. Applied rewrites 36.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. count-2 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-+.f64 N/A

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

      9. count-2 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 36.1%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 36.1%

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

      19. lift-*.f64 N/A

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

      20. count-2-rev N/A

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

      21. lift-+.f64 36.1%

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

   6. Applied rewrites 36.1%

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

   if 2.0000000000000001e148 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 50.6%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 15.2%

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

   4. Applied rewrites 15.2%

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

   5. Taylor expanded in Om around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 16.9%

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

   7. Applied rewrites 16.9%

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

Alternative 7: 55.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_1 := \sqrt{\left(n + n\right) \cdot \left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{U* \cdot \left(\ell \cdot n\right)}{Om}, t\right) \cdot U\right)}\\ \mathbf{if}\;n \leq -1.75 \cdot 10^{-231}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{+20}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-4, \frac{\left(\ell \cdot n\right) \cdot \left(\ell \cdot U\right)}{Om}, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* (+ n n) (* (fma (/ l Om) (/ (* U* (* l n)) Om) t) U)))))
   (if (<= n -1.75e-231)
     t_1
     (if (<= n 1.1e+20)
       (sqrt (fma -4.0 (/ (* (* l n) (* l U)) Om) (* 2.0 (* U (* n t)))))
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((n + n) * (fma((l / Om), ((U_42_ * (l * n)) / Om), t) * U)));
	double tmp;
	if (n <= -1.75e-231) {
		tmp = t_1;
	} else if (n <= 1.1e+20) {
		tmp = sqrt(fma(-4.0, (((l * n) * (l * U)) / Om), (2.0 * (U * (n * t)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(n + n) * Float64(fma(Float64(l / Om), Float64(Float64(U_42_ * Float64(l * n)) / Om), t) * U)))
	tmp = 0.0
	if (n <= -1.75e-231)
		tmp = t_1;
	elseif (n <= 1.1e+20)
		tmp = sqrt(fma(-4.0, Float64(Float64(Float64(l * n) * Float64(l * U)) / Om), Float64(2.0 * Float64(U * Float64(n * t)))));
	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[(N[(N[(l / Om), $MachinePrecision] * N[(N[(U$42$ * N[(l * n), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -1.75e-231], t$95$1, If[LessEqual[n, 1.1e+20], N[Sqrt[N[(-4.0 * N[(N[(N[(l * n), $MachinePrecision] * N[(l * U), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.7500000000000001e-231 or 1.1e20 < n

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 52.2%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 52.2%

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.9%

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

   5. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.9%

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

   7. Applied rewrites 56.2%

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

   8. Taylor expanded in U* around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 50.3%

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

   10. Applied rewrites 50.3%

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

   if -1.7500000000000001e-231 < n < 1.1e20

   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 Om around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 43.3%

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

   4. Applied rewrites 43.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 47.0%

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

   6. Applied rewrites 47.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 48.5%

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

   8. Applied rewrites 48.5%

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

Alternative 8: 50.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|\ell\right|}{Om}\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\left|\ell\right| \cdot \left|\ell\right|}{Om}\right) - \left(n \cdot {t\_1}^{2}\right) \cdot \left(U - U*\right)\right)}\\ t_3 := \sqrt{\left(n + n\right) \cdot \left(\mathsf{fma}\left(t\_1, -2 \cdot \left|\ell\right|, t\right) \cdot U\right)}\\ \mathbf{if}\;t\_2 \leq 10^{+20}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \frac{2}{Om}\right)\right)}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (fabs l) Om))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* (fabs l) (fabs l)) Om)))
            (* (* n (pow t_1 2.0)) (- U U*))))))
        (t_3 (sqrt (* (+ n n) (* (fma t_1 (* -2.0 (fabs l)) t) U)))))
   (if (<= t_2 1e+20)
     t_3
     (if (<= t_2 5e+144)
       (sqrt (* (* U (+ n n)) t))
       (if (<= t_2 INFINITY)
         t_3
         (* (fabs l) (sqrt (* -2.0 (* U (* n (/ 2.0 Om)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = fabs(l) / Om;
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((fabs(l) * fabs(l)) / Om))) - ((n * pow(t_1, 2.0)) * (U - U_42_)))));
	double t_3 = sqrt(((n + n) * (fma(t_1, (-2.0 * fabs(l)), t) * U)));
	double tmp;
	if (t_2 <= 1e+20) {
		tmp = t_3;
	} else if (t_2 <= 5e+144) {
		tmp = sqrt(((U * (n + n)) * t));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = fabs(l) * sqrt((-2.0 * (U * (n * (2.0 / Om)))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(abs(l) / Om)
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(abs(l) * abs(l)) / Om))) - Float64(Float64(n * (t_1 ^ 2.0)) * Float64(U - U_42_)))))
	t_3 = sqrt(Float64(Float64(n + n) * Float64(fma(t_1, Float64(-2.0 * abs(l)), t) * U)))
	tmp = 0.0
	if (t_2 <= 1e+20)
		tmp = t_3;
	elseif (t_2 <= 5e+144)
		tmp = sqrt(Float64(Float64(U * Float64(n + n)) * t));
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(abs(l) * sqrt(Float64(-2.0 * Float64(U * Float64(n * Float64(2.0 / Om))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[Abs[l], $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(N[(t$95$1 * N[(-2.0 * N[Abs[l], $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 1e+20], t$95$3, If[LessEqual[t$95$2, 5e+144], N[Sqrt[N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$3, N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(-2.0 * N[(U * N[(n * N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (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*))))) < 1e20 or 4.9999999999999999e144 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 50.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 52.2%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 52.2%

