Toniolo and Linder, Equation (13)

Percentage Accurate: 49.5% → 61.7%

Time: 12.7s

Alternatives: 8

Speedup: 2.9×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 49.5% 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: 61.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot \left(U - U*\right)\\ t_2 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot t\_2\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(\frac{\left(-U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{t\_1}{Om}\right) - 2 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om} + U \cdot t\right)}\\ \mathbf{elif}\;t\_3 \leq 10^{+289}:\\ \;\;\;\;\sqrt{t\_3}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(t - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right) - \left(U - U*\right) \cdot \left(t\_2 \cdot n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot U + \frac{U \cdot t\_1}{Om}\right)}{-Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (- U U*)))
        (t_2 (pow (/ l Om) 2.0))
        (t_3
         (*
          (* (* 2.0 n) U)
          (- (- t (* 2.0 (/ (* l l) Om))) (* (* n t_2) (- U U*))))))
   (if (<= t_3 0.0)
     (sqrt
      (*
       (* n 2.0)
       (+
        (/ (- (* (- U) (* (* l l) (/ t_1 Om))) (* 2.0 (* U (* l l)))) Om)
        (* U t))))
     (if (<= t_3 1e+289)
       (sqrt t_3)
       (if (<= t_3 INFINITY)
         (sqrt
          (*
           (* n 2.0)
           (* U (- (- t (* (* l (/ l Om)) 2.0)) (* (- U U*) (* t_2 n))))))
         (sqrt
          (*
           (* n 2.0)
           (/ (* (* l l) (+ (* 2.0 U) (/ (* U t_1) Om))) (- Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * (U - U_42_);
	double t_2 = pow((l / Om), 2.0);
	double t_3 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * t_2) * (U - U_42_)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt(((n * 2.0) * ((((-U * ((l * l) * (t_1 / Om))) - (2.0 * (U * (l * l)))) / Om) + (U * t))));
	} else if (t_3 <= 1e+289) {
		tmp = sqrt(t_3);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((n * 2.0) * (U * ((t - ((l * (l / Om)) * 2.0)) - ((U - U_42_) * (t_2 * n))))));
	} else {
		tmp = sqrt(((n * 2.0) * (((l * l) * ((2.0 * U) + ((U * t_1) / Om))) / -Om)));
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * (U - U_42_);
	double t_2 = Math.pow((l / Om), 2.0);
	double t_3 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * t_2) * (U - U_42_)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = Math.sqrt(((n * 2.0) * ((((-U * ((l * l) * (t_1 / Om))) - (2.0 * (U * (l * l)))) / Om) + (U * t))));
	} else if (t_3 <= 1e+289) {
		tmp = Math.sqrt(t_3);
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((n * 2.0) * (U * ((t - ((l * (l / Om)) * 2.0)) - ((U - U_42_) * (t_2 * n))))));
	} else {
		tmp = Math.sqrt(((n * 2.0) * (((l * l) * ((2.0 * U) + ((U * t_1) / Om))) / -Om)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = n * (U - U_42_)
	t_2 = math.pow((l / Om), 2.0)
	t_3 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * t_2) * (U - U_42_)))
	tmp = 0
	if t_3 <= 0.0:
		tmp = math.sqrt(((n * 2.0) * ((((-U * ((l * l) * (t_1 / Om))) - (2.0 * (U * (l * l)))) / Om) + (U * t))))
	elif t_3 <= 1e+289:
		tmp = math.sqrt(t_3)
	elif t_3 <= math.inf:
		tmp = math.sqrt(((n * 2.0) * (U * ((t - ((l * (l / Om)) * 2.0)) - ((U - U_42_) * (t_2 * n))))))
	else:
		tmp = math.sqrt(((n * 2.0) * (((l * l) * ((2.0 * U) + ((U * t_1) / Om))) / -Om)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * Float64(U - U_42_))
	t_2 = Float64(l / Om) ^ 2.0
	t_3 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * t_2) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(Float64(Float64(Float64(Float64(-U) * Float64(Float64(l * l) * Float64(t_1 / Om))) - Float64(2.0 * Float64(U * Float64(l * l)))) / Om) + Float64(U * t))));
	elseif (t_3 <= 1e+289)
		tmp = sqrt(t_3);
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(Float64(t - Float64(Float64(l * Float64(l / Om)) * 2.0)) - Float64(Float64(U - U_42_) * Float64(t_2 * n))))));
	else
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(Float64(Float64(l * l) * Float64(Float64(2.0 * U) + Float64(Float64(U * t_1) / Om))) / Float64(-Om))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = n * (U - U_42_);
	t_2 = (l / Om) ^ 2.0;
	t_3 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * t_2) * (U - U_42_)));
	tmp = 0.0;
	if (t_3 <= 0.0)
		tmp = sqrt(((n * 2.0) * ((((-U * ((l * l) * (t_1 / Om))) - (2.0 * (U * (l * l)))) / Om) + (U * t))));
	elseif (t_3 <= 1e+289)
		tmp = sqrt(t_3);
	elseif (t_3 <= Inf)
		tmp = sqrt(((n * 2.0) * (U * ((t - ((l * (l / Om)) * 2.0)) - ((U - U_42_) * (t_2 * n))))));
	else
		tmp = sqrt(((n * 2.0) * (((l * l) * ((2.0 * U) + ((U * t_1) / Om))) / -Om)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = 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 * t$95$2), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(N[(N[(N[((-U) * N[(N[(l * l), $MachinePrecision] * N[(t$95$1 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(U * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 1e+289], N[Sqrt[t$95$3], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(N[(t - N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(t$95$2 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 * U), $MachinePrecision] + N[(N[(U * t$95$1), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-Om)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 14.8%

