Toniolo and Linder, Equation (13)

Percentage Accurate: 50.2% → 61.8%

Time: 9.4s

Alternatives: 17

Speedup: 2.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 61.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 11.7%

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

   4. Applied rewrites 37.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 37.9

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

   6. Applied rewrites 37.9%

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

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

   1. Initial program 69.1%

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

   4. Applied rewrites 73.5%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 73.5

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

   6. Applied rewrites 73.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in l around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 30.8%

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

Alternative 2: 41.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := \sqrt{t\_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;-1 \cdot \left(n \cdot \left(-1 \cdot \sqrt{\frac{U \cdot t}{n} \cdot 2}\right)\right)\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{t\_1 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-2 \cdot U\right) \cdot \left(2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot n}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (sqrt
          (*
           t_1
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_2 0.0)
     (* -1.0 (* n (* -1.0 (sqrt (* (/ (* U t) n) 2.0)))))
     (if (<= t_2 4e+145)
       (sqrt (* t_1 t))
       (sqrt (* (* -2.0 U) (* 2.0 (/ (* (* l l) n) 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 = sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = -1.0 * (n * (-1.0 * sqrt((((U * t) / n) * 2.0))));
	} else if (t_2 <= 4e+145) {
		tmp = sqrt((t_1 * t));
	} else {
		tmp = sqrt(((-2.0 * U) * (2.0 * (((l * l) * 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) :: t_2
    real(8) :: tmp
    t_1 = (2.0d0 * n) * u
    t_2 = sqrt((t_1 * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))))
    if (t_2 <= 0.0d0) then
        tmp = (-1.0d0) * (n * ((-1.0d0) * sqrt((((u * t) / n) * 2.0d0))))
    else if (t_2 <= 4d+145) then
        tmp = sqrt((t_1 * t))
    else
        tmp = sqrt((((-2.0d0) * u) * (2.0d0 * (((l * l) * 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 = (2.0 * n) * U;
	double t_2 = Math.sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = -1.0 * (n * (-1.0 * Math.sqrt((((U * t) / n) * 2.0))));
	} else if (t_2 <= 4e+145) {
		tmp = Math.sqrt((t_1 * t));
	} else {
		tmp = Math.sqrt(((-2.0 * U) * (2.0 * (((l * l) * n) / Om))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (2.0 * n) * U
	t_2 = math.sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))))
	tmp = 0
	if t_2 <= 0.0:
		tmp = -1.0 * (n * (-1.0 * math.sqrt((((U * t) / n) * 2.0))))
	elif t_2 <= 4e+145:
		tmp = math.sqrt((t_1 * t))
	else:
		tmp = math.sqrt(((-2.0 * U) * (2.0 * (((l * l) * n) / Om))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(-1.0 * Float64(n * Float64(-1.0 * sqrt(Float64(Float64(Float64(U * t) / n) * 2.0)))));
	elseif (t_2 <= 4e+145)
		tmp = sqrt(Float64(t_1 * t));
	else
		tmp = sqrt(Float64(Float64(-2.0 * U) * Float64(2.0 * Float64(Float64(Float64(l * l) * n) / Om))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (2.0 * n) * U;
	t_2 = sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)))));
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = -1.0 * (n * (-1.0 * sqrt((((U * t) / n) * 2.0))));
	elseif (t_2 <= 4e+145)
		tmp = sqrt((t_1 * t));
	else
		tmp = sqrt(((-2.0 * U) * (2.0 * (((l * l) * n) / Om))));
	end
	tmp_2 = 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[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(-1.0 * N[(n * N[(-1.0 * N[Sqrt[N[(N[(N[(U * t), $MachinePrecision] / n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+145], N[Sqrt[N[(t$95$1 * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(-2.0 * U), $MachinePrecision] * N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] * 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 11.7%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 30.5%

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

   6. Taylor expanded in n around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

   8. Applied rewrites 19.1%

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

   9. Taylor expanded in t around -inf

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

       1. sqrt-unprod N/A

          \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \left(\sqrt{\frac{U \cdot t}{n}} \cdot \sqrt{-2 \cdot -1}\right)\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \left(\sqrt{\frac{U \cdot t}{n}} \cdot \sqrt{2}\right)\right)\right) \]