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.9%

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

   5. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.9%

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

   7. Applied rewrites 56.2%

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

   8. Taylor expanded in n around 0

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

      1. lower-*.f64 47.1%

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

   10. Applied rewrites 47.1%

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

   if 1e20 < (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*))))) < 4.9999999999999999e144

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 36.3%

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

   4. Applied rewrites 36.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. count-2 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-+.f64 N/A

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

      9. count-2 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 36.1%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 36.1%

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

      19. lift-*.f64 N/A

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

      20. count-2-rev N/A

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

      21. lift-+.f64 36.1%

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

   6. Applied rewrites 36.1%

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

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

   1. Initial program 50.6%

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

   2. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 15.2%

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

   4. Applied rewrites 15.2%

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

   5. Taylor expanded in n around 0

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

      1. lower-/.f64 10.1%

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

   7. Applied rewrites 10.1%

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

Alternative 9: 50.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\frac{\ell}{Om}, -2 \cdot \ell, t\right) \cdot U\\ \mathbf{if}\;n \leq -3.8 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{t\_1}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (fma (/ l Om) (* -2.0 l) t) U)))
   (if (<= n -3.8e-303) (sqrt (* (+ n n) t_1)) (* (sqrt (+ n n)) (sqrt t_1)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = fma((l / Om), (-2.0 * l), t) * U;
	double tmp;
	if (n <= -3.8e-303) {
		tmp = sqrt(((n + n) * t_1));
	} else {
		tmp = sqrt((n + n)) * sqrt(t_1);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(fma(Float64(l / Om), Float64(-2.0 * l), t) * U)
	tmp = 0.0
	if (n <= -3.8e-303)
		tmp = sqrt(Float64(Float64(n + n) * t_1));
	else
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(t_1));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(l / Om), $MachinePrecision] * N[(-2.0 * l), $MachinePrecision] + t), $MachinePrecision] * U), $MachinePrecision]}, If[LessEqual[n, -3.8e-303], N[Sqrt[N[(N[(n + n), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < -3.80000000000000009e-303

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   3. Applied rewrites 52.2%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 52.2%

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.9%

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

   5. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.9%

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

   7. Applied rewrites 56.2%

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

   8. Taylor expanded in n around 0

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

      1. lower-*.f64 47.1%

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

   10. Applied rewrites 47.1%

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

   if -3.80000000000000009e-303 < n

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right)}\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      8. sub-negate-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \color{blue}{\left(U* - U\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} \cdot \left(U* - U\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U* - U\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      11. lift-pow.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}} \cdot n\right) \cdot \left(U* - U\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      12. unpow2 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot n\right) \cdot \left(U* - U\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      13. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \cdot \left(U* - U\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      14. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]

   3. Applied rewrites 52.2%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)}} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \color{blue}{-2 \cdot \frac{\ell \cdot \ell}{Om} + t}\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \color{blue}{\frac{\ell \cdot \ell}{Om} \cdot -2} + t\right)} \]

      3. lower-fma.f64 52.2%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \color{blue}{\mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right)}\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\color{blue}{\frac{\ell \cdot \ell}{Om}}, -2, t\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\frac{\color{blue}{\ell \cdot \ell}}{Om}, -2, t\right)\right)} \]

      6. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\color{blue}{\ell \cdot \frac{\ell}{Om}}, -2, t\right)\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \color{blue}{\frac{\ell}{Om}}, -2, t\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\color{blue}{\frac{\ell}{Om} \cdot \ell}, -2, t\right)\right)} \]

      9. lower-*.f64 55.9%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\color{blue}{\frac{\ell}{Om} \cdot \ell}, -2, t\right)\right)} \]

   5. Applied rewrites 55.9%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \color{blue}{\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)}\right)} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)\right)}} \]

      5. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)}} \]

      6. lower-unsound-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)}} \]

      7. lower-unsound-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot n}} \cdot \sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot n}} \cdot \sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)} \]

      9. count-2-rev N/A

         \[\leadsto \sqrt{\color{blue}{n + n}} \cdot \sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \sqrt{\color{blue}{n + n}} \cdot \sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)} \]

      11. lower-unsound-sqrt.f64 N/A

          \[\leadsto \sqrt{n + n} \cdot \color{blue}{\sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)}} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right) \cdot U}} \]

      13. lower-*.f64 33.6%

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right) \cdot U}} \]

   7. Applied rewrites 32.7%

      \[\leadsto \color{blue}{\sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\left(U* - U\right) \cdot n, \frac{\ell}{Om}, -2 \cdot \ell\right), t\right) \cdot U}} \]