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

   4. Applied rewrites 44.4%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 34.4%

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

   8. Taylor expanded in Om around -inf

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

      1. lower-+.f64 N/A

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

   10. Applied rewrites 46.7%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1.0000000000000001e289

   1. Initial program 98.7%

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

   if 1.0000000000000001e289 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 33.3%

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

   4. Applied rewrites 45.5%

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

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

   1. Initial program 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 6.6%

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

   5. Taylor expanded in Om around -inf

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

      1. lower-+.f64 N/A

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift--.f64 39.6

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

   10. Applied rewrites 39.6%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(\frac{\left(-U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right) - 2 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om} + U \cdot t\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 10^{+289}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(t - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot U + \frac{U \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)}{-Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot \left(U - U*\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t - 2 \cdot \frac{\ell \cdot \ell}{Om}\\ t_4 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_5 := t\_2 \cdot \left(t\_3 - \left(n \cdot t\_4\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_5 \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(\frac{\left(-U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{t\_1}{Om}\right) - 2 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om} + U \cdot t\right)}\\ \mathbf{elif}\;t\_5 \leq 10^{+289}:\\ \;\;\;\;\sqrt{t\_2 \cdot \left(t\_3 - \left(n \cdot \mathsf{ratio\_of\_squares}\left(\ell, Om\right)\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(t - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right) - \left(U - U*\right) \cdot \left(t\_4 \cdot n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot U + \frac{U \cdot t\_1}{Om}\right)}{-Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (- U U*)))
        (t_2 (* (* 2.0 n) U))
        (t_3 (- t (* 2.0 (/ (* l l) Om))))
        (t_4 (pow (/ l Om) 2.0))
        (t_5 (* t_2 (- t_3 (* (* n t_4) (- U U*))))))
   (if (<= t_5 0.0)
     (sqrt
      (*
       (* n 2.0)
       (+
        (/ (- (* (- U) (* (* l l) (/ t_1 Om))) (* 2.0 (* U (* l l)))) Om)
        (* U t))))
     (if (<= t_5 1e+289)
       (sqrt (* t_2 (- t_3 (* (* n (ratio-of-squares l Om)) (- U U*)))))
       (if (<= t_5 INFINITY)
         (sqrt
          (*
           (* n 2.0)
           (* U (- (- t (* (* l (/ l Om)) 2.0)) (* (- U U*) (* t_4 n))))))
         (sqrt
          (*
           (* n 2.0)
           (/ (* (* l l) (+ (* 2.0 U) (/ (* U t_1) Om))) (- Om)))))))))