       3. sqrt-prod N/A

          \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \sqrt{\frac{U \cdot t}{n} \cdot 2}\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \sqrt{\frac{U \cdot t}{n} \cdot 2}\right)\right) \]

       5. lift-/.f64 N/A

          \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \sqrt{\frac{U \cdot t}{n} \cdot 2}\right)\right) \]

       6. lift-*.f64 N/A

          \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \sqrt{\frac{U \cdot t}{n} \cdot 2}\right)\right) \]

       7. lift-*.f64 N/A

          \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \sqrt{\frac{U \cdot t}{n} \cdot 2}\right)\right) \]

       8. lift-sqrt.f64 33.5

          \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \sqrt{\frac{U \cdot t}{n} \cdot 2}\right)\right) \]

   11. Applied rewrites 33.5%

       \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \sqrt{\frac{U \cdot t}{n} \cdot 2}\right)\right) \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 4e145

   1. Initial program 97.7%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 76.1%

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

   if 4e145 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 21.3%

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

   3. Taylor expanded in l around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 27.6%

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

   6. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 15.7

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

   8. Applied rewrites 15.7%

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

Alternative 3: 54.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 10.3%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 30.6%

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

   6. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 34.0

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

   8. Applied rewrites 34.0%

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

   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*)))) < +inf.0

   1. Initial program 69.1%

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

   3. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 63.6

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

   5. Applied rewrites 63.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 2.6%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 29.2

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

   8. Applied rewrites 29.2%

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

   9. Taylor expanded in n around inf

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

       1. pow2 N/A

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

       2. lower-*.f64 N/A

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

       3. times-frac N/A

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

       4. lower-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift--.f64 34.1

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

   11. Applied rewrites 34.1%

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

Alternative 4: 41.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := \sqrt{t\_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;-1 \cdot \left(n \cdot \left(-1 \cdot \sqrt{\frac{U \cdot t}{n} \cdot 2}\right)\right)\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{t\_1 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\ell \cdot \ell\right) \cdot \left(-4 \cdot \frac{U}{Om}\right)\right) \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (sqrt
          (*
           t_1
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_2 0.0)
     (* -1.0 (* n (* -1.0 (sqrt (* (/ (* U t) n) 2.0)))))
     (if (<= t_2 4e+145)
       (sqrt (* t_1 t))
       (sqrt (* (* (* l l) (* -4.0 (/ U Om))) n))))))

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 = sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = -1.0 * (n * (-1.0 * sqrt((((U * t) / n) * 2.0))));
	} else if (t_2 <= 4e+145) {
		tmp = sqrt((t_1 * t));
	} else {
		tmp = sqrt((((l * l) * (-4.0 * (U / Om))) * 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) :: t_2
    real(8) :: tmp
    t_1 = (2.0d0 * n) * u
    t_2 = sqrt((t_1 * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))))
    if (t_2 <= 0.0d0) then
        tmp = (-1.0d0) * (n * ((-1.0d0) * sqrt((((u * t) / n) * 2.0d0))))
    else if (t_2 <= 4d+145) then
        tmp = sqrt((t_1 * t))
    else
        tmp = sqrt((((l * l) * ((-4.0d0) * (u / om))) * 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 = (2.0 * n) * U;
	double t_2 = Math.sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = -1.0 * (n * (-1.0 * Math.sqrt((((U * t) / n) * 2.0))));
	} else if (t_2 <= 4e+145) {
		tmp = Math.sqrt((t_1 * t));
	} else {
		tmp = Math.sqrt((((l * l) * (-4.0 * (U / Om))) * n));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (2.0 * n) * U
	t_2 = math.sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))))
	tmp = 0
	if t_2 <= 0.0:
		tmp = -1.0 * (n * (-1.0 * math.sqrt((((U * t) / n) * 2.0))))
	elif t_2 <= 4e+145:
		tmp = math.sqrt((t_1 * t))
	else:
		tmp = math.sqrt((((l * l) * (-4.0 * (U / Om))) * n))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(-1.0 * Float64(n * Float64(-1.0 * sqrt(Float64(Float64(Float64(U * t) / n) * 2.0)))));
	elseif (t_2 <= 4e+145)
		tmp = sqrt(Float64(t_1 * t));
	else
		tmp = sqrt(Float64(Float64(Float64(l * l) * Float64(-4.0 * Float64(U / Om))) * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (2.0 * n) * U;
	t_2 = sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)))));
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = -1.0 * (n * (-1.0 * sqrt((((U * t) / n) * 2.0))));
	elseif (t_2 <= 4e+145)
		tmp = sqrt((t_1 * t));
	else
		tmp = sqrt((((l * l) * (-4.0 * (U / Om))) * n));
	end
	tmp_2 = 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[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(-1.0 * N[(n * N[(-1.0 * N[Sqrt[N[(N[(N[(U * t), $MachinePrecision] / n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+145], N[Sqrt[N[(t$95$1 * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(l * l), $MachinePrecision] * N[(-4.0 * N[(U / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $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 11.7%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 30.5%