   8. Taylor expanded in n around 0

      \[\leadsto \sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{-2 \cdot \ell}, t\right) \cdot U} \]
   9. Step-by-step derivation

      1. lower-*.f64 28.0%

         \[\leadsto \sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(\frac{\ell}{Om}, -2 \cdot \color{blue}{\ell}, t\right) \cdot U} \]

   10. Applied rewrites 28.0%

       \[\leadsto \sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{-2 \cdot \ell}, t\right) \cdot U} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 45.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\left|\ell\right| \cdot \left|\ell\right|}{Om}\right) - \left(n \cdot {\left(\frac{\left|\ell\right|}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \frac{2}{Om}\right)\right)}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* (fabs l) (fabs l)) Om)))
            (* (* n (pow (/ (fabs l) Om) 2.0)) (- U U*)))))))
   (if (<= t_1 0.0)
     (* (sqrt n) (sqrt (* 2.0 (* U t))))
     (if (<= t_1 2e+148)
       (sqrt (* (* U (+ n n)) t))
       (* (fabs l) (sqrt (* -2.0 (* U (* n (/ 2.0 Om))))))))))

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 * ((fabs(l) * fabs(l)) / Om))) - ((n * pow((fabs(l) / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt(n) * sqrt((2.0 * (U * t)));
	} else if (t_1 <= 2e+148) {
		tmp = sqrt(((U * (n + n)) * t));
	} else {
		tmp = fabs(l) * sqrt((-2.0 * (U * (n * (2.0 / Om)))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((abs(l) * abs(l)) / om))) - ((n * ((abs(l) / om) ** 2.0d0)) * (u - u_42)))))
    if (t_1 <= 0.0d0) then
        tmp = sqrt(n) * sqrt((2.0d0 * (u * t)))
    else if (t_1 <= 2d+148) then
        tmp = sqrt(((u * (n + n)) * t))
    else
        tmp = abs(l) * sqrt(((-2.0d0) * (u * (n * (2.0d0 / om)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((Math.abs(l) * Math.abs(l)) / Om))) - ((n * Math.pow((Math.abs(l) / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.sqrt(n) * Math.sqrt((2.0 * (U * t)));
	} else if (t_1 <= 2e+148) {
		tmp = Math.sqrt(((U * (n + n)) * t));
	} else {
		tmp = Math.abs(l) * Math.sqrt((-2.0 * (U * (n * (2.0 / Om)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((math.fabs(l) * math.fabs(l)) / Om))) - ((n * math.pow((math.fabs(l) / Om), 2.0)) * (U - U_42_)))))
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.sqrt(n) * math.sqrt((2.0 * (U * t)))
	elif t_1 <= 2e+148:
		tmp = math.sqrt(((U * (n + n)) * t))
	else:
		tmp = math.fabs(l) * math.sqrt((-2.0 * (U * (n * (2.0 / Om)))))
	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(abs(l) * abs(l)) / Om))) - Float64(Float64(n * (Float64(abs(l) / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(sqrt(n) * sqrt(Float64(2.0 * Float64(U * t))));
	elseif (t_1 <= 2e+148)
		tmp = sqrt(Float64(Float64(U * Float64(n + n)) * t));
	else
		tmp = Float64(abs(l) * sqrt(Float64(-2.0 * Float64(U * Float64(n * Float64(2.0 / Om))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((abs(l) * abs(l)) / Om))) - ((n * ((abs(l) / Om) ^ 2.0)) * (U - U_42_)))));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = sqrt(n) * sqrt((2.0 * (U * t)));
	elseif (t_1 <= 2e+148)
		tmp = sqrt(((U * (n + n)) * t));
	else
		tmp = abs(l) * sqrt((-2.0 * (U * (n * (2.0 / Om)))));
	end
	tmp_2 = 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[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(N[Abs[l], $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+148], N[Sqrt[N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(-2.0 * N[(U * N[(n * N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 0.0

   1. Initial program 50.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 24.9%

      \[\leadsto \color{blue}{\sqrt{n} \cdot \sqrt{\left(U + U\right) \cdot \mathsf{fma}\left(\left(U* - U\right) \cdot n, \ell \cdot \frac{\ell}{Om \cdot Om}, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)}} \]

   4. Taylor expanded in t around inf

      \[\leadsto \sqrt{n} \cdot \sqrt{\color{blue}{2 \cdot \left(U \cdot t\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{2 \cdot \color{blue}{\left(U \cdot t\right)}} \]

      2. lower-*.f64 20.8%

         \[\leadsto \sqrt{n} \cdot \sqrt{2 \cdot \left(U \cdot \color{blue}{t}\right)} \]

   6. Applied rewrites 20.8%

      \[\leadsto \sqrt{n} \cdot \sqrt{\color{blue}{2 \cdot \left(U \cdot t\right)}} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2.0000000000000001e148

   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. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 36.3%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 36.3%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      4. count-2 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(n \cdot \color{blue}{t}\right)} \]