Derivation

1. Split input into 4 regimes

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

   1. Initial program 14.8%

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

   4. Applied rewrites 44.4%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 34.4%

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

   8. Taylor expanded in Om around -inf

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

      1. lower-+.f64 N/A

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

   10. Applied rewrites 46.7%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1.0000000000000001e289

   1. Initial program 98.7%

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

   3. Taylor expanded in l around 0

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

      1. pow2 N/A

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

      2. unpow2 N/A

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

      3. lower-ratio-of-squares.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 1.0000000000000001e289 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 33.3%

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

   4. Applied rewrites 45.5%

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

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

   1. Initial program 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 6.6%

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

   5. Taylor expanded in Om around -inf

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

      1. lower-+.f64 N/A

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift--.f64 39.6

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

   10. Applied rewrites 39.6%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(\frac{\left(-U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right) - 2 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om} + U \cdot t\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 10^{+289}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \mathsf{ratio\_of\_squares}\left(\ell, Om\right)\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(t - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot U + \frac{U \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)}{-Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot \left(U - U*\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t - 2 \cdot \frac{\ell \cdot \ell}{Om}\\ t_4 := t\_2 \cdot \left(t\_3 - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(\frac{\left(-U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{t\_1}{Om}\right) - 2 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om} + U \cdot t\right)}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{t\_2 \cdot \left(t\_3 - \left(n \cdot \mathsf{ratio\_of\_squares}\left(\ell, Om\right)\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot U + \frac{U \cdot t\_1}{Om}\right)}{-Om}}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (- U U*)))
        (t_2 (* (* 2.0 n) U))
        (t_3 (- t (* 2.0 (/ (* l l) Om))))
        (t_4 (* t_2 (- t_3 (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_4 0.0)
     (sqrt
      (*
       (* n 2.0)
       (+
        (/ (- (* (- U) (* (* l l) (/ t_1 Om))) (* 2.0 (* U (* l l)))) Om)
        (* U t))))
     (if (<= t_4 INFINITY)
       (sqrt (* t_2 (- t_3 (* (* n (ratio-of-squares l Om)) (- U U*)))))
       (sqrt
        (*
         (* n 2.0)
         (/ (* (* l l) (+ (* 2.0 U) (/ (* U t_1) Om))) (- Om))))))))

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 14.8%

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

   4. Applied rewrites 44.4%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 34.4%

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

   8. Taylor expanded in Om around -inf

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

      1. lower-+.f64 N/A

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

   10. Applied rewrites 46.7%

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

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

   1. Initial program 73.5%

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

   3. Taylor expanded in l around 0

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

      1. pow2 N/A

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

      2. unpow2 N/A

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

      3. lower-ratio-of-squares.f64 73.5

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

   5. Applied rewrites 73.5%

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

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

   1. Initial program 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 6.6%

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

   5. Taylor expanded in Om around -inf

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

      1. lower-+.f64 N/A

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift--.f64 39.6

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

   10. Applied rewrites 39.6%

       \[\leadsto \sqrt{\left(n \cdot 2\right) \cdot \left(-1 \cdot \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot U + \frac{U \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)}{Om}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 0:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(\frac{\left(-U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{n \cdot \left(U - U*\right)}{Om}\right) - 2 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)}{Om} + U \cdot t\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \mathsf{ratio\_of\_squares}\left(\ell, Om\right)\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot U + \frac{U \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)}{-Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.6% accurate, 0.8× speedup

Math

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

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)
     (sqrt (* (* n 2.0) (* U t)))
     (sqrt (* t_1 (- t (* (* n (ratio-of-squares l Om)) (- U U*))))))))

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 16.0%

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

   4. Applied rewrites 47.9%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 37.3%

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

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

   1. Initial program 61.4%

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

   3. Taylor expanded in l around 0

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

      1. pow2 N/A

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

      2. unpow2 N/A

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

      3. lower-ratio-of-squares.f64 61.4

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

   5. Applied rewrites 61.4%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 64.0%

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

Alternative 5: 52.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t - 2 \cdot \frac{\ell \cdot \ell}{Om}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 5 \cdot 10^{-272}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t - \left(n \cdot \mathsf{ratio\_of\_squares}\left(\ell, Om\right)\right) \cdot \left(U - U*\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U)) (t_2 (- t (* 2.0 (/ (* l l) Om)))))
   (if (<= (* t_1 (- t_2 (* (* n (pow (/ l Om) 2.0)) (- U U*)))) 5e-272)
     (sqrt (* 2.0 (* U (* n t_2))))
     (sqrt (* t_1 (- t (* (* n (ratio-of-squares l Om)) (- U U*))))))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.99999999999999982e-272