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

   6. Taylor expanded in n around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

   8. Applied rewrites 19.1%

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

   9. Taylor expanded in t around -inf

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

       1. sqrt-unprod N/A

          \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \left(\sqrt{\frac{U \cdot t}{n}} \cdot \sqrt{-2 \cdot -1}\right)\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \left(\sqrt{\frac{U \cdot t}{n}} \cdot \sqrt{2}\right)\right)\right) \]

       3. sqrt-prod N/A

          \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \sqrt{\frac{U \cdot t}{n} \cdot 2}\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \sqrt{\frac{U \cdot t}{n} \cdot 2}\right)\right) \]

       5. lift-/.f64 N/A

          \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \sqrt{\frac{U \cdot t}{n} \cdot 2}\right)\right) \]

       6. lift-*.f64 N/A

          \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \sqrt{\frac{U \cdot t}{n} \cdot 2}\right)\right) \]

       7. lift-*.f64 N/A

          \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \sqrt{\frac{U \cdot t}{n} \cdot 2}\right)\right) \]

       8. lift-sqrt.f64 33.5

          \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \sqrt{\frac{U \cdot t}{n} \cdot 2}\right)\right) \]

   11. Applied rewrites 33.5%

       \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \sqrt{\frac{U \cdot t}{n} \cdot 2}\right)\right) \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 4e145

   1. Initial program 97.7%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 76.1%

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

   if 4e145 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 21.3%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 17.2%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 25.5

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

   8. Applied rewrites 25.5%

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

   9. Taylor expanded in n around 0

      \[\leadsto \sqrt{\left(\left(\ell \cdot \ell\right) \cdot \left(-4 \cdot \frac{U}{Om}\right)\right) \cdot n} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(\ell \cdot \ell\right) \cdot \left(-4 \cdot \frac{U}{Om}\right)\right) \cdot n} \]

       2. lift-/.f64 14.6

          \[\leadsto \sqrt{\left(\left(\ell \cdot \ell\right) \cdot \left(-4 \cdot \frac{U}{Om}\right)\right) \cdot n} \]

   11. Applied rewrites 14.6%

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

Alternative 5: 41.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := \sqrt{t\_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;U \cdot \sqrt{\frac{n \cdot t}{U} \cdot 2}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{t\_1 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\ell \cdot \ell\right) \cdot \left(-4 \cdot \frac{U}{Om}\right)\right) \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (sqrt
          (*
           t_1
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_2 0.0)
     (* U (sqrt (* (/ (* n t) U) 2.0)))
     (if (<= t_2 4e+145)
       (sqrt (* t_1 t))
       (sqrt (* (* (* l l) (* -4.0 (/ U Om))) n))))))