      7. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(U + U\right) \cdot n\right) \cdot \color{blue}{t}} \]

      8. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(\left(U + U\right) \cdot n\right) \cdot t} \]

      9. count-2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot U\right) \cdot n\right) \cdot t} \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(U \cdot 2\right) \cdot n\right) \cdot t} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      15. lower-*.f64 36.1%

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      16. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      17. *-commutative N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      18. lower-*.f64 36.1%

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      19. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      20. count-2-rev N/A

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      21. lift-+.f64 36.1%

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   6. Applied rewrites 36.1%

      \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot \color{blue}{t}} \]

   if 2.0000000000000001e148 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 50.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      11. lower-pow.f64 15.2%

          \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

   4. Applied rewrites 15.2%

      \[\leadsto \color{blue}{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]

   5. Taylor expanded in n around 0

      \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \frac{2}{Om}\right)\right)} \]
   6. Step-by-step derivation

      1. lower-/.f64 10.1%

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \frac{2}{Om}\right)\right)} \]

   7. Applied rewrites 10.1%

      \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \frac{2}{Om}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 44.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\left|\ell\right| \cdot \left|\ell\right|}{Om}\right) - \left(n \cdot {\left(\frac{\left|\ell\right|}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \sqrt{-4 \cdot \frac{U \cdot n}{Om}}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* (fabs l) (fabs l)) Om)))
            (* (* n (pow (/ (fabs l) Om) 2.0)) (- U U*)))))))
   (if (<= t_1 0.0)
     (* (sqrt n) (sqrt (* 2.0 (* U t))))
     (if (<= t_1 2e+148)
       (sqrt (* (* U (+ n n)) t))
       (* (fabs l) (sqrt (* -4.0 (/ (* U n) Om))))))))

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 * ((fabs(l) * fabs(l)) / Om))) - ((n * pow((fabs(l) / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt(n) * sqrt((2.0 * (U * t)));
	} else if (t_1 <= 2e+148) {
		tmp = sqrt(((U * (n + n)) * t));
	} else {
		tmp = fabs(l) * sqrt((-4.0 * ((U * n) / Om)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((abs(l) * abs(l)) / om))) - ((n * ((abs(l) / om) ** 2.0d0)) * (u - u_42)))))
    if (t_1 <= 0.0d0) then
        tmp = sqrt(n) * sqrt((2.0d0 * (u * t)))
    else if (t_1 <= 2d+148) then
        tmp = sqrt(((u * (n + n)) * t))
    else
        tmp = abs(l) * sqrt(((-4.0d0) * ((u * n) / om)))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((Math.abs(l) * Math.abs(l)) / Om))) - ((n * Math.pow((Math.abs(l) / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.sqrt(n) * Math.sqrt((2.0 * (U * t)));
	} else if (t_1 <= 2e+148) {
		tmp = Math.sqrt(((U * (n + n)) * t));
	} else {
		tmp = Math.abs(l) * Math.sqrt((-4.0 * ((U * n) / Om)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((math.fabs(l) * math.fabs(l)) / Om))) - ((n * math.pow((math.fabs(l) / Om), 2.0)) * (U - U_42_)))))
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.sqrt(n) * math.sqrt((2.0 * (U * t)))
	elif t_1 <= 2e+148:
		tmp = math.sqrt(((U * (n + n)) * t))
	else:
		tmp = math.fabs(l) * math.sqrt((-4.0 * ((U * n) / Om)))
	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(abs(l) * abs(l)) / Om))) - Float64(Float64(n * (Float64(abs(l) / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(sqrt(n) * sqrt(Float64(2.0 * Float64(U * t))));
	elseif (t_1 <= 2e+148)
		tmp = sqrt(Float64(Float64(U * Float64(n + n)) * t));
	else
		tmp = Float64(abs(l) * sqrt(Float64(-4.0 * Float64(Float64(U * n) / Om))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((abs(l) * abs(l)) / Om))) - ((n * ((abs(l) / Om) ^ 2.0)) * (U - U_42_)))));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = sqrt(n) * sqrt((2.0 * (U * t)));
	elseif (t_1 <= 2e+148)
		tmp = sqrt(((U * (n + n)) * t));
	else
		tmp = abs(l) * sqrt((-4.0 * ((U * n) / Om)));
	end
	tmp_2 = 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[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(N[Abs[l], $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+148], N[Sqrt[N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(-4.0 * N[(N[(U * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 0.0

   1. Initial program 50.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 24.9%

      \[\leadsto \color{blue}{\sqrt{n} \cdot \sqrt{\left(U + U\right) \cdot \mathsf{fma}\left(\left(U* - U\right) \cdot n, \ell \cdot \frac{\ell}{Om \cdot Om}, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)}} \]

   4. Taylor expanded in t around inf

      \[\leadsto \sqrt{n} \cdot \sqrt{\color{blue}{2 \cdot \left(U \cdot t\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{2 \cdot \color{blue}{\left(U \cdot t\right)}} \]