   1. Initial program 25.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. 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 40.8

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

   5. Applied rewrites 40.8%

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

   6. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-*.f64 N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 49.5

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

   8. Applied rewrites 49.5%

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

   if 4.99999999999999982e-272 < (*.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 61.3%

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

   3. Taylor expanded in l around 0

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

      1. pow2 N/A

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

      2. unpow2 N/A

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

      3. lower-ratio-of-squares.f64 61.3

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

   5. Applied rewrites 61.3%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 64.3%

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

Alternative 6: 38.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ \mathbf{if}\;\sqrt{t\_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 2 \cdot 10^{-136}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1 \cdot t}\\ \end{array} \end{array} \]

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*)))))
        2e-136)
     (sqrt (* (* (* t n) U) 2.0))
     (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_))))) <= 2e-136) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} 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))))) <= 2d-136) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    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_))))) <= 2e-136) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} 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_))))) <= 2e-136:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	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_))))) <= 2e-136)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	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_))))) <= 2e-136)
		tmp = sqrt((((t * n) * U) * 2.0));
	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], 2e-136], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $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*))))) < 2e-136

   1. Initial program 27.7%

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

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   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 43.8

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

   5. Applied rewrites 43.8%

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

   if 2e-136 < (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 60.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. 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 48.1%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 2 \cdot 10^{-136}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot \mathsf{ratio\_of\_squares}\left(\ell, Om\right)\\ t_2 := \left(2 \cdot n\right) \cdot U\\ \mathbf{if}\;n \leq -0.108:\\ \;\;\;\;\sqrt{t\_2 \cdot \left(t - t\_1 \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;n \leq 9 \cdot 10^{-53}:\\ \;\;\;\;\sqrt{\left(\left(\left(t - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_2 \cdot \left(t - t\_1 \cdot \left(-U*\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (ratio-of-squares l Om))) (t_2 (* (* 2.0 n) U)))
   (if (<= n -0.108)
     (sqrt (* t_2 (- t (* t_1 (- U U*)))))
     (if (<= n 9e-53)
       (sqrt (* (* (* (- t (* (* l (/ l Om)) 2.0)) n) U) 2.0))
       (sqrt (* t_2 (- t (* t_1 (- U*)))))))))

Derivation

1. Split input into 3 regimes

2. if n < -0.107999999999999999

   1. Initial program 62.9%

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

   3. Taylor expanded in l around 0

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

      1. pow2 N/A

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

      2. unpow2 N/A

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

      3. lower-ratio-of-squares.f64 62.8

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

   5. Applied rewrites 62.8%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 72.9%

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

   if -0.107999999999999999 < n < 8.9999999999999997e-53

   1. Initial program 45.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. 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. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-/.f64 57.9

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

   5. Applied rewrites 57.9%

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

   if 8.9999999999999997e-53 < n

   1. Initial program 63.5%

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

   3. Taylor expanded in l around 0

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

      1. pow2 N/A

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

      2. unpow2 N/A

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

      3. lower-ratio-of-squares.f64 63.5

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

   5. Applied rewrites 63.5%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 67.8%

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

   9. Taylor expanded in U around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(n \cdot \mathsf{ratio\_of\_squares}\left(\ell, Om\right)\right) \cdot \color{blue}{\left(-1 \cdot U*\right)}\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 67.8

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

   11. Applied rewrites 67.8%

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(n \cdot \mathsf{ratio\_of\_squares}\left(\ell, Om\right)\right) \cdot \color{blue}{\left(-1 \cdot U*\right)}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -0.108:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(n \cdot \mathsf{ratio\_of\_squares}\left(\ell, Om\right)\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;n \leq 9 \cdot 10^{-53}:\\ \;\;\;\;\sqrt{\left(\left(\left(t - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(n \cdot \mathsf{ratio\_of\_squares}\left(\ell, Om\right)\right) \cdot \left(-U*\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.7% 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 55.1%

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

3. Taylor expanded in 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 44.2

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

5. Applied rewrites 44.2%

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

Reproduce

herbie shell --seed 2025058 
(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*))))))