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 = sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = U * sqrt((((n * t) / U) * 2.0));
	} else if (t_2 <= 4e+145) {
		tmp = sqrt((t_1 * t));
	} else {
		tmp = sqrt((((l * l) * (-4.0 * (U / Om))) * 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) :: t_2
    real(8) :: tmp
    t_1 = (2.0d0 * n) * u
    t_2 = sqrt((t_1 * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))))
    if (t_2 <= 0.0d0) then
        tmp = u * sqrt((((n * t) / u) * 2.0d0))
    else if (t_2 <= 4d+145) then
        tmp = sqrt((t_1 * t))
    else
        tmp = sqrt((((l * l) * ((-4.0d0) * (u / om))) * 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 = (2.0 * n) * U;
	double t_2 = Math.sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = U * Math.sqrt((((n * t) / U) * 2.0));
	} else if (t_2 <= 4e+145) {
		tmp = Math.sqrt((t_1 * t));
	} else {
		tmp = Math.sqrt((((l * l) * (-4.0 * (U / Om))) * n));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (2.0 * n) * U
	t_2 = math.sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))))
	tmp = 0
	if t_2 <= 0.0:
		tmp = U * math.sqrt((((n * t) / U) * 2.0))
	elif t_2 <= 4e+145:
		tmp = math.sqrt((t_1 * t))
	else:
		tmp = math.sqrt((((l * l) * (-4.0 * (U / Om))) * n))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(U * sqrt(Float64(Float64(Float64(n * t) / U) * 2.0)));
	elseif (t_2 <= 4e+145)
		tmp = sqrt(Float64(t_1 * t));
	else
		tmp = sqrt(Float64(Float64(Float64(l * l) * Float64(-4.0 * Float64(U / Om))) * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (2.0 * n) * U;
	t_2 = sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)))));
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = U * sqrt((((n * t) / U) * 2.0));
	elseif (t_2 <= 4e+145)
		tmp = sqrt((t_1 * t));
	else
		tmp = sqrt((((l * l) * (-4.0 * (U / Om))) * n));
	end
	tmp_2 = 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[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(U * N[Sqrt[N[(N[(N[(n * t), $MachinePrecision] / U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+145], N[Sqrt[N[(t$95$1 * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(l * l), $MachinePrecision] * N[(-4.0 * N[(U / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $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 11.7%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 30.5%

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

   6. Taylor expanded in n around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

   8. Applied rewrites 19.1%

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

   9. Taylor expanded in U around -inf

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

   11. Applied rewrites 0.5%

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

   12. Taylor expanded in t around inf

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

       1. sqrt-unprod N/A

          \[\leadsto U \cdot \sqrt{\frac{n \cdot t}{U} \cdot 2} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto U \cdot \sqrt{\frac{n \cdot t}{U} \cdot 2} \]

       3. lower-*.f64 N/A

          \[\leadsto U \cdot \sqrt{\frac{n \cdot t}{U} \cdot 2} \]

       4. lower-/.f64 N/A

          \[\leadsto U \cdot \sqrt{\frac{n \cdot t}{U} \cdot 2} \]

       5. lift-*.f64 32.6

          \[\leadsto U \cdot \sqrt{\frac{n \cdot t}{U} \cdot 2} \]

   14. Applied rewrites 32.6%

       \[\leadsto U \cdot \sqrt{\frac{n \cdot t}{U} \cdot 2} \]

   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*))))) < 4e145

   1. Initial program 97.7%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 76.1%

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

   if 4e145 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 21.3%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 17.2%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 25.5

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

   8. Applied rewrites 25.5%

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

   9. Taylor expanded in n around 0

      \[\leadsto \sqrt{\left(\left(\ell \cdot \ell\right) \cdot \left(-4 \cdot \frac{U}{Om}\right)\right) \cdot n} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(\ell \cdot \ell\right) \cdot \left(-4 \cdot \frac{U}{Om}\right)\right) \cdot n} \]

       2. lift-/.f64 14.6

          \[\leadsto \sqrt{\left(\left(\ell \cdot \ell\right) \cdot \left(-4 \cdot \frac{U}{Om}\right)\right) \cdot n} \]