      2. lower-*.f64 20.8%

         \[\leadsto \sqrt{n} \cdot \sqrt{2 \cdot \left(U \cdot \color{blue}{t}\right)} \]

   6. Applied rewrites 20.8%

      \[\leadsto \sqrt{n} \cdot \sqrt{\color{blue}{2 \cdot \left(U \cdot t\right)}} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2.0000000000000001e148

   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. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 36.3%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 36.3%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      4. count-2 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(n \cdot \color{blue}{t}\right)} \]

      7. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(U + U\right) \cdot n\right) \cdot \color{blue}{t}} \]

      8. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(\left(U + U\right) \cdot n\right) \cdot t} \]

      9. count-2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot U\right) \cdot n\right) \cdot t} \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(U \cdot 2\right) \cdot n\right) \cdot t} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      15. lower-*.f64 36.1%

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      16. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      17. *-commutative N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      18. lower-*.f64 36.1%

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      19. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      20. count-2-rev N/A

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      21. lift-+.f64 36.1%

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   6. Applied rewrites 36.1%

      \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot \color{blue}{t}} \]

   if 2.0000000000000001e148 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 50.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      11. lower-pow.f64 15.2%

          \[\leadsto \ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

   4. Applied rewrites 15.2%

      \[\leadsto \color{blue}{\ell \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]

   5. Taylor expanded in n around 0

      \[\leadsto \ell \cdot \sqrt{-4 \cdot \frac{U \cdot n}{Om}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-4 \cdot \frac{U \cdot n}{Om}} \]

      2. lower-/.f64 N/A

         \[\leadsto \ell \cdot \sqrt{-4 \cdot \frac{U \cdot n}{Om}} \]

      3. lower-*.f64 10.1%

         \[\leadsto \ell \cdot \sqrt{-4 \cdot \frac{U \cdot n}{Om}} \]

   7. Applied rewrites 10.1%

      \[\leadsto \ell \cdot \sqrt{-4 \cdot \frac{U \cdot n}{Om}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 40.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\left(\left(U + U\right) \cdot t\right) \cdot n\right|}\\ \end{array} \]

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 0.0)
     (* (sqrt n) (sqrt (* 2.0 (* U t))))
     (if (<= t_1 5e+144)
       (sqrt (* (* U (+ n n)) t))
       (sqrt (fabs (* (* (+ U U) t) n)))))))

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 <= 0.0) {
		tmp = sqrt(n) * sqrt((2.0 * (U * t)));
	} else if (t_1 <= 5e+144) {
		tmp = sqrt(((U * (n + n)) * t));
	} else {
		tmp = sqrt(fabs((((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) :: t_1
    real(8) :: tmp
    t_1 = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))))
    if (t_1 <= 0.0d0) then
        tmp = sqrt(n) * sqrt((2.0d0 * (u * t)))
    else if (t_1 <= 5d+144) then
        tmp = sqrt(((u * (n + n)) * t))
    else
        tmp = sqrt(abs((((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 t_1 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.sqrt(n) * Math.sqrt((2.0 * (U * t)));
	} else if (t_1 <= 5e+144) {
		tmp = Math.sqrt(((U * (n + n)) * t));
	} else {
		tmp = Math.sqrt(Math.abs((((U + U) * t) * n)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))))
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.sqrt(n) * math.sqrt((2.0 * (U * t)))
	elif t_1 <= 5e+144:
		tmp = math.sqrt(((U * (n + n)) * t))
	else:
		tmp = math.sqrt(math.fabs((((U + U) * t) * n)))
	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 <= 0.0)
		tmp = Float64(sqrt(n) * sqrt(Float64(2.0 * Float64(U * t))));
	elseif (t_1 <= 5e+144)
		tmp = sqrt(Float64(Float64(U * Float64(n + n)) * t));
	else
		tmp = sqrt(abs(Float64(Float64(Float64(U + U) * t) * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)))));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = sqrt(n) * sqrt((2.0 * (U * t)));
	elseif (t_1 <= 5e+144)
		tmp = sqrt(((U * (n + n)) * t));
	else
		tmp = sqrt(abs((((U + U) * t) * n)));
	end
	tmp_2 = 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, 0.0], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+144], N[Sqrt[N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(N[(N[(U + U), $MachinePrecision] * t), $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 50.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 24.9%

      \[\leadsto \color{blue}{\sqrt{n} \cdot \sqrt{\left(U + U\right) \cdot \mathsf{fma}\left(\left(U* - U\right) \cdot n, \ell \cdot \frac{\ell}{Om \cdot Om}, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)}} \]

   4. Taylor expanded in t around inf

      \[\leadsto \sqrt{n} \cdot \sqrt{\color{blue}{2 \cdot \left(U \cdot t\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{n} \cdot \sqrt{2 \cdot \color{blue}{\left(U \cdot t\right)}} \]

      2. lower-*.f64 20.8%

         \[\leadsto \sqrt{n} \cdot \sqrt{2 \cdot \left(U \cdot \color{blue}{t}\right)} \]

   6. Applied rewrites 20.8%

      \[\leadsto \sqrt{n} \cdot \sqrt{\color{blue}{2 \cdot \left(U \cdot t\right)}} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 4.9999999999999999e144

   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. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 36.3%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 36.3%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      4. count-2 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(n \cdot \color{blue}{t}\right)} \]

      7. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(U + U\right) \cdot n\right) \cdot \color{blue}{t}} \]

      8. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(\left(U + U\right) \cdot n\right) \cdot t} \]

      9. count-2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot U\right) \cdot n\right) \cdot t} \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(U \cdot 2\right) \cdot n\right) \cdot t} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      15. lower-*.f64 36.1%

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      16. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      17. *-commutative N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      18. lower-*.f64 36.1%

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      19. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      20. count-2-rev N/A

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      21. lift-+.f64 36.1%

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   6. Applied rewrites 36.1%

      \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot \color{blue}{t}} \]

   if 4.9999999999999999e144 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 50.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   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. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 36.3%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 36.3%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      4. sqr-abs-rev N/A

         \[\leadsto \sqrt{\color{blue}{\left|\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right| \cdot \left|\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right|}} \]

   6. Applied rewrites 37.5%

      \[\leadsto \sqrt{\color{blue}{\left|\left(\left(U + U\right) \cdot t\right) \cdot n\right|}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 40.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|\left(\left(U + U\right) \cdot t\right) \cdot n\right|}\\ \end{array} \]

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 0.0)
     (* (sqrt (+ n n)) (sqrt (* U t)))
     (if (<= t_1 5e+144)
       (sqrt (* (* U (+ n n)) t))
       (sqrt (fabs (* (* (+ U U) t) n)))))))

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 <= 0.0) {
		tmp = sqrt((n + n)) * sqrt((U * t));
	} else if (t_1 <= 5e+144) {
		tmp = sqrt(((U * (n + n)) * t));
	} else {
		tmp = sqrt(fabs((((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) :: t_1
    real(8) :: tmp
    t_1 = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))))
    if (t_1 <= 0.0d0) then
        tmp = sqrt((n + n)) * sqrt((u * t))
    else if (t_1 <= 5d+144) then
        tmp = sqrt(((u * (n + n)) * t))
    else
        tmp = sqrt(abs((((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 t_1 = Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.sqrt((n + n)) * Math.sqrt((U * t));
	} else if (t_1 <= 5e+144) {
		tmp = Math.sqrt(((U * (n + n)) * t));
	} else {
		tmp = Math.sqrt(Math.abs((((U + U) * t) * n)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))))
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.sqrt((n + n)) * math.sqrt((U * t))
	elif t_1 <= 5e+144:
		tmp = math.sqrt(((U * (n + n)) * t))
	else:
		tmp = math.sqrt(math.fabs((((U + U) * t) * n)))
	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 <= 0.0)
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(Float64(U * t)));
	elseif (t_1 <= 5e+144)
		tmp = sqrt(Float64(Float64(U * Float64(n + n)) * t));
	else
		tmp = sqrt(abs(Float64(Float64(Float64(U + U) * t) * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)))));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = sqrt((n + n)) * sqrt((U * t));
	elseif (t_1 <= 5e+144)
		tmp = sqrt(((U * (n + n)) * t));
	else
		tmp = sqrt(abs((((U + U) * t) * n)));
	end
	tmp_2 = 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, 0.0], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+144], N[Sqrt[N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Abs[N[(N[(N[(U + U), $MachinePrecision] * t), $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 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. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right) \cdot \left(U - U*\right)\right)}} \]

      4. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right) \cdot \left(U - U*\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right)}\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      8. sub-negate-rev N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \color{blue}{\left(U* - U\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} \cdot \left(U* - U\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} \cdot \left(U* - U\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      11. lift-pow.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\color{blue}{{\left(\frac{\ell}{Om}\right)}^{2}} \cdot n\right) \cdot \left(U* - U\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      12. unpow2 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot n\right) \cdot \left(U* - U\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      13. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \cdot \left(U* - U\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      14. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\frac{\ell}{Om} \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]

   3. Applied rewrites 52.2%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)}} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \color{blue}{-2 \cdot \frac{\ell \cdot \ell}{Om} + t}\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \color{blue}{\frac{\ell \cdot \ell}{Om} \cdot -2} + t\right)} \]

      3. lower-fma.f64 52.2%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \color{blue}{\mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right)}\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\color{blue}{\frac{\ell \cdot \ell}{Om}}, -2, t\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\frac{\color{blue}{\ell \cdot \ell}}{Om}, -2, t\right)\right)} \]

      6. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\color{blue}{\ell \cdot \frac{\ell}{Om}}, -2, t\right)\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\ell \cdot \color{blue}{\frac{\ell}{Om}}, -2, t\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\color{blue}{\frac{\ell}{Om} \cdot \ell}, -2, t\right)\right)} \]

      9. lower-*.f64 55.9%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\color{blue}{\frac{\ell}{Om} \cdot \ell}, -2, t\right)\right)} \]

   5. Applied rewrites 55.9%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \color{blue}{\mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)}\right)} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)\right)}} \]