   11. Applied rewrites 14.6%

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

Alternative 6: 41.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := \sqrt{t\_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;U \cdot \sqrt{\frac{n \cdot t}{U} \cdot 2}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{t\_1 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-4 \cdot \frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (sqrt
          (*
           t_1
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_2 0.0)
     (* U (sqrt (* (/ (* n t) U) 2.0)))
     (if (<= t_2 4e+145)
       (sqrt (* t_1 t))
       (sqrt (* -4.0 (/ (* U (* (* l l) n)) 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 = sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = U * sqrt((((n * t) / U) * 2.0));
	} else if (t_2 <= 4e+145) {
		tmp = sqrt((t_1 * t));
	} else {
		tmp = sqrt((-4.0 * ((U * ((l * l) * 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) :: t_2
    real(8) :: tmp
    t_1 = (2.0d0 * n) * u
    t_2 = sqrt((t_1 * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))))
    if (t_2 <= 0.0d0) then
        tmp = u * sqrt((((n * t) / u) * 2.0d0))
    else if (t_2 <= 4d+145) then
        tmp = sqrt((t_1 * t))
    else
        tmp = sqrt(((-4.0d0) * ((u * ((l * l) * 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 = (2.0 * n) * U;
	double t_2 = Math.sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = U * Math.sqrt((((n * t) / U) * 2.0));
	} else if (t_2 <= 4e+145) {
		tmp = Math.sqrt((t_1 * t));
	} else {
		tmp = Math.sqrt((-4.0 * ((U * ((l * l) * n)) / Om)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (2.0 * n) * U
	t_2 = math.sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))))
	tmp = 0
	if t_2 <= 0.0:
		tmp = U * math.sqrt((((n * t) / U) * 2.0))
	elif t_2 <= 4e+145:
		tmp = math.sqrt((t_1 * t))
	else:
		tmp = math.sqrt((-4.0 * ((U * ((l * l) * n)) / Om)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(U * sqrt(Float64(Float64(Float64(n * t) / U) * 2.0)));
	elseif (t_2 <= 4e+145)
		tmp = sqrt(Float64(t_1 * t));
	else
		tmp = sqrt(Float64(-4.0 * Float64(Float64(U * Float64(Float64(l * l) * n)) / Om)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (2.0 * n) * U;
	t_2 = sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)))));
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = U * sqrt((((n * t) / U) * 2.0));
	elseif (t_2 <= 4e+145)
		tmp = sqrt((t_1 * t));
	else
		tmp = sqrt((-4.0 * ((U * ((l * l) * n)) / Om)));
	end
	tmp_2 = 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[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(U * N[Sqrt[N[(N[(N[(n * t), $MachinePrecision] / U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+145], N[Sqrt[N[(t$95$1 * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-4.0 * N[(N[(U * N[(N[(l * l), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / Om), $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 11.7%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 30.5%

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

   6. Taylor expanded in n around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

   8. Applied rewrites 19.1%

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

   9. Taylor expanded in U around -inf

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

   11. Applied rewrites 0.5%

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

   12. Taylor expanded in t around inf

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

       1. sqrt-unprod N/A

          \[\leadsto U \cdot \sqrt{\frac{n \cdot t}{U} \cdot 2} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto U \cdot \sqrt{\frac{n \cdot t}{U} \cdot 2} \]

       3. lower-*.f64 N/A

          \[\leadsto U \cdot \sqrt{\frac{n \cdot t}{U} \cdot 2} \]

       4. lower-/.f64 N/A

          \[\leadsto U \cdot \sqrt{\frac{n \cdot t}{U} \cdot 2} \]

       5. lift-*.f64 32.6

          \[\leadsto U \cdot \sqrt{\frac{n \cdot t}{U} \cdot 2} \]

   14. Applied rewrites 32.6%

       \[\leadsto U \cdot \sqrt{\frac{n \cdot t}{U} \cdot 2} \]

   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*))))) < 4e145

   1. Initial program 97.7%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 76.1%

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

   if 4e145 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 21.3%

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

   3. Taylor expanded in l around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 27.6%

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

   6. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

         \[\leadsto \sqrt{-4 \cdot \frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{Om}} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{-4 \cdot \frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{Om}} \]

      6. lift-*.f64 15.1

         \[\leadsto \sqrt{-4 \cdot \frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{Om}} \]