      5. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)}} \]

      6. lower-unsound-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)}} \]

      7. lower-unsound-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot n}} \cdot \sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot n}} \cdot \sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)} \]

      9. count-2-rev N/A

         \[\leadsto \sqrt{\color{blue}{n + n}} \cdot \sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \sqrt{\color{blue}{n + n}} \cdot \sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)} \]

      11. lower-unsound-sqrt.f64 N/A

          \[\leadsto \sqrt{n + n} \cdot \color{blue}{\sqrt{U \cdot \mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)}} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right) \cdot U}} \]

      13. lower-*.f64 33.6%

          \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right) \cdot U}} \]

   7. Applied rewrites 32.7%

      \[\leadsto \color{blue}{\sqrt{n + n} \cdot \sqrt{\mathsf{fma}\left(\frac{\ell}{Om}, \mathsf{fma}\left(\left(U* - U\right) \cdot n, \frac{\ell}{Om}, -2 \cdot \ell\right), t\right) \cdot U}} \]

   8. Taylor expanded in t around inf

      \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{U \cdot t}} \]
   9. Step-by-step derivation

      1. lower-*.f64 20.8%

         \[\leadsto \sqrt{n + n} \cdot \sqrt{U \cdot \color{blue}{t}} \]

   10. Applied rewrites 20.8%

       \[\leadsto \sqrt{n + n} \cdot \sqrt{\color{blue}{U \cdot t}} \]

   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*))))) < 4.9999999999999999e144

   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. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 36.3%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 36.3%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      4. count-2 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(n \cdot \color{blue}{t}\right)} \]

      7. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(U + U\right) \cdot n\right) \cdot \color{blue}{t}} \]

      8. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(\left(U + U\right) \cdot n\right) \cdot t} \]

      9. count-2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot U\right) \cdot n\right) \cdot t} \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(U \cdot 2\right) \cdot n\right) \cdot t} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      15. lower-*.f64 36.1%

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      16. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      17. *-commutative N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      18. lower-*.f64 36.1%

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      19. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      20. count-2-rev N/A

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      21. lift-+.f64 36.1%

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   6. Applied rewrites 36.1%

      \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot \color{blue}{t}} \]

   if 4.9999999999999999e144 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 50.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   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. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 36.3%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 36.3%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      4. sqr-abs-rev N/A

         \[\leadsto \sqrt{\color{blue}{\left|\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right| \cdot \left|\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right|}} \]

   6. Applied rewrites 37.5%

      \[\leadsto \sqrt{\color{blue}{\left|\left(\left(U + U\right) \cdot t\right) \cdot n\right|}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 38.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;U \leq 6.2 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{\left|\left(\left(U + U\right) \cdot t\right) \cdot n\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U 6.2e+97)
   (sqrt (fabs (* (* (+ U U) t) n)))
   (sqrt (* (* U (+ n n)) t))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 6.2e+97) {
		tmp = sqrt(fabs((((U + U) * t) * n)));
	} else {
		tmp = sqrt(((U * (n + n)) * 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 (u <= 6.2d+97) then
        tmp = sqrt(abs((((u + u) * t) * n)))
    else
        tmp = sqrt(((u * (n + n)) * 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 (U <= 6.2e+97) {
		tmp = Math.sqrt(Math.abs((((U + U) * t) * n)));
	} else {
		tmp = Math.sqrt(((U * (n + n)) * t));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= 6.2e+97:
		tmp = math.sqrt(math.fabs((((U + U) * t) * n)))
	else:
		tmp = math.sqrt(((U * (n + n)) * t))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= 6.2e+97)
		tmp = sqrt(abs(Float64(Float64(Float64(U + U) * t) * n)));
	else
		tmp = sqrt(Float64(Float64(U * Float64(n + n)) * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= 6.2e+97)
		tmp = sqrt(abs((((U + U) * t) * n)));
	else
		tmp = sqrt(((U * (n + n)) * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, 6.2e+97], N[Sqrt[N[Abs[N[(N[(N[(U + U), $MachinePrecision] * t), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 6.19999999999999962e97

   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. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 36.3%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 36.3%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \cdot \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \cdot \color{blue}{\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}}} \]

      4. sqr-abs-rev N/A

         \[\leadsto \sqrt{\color{blue}{\left|\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right| \cdot \left|\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\right|}} \]

   6. Applied rewrites 37.5%

      \[\leadsto \sqrt{\color{blue}{\left|\left(\left(U + U\right) \cdot t\right) \cdot n\right|}} \]

   if 6.19999999999999962e97 < 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. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 36.3%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 36.3%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      4. count-2 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(n \cdot \color{blue}{t}\right)} \]

      7. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(U + U\right) \cdot n\right) \cdot \color{blue}{t}} \]

      8. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(\left(U + U\right) \cdot n\right) \cdot t} \]

      9. count-2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot U\right) \cdot n\right) \cdot t} \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(U \cdot 2\right) \cdot n\right) \cdot t} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      15. lower-*.f64 36.1%