   8. Applied rewrites 15.1%

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

Alternative 7: 54.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 10.3%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 30.6%

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

   6. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 34.0

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

   8. Applied rewrites 34.0%

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

   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*)))) < +inf.0

   1. Initial program 69.1%

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

   3. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 63.6

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

   5. Applied rewrites 63.6%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 8.3%

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

   5. Taylor expanded in U* around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 29.5

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

   7. Applied rewrites 29.5%

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

Alternative 8: 54.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 10.3%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 30.6%

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

   6. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 34.0

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

   8. Applied rewrites 34.0%

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

   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*)))) < +inf.0

   1. Initial program 69.1%

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

   3. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 63.6

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

   5. Applied rewrites 63.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 2.6%

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

   6. Taylor expanded in U* around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 30.2

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

   8. Applied rewrites 30.2%

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

Alternative 9: 57.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 60.2%

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

   4. Applied rewrites 62.5%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 62.5

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

   6. Applied rewrites 62.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in l around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 30.8%

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

Alternative 10: 50.4% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U)))
   (if (<=
        (sqrt
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*)))))
        0.0)
     (* -1.0 (* n (* -1.0 (sqrt (* (/ (* U t) n) 2.0)))))
     (sqrt (* t_1 (fma -2.0 (* l (/ l Om)) t))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double tmp;
	if (sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_))))) <= 0.0) {
		tmp = -1.0 * (n * (-1.0 * sqrt((((U * t) / n) * 2.0))));
	} else {
		tmp = sqrt((t_1 * fma(-2.0, (l * (l / Om)), t)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	tmp = 0.0
	if (sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))) <= 0.0)
		tmp = Float64(-1.0 * Float64(n * Float64(-1.0 * sqrt(Float64(Float64(Float64(U * t) / n) * 2.0)))));
	else
		tmp = sqrt(Float64(t_1 * fma(-2.0, Float64(l * Float64(l / Om)), t)));
	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]}, If[LessEqual[N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(-1.0 * N[(n * N[(-1.0 * N[Sqrt[N[(N[(N[(U * t), $MachinePrecision] / n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(t$95$1 * N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 11.7%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 30.5%

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

   6. Taylor expanded in n around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

   8. Applied rewrites 19.1%

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

   9. Taylor expanded in t around -inf

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

       1. sqrt-unprod N/A

          \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \left(\sqrt{\frac{U \cdot t}{n}} \cdot \sqrt{-2 \cdot -1}\right)\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \left(\sqrt{\frac{U \cdot t}{n}} \cdot \sqrt{2}\right)\right)\right) \]

       3. sqrt-prod N/A

          \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \sqrt{\frac{U \cdot t}{n} \cdot 2}\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \sqrt{\frac{U \cdot t}{n} \cdot 2}\right)\right) \]

       5. lift-/.f64 N/A

          \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \sqrt{\frac{U \cdot t}{n} \cdot 2}\right)\right) \]

       6. lift-*.f64 N/A

          \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \sqrt{\frac{U \cdot t}{n} \cdot 2}\right)\right) \]

       7. lift-*.f64 N/A

          \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \sqrt{\frac{U \cdot t}{n} \cdot 2}\right)\right) \]

       8. lift-sqrt.f64 33.5

          \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \sqrt{\frac{U \cdot t}{n} \cdot 2}\right)\right) \]

   11. Applied rewrites 33.5%

       \[\leadsto -1 \cdot \left(n \cdot \left(-1 \cdot \sqrt{\frac{U \cdot t}{n} \cdot 2}\right)\right) \]

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

   1. Initial program 56.0%

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

   3. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 52.9

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

   5. Applied rewrites 52.9%

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

Alternative 11: 39.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double tmp;
	if (sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_))))) <= 0.0) {
		tmp = sqrt(((2.0 * (U * t)) * n));
	} else {
		tmp = sqrt((t_1 * 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) :: t_1
    real(8) :: tmp
    t_1 = (2.0d0 * n) * u
    if (sqrt((t_1 * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42))))) <= 0.0d0) then
        tmp = sqrt(((2.0d0 * (u * t)) * n))
    else
        tmp = sqrt((t_1 * 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 t_1 = (2.0 * n) * U;
	double tmp;
	if (Math.sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_))))) <= 0.0) {
		tmp = Math.sqrt(((2.0 * (U * t)) * n));
	} else {
		tmp = Math.sqrt((t_1 * t));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (2.0 * n) * U
	tmp = 0
	if math.sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_))))) <= 0.0:
		tmp = math.sqrt(((2.0 * (U * t)) * n))
	else:
		tmp = math.sqrt((t_1 * t))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	tmp = 0.0
	if (sqrt(Float64(t_1 * 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(2.0 * Float64(U * t)) * n));
	else
		tmp = sqrt(Float64(t_1 * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (2.0 * n) * U;
	tmp = 0.0;
	if (sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_))))) <= 0.0)
		tmp = sqrt(((2.0 * (U * t)) * n));
	else
		tmp = sqrt((t_1 * t));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 11.7%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 32.4%

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

   6. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lift-*.f64 30.6

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

   8. Applied rewrites 30.6%

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

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

   1. Initial program 56.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 40.6%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 39.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot t\right)\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(\left(U \cdot n\right) \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<=
      (sqrt
       (*
        (* (* 2.0 n) U)
        (-
         (- t (* 2.0 (/ (* l l) Om)))
         (* (* n (pow (/ l Om) 2.0)) (- U U*)))))
      0.0)
   (sqrt (* (* 2.0 (* U t)) n))
   (sqrt (* t (* (* U n) 2.0)))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42))))) <= 0.0d0) then
        tmp = sqrt(((2.0d0 * (u * t)) * n))
    else
        tmp = sqrt((t * ((u * n) * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_))))) <= 0.0) {
		tmp = Math.sqrt(((2.0 * (U * t)) * n));
	} else {
		tmp = Math.sqrt((t * ((U * n) * 2.0)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_))))) <= 0.0:
		tmp = math.sqrt(((2.0 * (U * t)) * n))
	else:
		tmp = math.sqrt((t * ((U * n) * 2.0)))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_))))) <= 0.0)
		tmp = sqrt(((2.0 * (U * t)) * n));
	else
		tmp = sqrt((t * ((U * n) * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 11.7%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 32.4%

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

   6. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lift-*.f64 30.6

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

   8. Applied rewrites 30.6%

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

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

   1. Initial program 56.0%

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 40.5%

      \[\leadsto \sqrt{\color{blue}{t} \cdot \left(\left(U \cdot n\right) \cdot 2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 56.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(t - \left(U - U*\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right)\right) \cdot \left(\left(U \cdot n\right) \cdot 2\right)}\\ \mathbf{if}\;n \leq -5.6 \cdot 10^{-102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (* (- t (* (- U U*) (* (* (/ l Om) (/ l Om)) n))) (* (* U n) 2.0)))))
   (if (<= n -5.6e-102)
     t_1
     (if (<= n 4.5e-15)
       (sqrt (* (* (* (fma -2.0 (* l (/ l Om)) t) n) U) 2.0))
       t_1))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt(((t - ((U - U_42_) * (((l / Om) * (l / Om)) * n))) * ((U * n) * 2.0)));
	double tmp;
	if (n <= -5.6e-102) {
		tmp = t_1;
	} else if (n <= 4.5e-15) {
		tmp = sqrt((((fma(-2.0, (l * (l / Om)), t) * n) * U) * 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(t - Float64(Float64(U - U_42_) * Float64(Float64(Float64(l / Om) * Float64(l / Om)) * n))) * Float64(Float64(U * n) * 2.0)))
	tmp = 0.0
	if (n <= -5.6e-102)
		tmp = t_1;
	elseif (n <= 4.5e-15)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) * n) * U) * 2.0));
	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[(t - N[(N[(U - U$42$), $MachinePrecision] * N[(N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(U * n), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -5.6e-102], t$95$1, If[LessEqual[n, 4.5e-15], N[Sqrt[N[(N[(N[(N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if n < -5.60000000000000026e-102 or 4.4999999999999998e-15 < n