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      16. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      17. *-commutative N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      18. lower-*.f64 36.1%

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      19. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      20. count-2-rev N/A

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      21. lift-+.f64 36.1%

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   6. Applied rewrites 36.1%

      \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot \color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 38.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(\left(U + U\right) \cdot t\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t}\\ \end{array} \]

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*))))
      0.0)
   (sqrt (* (* (+ U U) t) n))
   (sqrt (* (* U (+ n n)) 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_)))) <= 0.0) {
		tmp = sqrt((((U + U) * t) * n));
	} else {
		tmp = sqrt(((U * (n + n)) * 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)))) <= 0.0d0) then
        tmp = sqrt((((u + u) * t) * n))
    else
        tmp = sqrt(((u * (n + n)) * 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_)))) <= 0.0) {
		tmp = Math.sqrt((((U + U) * t) * n));
	} else {
		tmp = Math.sqrt(((U * (n + n)) * 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_)))) <= 0.0:
		tmp = math.sqrt((((U + U) * t) * n))
	else:
		tmp = math.sqrt(((U * (n + n)) * 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_)))) <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(U + U) * t) * n));
	else
		tmp = sqrt(Float64(Float64(U * Float64(n + n)) * t));
	end
	return tmp
end

MATLAB

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_)))) <= 0.0)
		tmp = sqrt((((U + U) * t) * n));
	else
		tmp = sqrt(((U * (n + n)) * 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], 0.0], N[Sqrt[N[(N[(N[(U + U), $MachinePrecision] * t), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 0.0

   1. Initial program 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. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 36.3%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 36.3%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      4. count-2 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(n \cdot \color{blue}{t}\right)} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(t \cdot \color{blue}{n}\right)} \]

      8. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(U + U\right) \cdot t\right) \cdot \color{blue}{n}} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(U + U\right) \cdot t\right) \cdot \color{blue}{n}} \]

      10. lower-*.f64 35.2%

          \[\leadsto \sqrt{\left(\left(U + U\right) \cdot t\right) \cdot n} \]

   6. Applied rewrites 35.2%

      \[\leadsto \sqrt{\color{blue}{\left(\left(U + U\right) \cdot t\right) \cdot n}} \]

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 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. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. lower-*.f64 36.3%

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

   4. Applied rewrites 36.3%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

      4. count-2 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(U + U\right) \cdot \left(n \cdot \color{blue}{t}\right)} \]

      7. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(U + U\right) \cdot n\right) \cdot \color{blue}{t}} \]

      8. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(\left(U + U\right) \cdot n\right) \cdot t} \]

      9. count-2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot U\right) \cdot n\right) \cdot t} \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(U \cdot 2\right) \cdot n\right) \cdot t} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      15. lower-*.f64 36.1%

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

      16. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

      17. *-commutative N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      18. lower-*.f64 36.1%

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      19. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

      20. count-2-rev N/A

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

      21. lift-+.f64 36.1%

          \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   6. Applied rewrites 36.1%

      \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot \color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 36.1% accurate, 4.7× speedup

Math

\[\sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (sqrt (* (* U (+ n n)) t)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt(((U * (n + n)) * t));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt(((u * (n + n)) * t))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt(((U * (n + n)) * t));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt(((U * (n + n)) * t))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(U * Float64(n + n)) * t))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt(((U * (n + n)) * t));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.6%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

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. lower-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

   3. lower-*.f64 36.3%

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot \color{blue}{t}\right)\right)} \]

4. Applied rewrites 36.3%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot \left(U \cdot \color{blue}{\left(n \cdot t\right)}\right)} \]

   3. associate-*r* N/A

      \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   4. count-2 N/A

      \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

   5. lift-+.f64 N/A

      \[\leadsto \sqrt{\left(U + U\right) \cdot \left(\color{blue}{n} \cdot t\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(U + U\right) \cdot \left(n \cdot \color{blue}{t}\right)} \]

   7. associate-*r* N/A

      \[\leadsto \sqrt{\left(\left(U + U\right) \cdot n\right) \cdot \color{blue}{t}} \]

   8. lift-+.f64 N/A

      \[\leadsto \sqrt{\left(\left(U + U\right) \cdot n\right) \cdot t} \]

   9. count-2 N/A

      \[\leadsto \sqrt{\left(\left(2 \cdot U\right) \cdot n\right) \cdot t} \]

   10. *-commutative N/A

       \[\leadsto \sqrt{\left(\left(U \cdot 2\right) \cdot n\right) \cdot t} \]

   11. associate-*r* N/A

       \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

   12. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

   13. *-commutative N/A

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

   14. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

   15. lower-*.f64 36.1%

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

   16. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \]

   17. *-commutative N/A

       \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

   18. lower-*.f64 36.1%

       \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

   19. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t} \]

   20. count-2-rev N/A

       \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

   21. lift-+.f64 36.1%

       \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot t} \]

6. Applied rewrites 36.1%

   \[\leadsto \sqrt{\left(U \cdot \left(n + n\right)\right) \cdot \color{blue}{t}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025188 
(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*))))))