   1. Initial program 55.2%

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

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 58.5

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

   6. Applied rewrites 58.5%

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

   7. Taylor expanded in t around inf

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

   9. Applied rewrites 59.7%

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

   if -5.60000000000000026e-102 < n < 4.4999999999999998e-15

   1. Initial program 44.3%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 53.0

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

   5. Applied rewrites 53.0%

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

Alternative 14: 49.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{+198}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t} \cdot \sqrt{\left(U \cdot n\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 9e+198)
   (sqrt (* (* (* (fma -2.0 (* l (/ l Om)) t) n) U) 2.0))
   (* (sqrt t) (sqrt (* (* U n) 2.0)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 9e+198) {
		tmp = sqrt((((fma(-2.0, (l * (l / Om)), t) * n) * U) * 2.0));
	} else {
		tmp = sqrt(t) * sqrt(((U * n) * 2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 9e+198)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) * n) * U) * 2.0));
	else
		tmp = Float64(sqrt(t) * sqrt(Float64(Float64(U * n) * 2.0)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 9e+198], N[Sqrt[N[(N[(N[(N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[t], $MachinePrecision] * N[Sqrt[N[(N[(U * n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.00000000000000003e198

   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. Add Preprocessing

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 48.4

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

   5. Applied rewrites 48.4%

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

   if 9.00000000000000003e198 < t

   1. Initial program 47.0%

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 45.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

   9. Applied rewrites 58.2%

      \[\leadsto \color{blue}{\sqrt{t} \cdot \sqrt{\left(U \cdot n\right) \cdot 2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 39.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(U \cdot n\right) \cdot 2\\ \mathbf{if}\;t \leq 9.6 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{t \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t} \cdot \sqrt{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* U n) 2.0)))
   (if (<= t 9.6e-278) (sqrt (* t t_1)) (* (sqrt t) (sqrt t_1)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (U * n) * 2.0;
	double tmp;
	if (t <= 9.6e-278) {
		tmp = sqrt((t * t_1));
	} else {
		tmp = sqrt(t) * sqrt(t_1);
	}
	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 = (u * n) * 2.0d0
    if (t <= 9.6d-278) then
        tmp = sqrt((t * t_1))
    else
        tmp = sqrt(t) * sqrt(t_1)
    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 = (U * n) * 2.0;
	double tmp;
	if (t <= 9.6e-278) {
		tmp = Math.sqrt((t * t_1));
	} else {
		tmp = Math.sqrt(t) * Math.sqrt(t_1);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (U * n) * 2.0
	tmp = 0
	if t <= 9.6e-278:
		tmp = math.sqrt((t * t_1))
	else:
		tmp = math.sqrt(t) * math.sqrt(t_1)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(U * n) * 2.0)
	tmp = 0.0
	if (t <= 9.6e-278)
		tmp = sqrt(Float64(t * t_1));
	else
		tmp = Float64(sqrt(t) * sqrt(t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (U * n) * 2.0;
	tmp = 0.0;
	if (t <= 9.6e-278)
		tmp = sqrt((t * t_1));
	else
		tmp = sqrt(t) * sqrt(t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(U * n), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[t, 9.6e-278], N[Sqrt[N[(t * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[t], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 9.5999999999999999e-278

   1. Initial program 49.1%

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

   4. Applied rewrites 53.0%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 35.4%

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

   if 9.5999999999999999e-278 < t

   1. Initial program 51.5%

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

   4. Applied rewrites 55.2%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 38.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

   9. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\sqrt{t} \cdot \sqrt{\left(U \cdot n\right) \cdot 2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 36.5% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.2%

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

3. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 36.5

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

5. Applied rewrites 36.5%

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

Alternative 17: 35.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(2 \cdot \left(U \cdot t\right)\right) \cdot n} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.2%

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

3. Taylor expanded in n around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 38.9%

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

6. Taylor expanded in t around inf

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

   1. lower-*.f64 N/A

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

   2. lift-*.f64 35.8

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

8. Applied rewrites 35.8%

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

Reproduce

herbie shell --seed 2025091 
(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*))))))