Toniolo and Linder, Equation (13)

Percentage Accurate: 49.5% → 60.8%

Time: 10.8s

Alternatives: 14

Speedup: 2.0×

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: 60.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l / Om) * (l / Om);
	double t_2 = (U - U_42_) * n;
	double t_3 = (2.0 * n) * U;
	double t_4 = t - (2.0 * ((l * l) / Om));
	double t_5 = t_3 * (t_4 - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_5 <= 0.0) {
		tmp = sqrt(((n * 2.0) * ((((-U * ((l * l) * (t_2 / Om))) - (((l * l) * U) * 2.0)) / Om) + (t * U))));
	} else if (t_5 <= 1e+289) {
		tmp = sqrt((t_3 * (t_4 - ((n * t_1) * (U - U_42_)))));
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = sqrt(((n * 2.0) * (U * ((t - ((l * (l / Om)) * 2.0)) - ((U - U_42_) * (t_1 * n))))));
	} else {
		tmp = sqrt(((-2.0 * U) * (((l * l) * n) * ((t_2 / (Om * Om)) + (2.0 / Om)))));
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l / Om) * (l / Om);
	double t_2 = (U - U_42_) * n;
	double t_3 = (2.0 * n) * U;
	double t_4 = t - (2.0 * ((l * l) / Om));
	double t_5 = t_3 * (t_4 - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_5 <= 0.0) {
		tmp = Math.sqrt(((n * 2.0) * ((((-U * ((l * l) * (t_2 / Om))) - (((l * l) * U) * 2.0)) / Om) + (t * U))));
	} else if (t_5 <= 1e+289) {
		tmp = Math.sqrt((t_3 * (t_4 - ((n * t_1) * (U - U_42_)))));
	} else if (t_5 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((n * 2.0) * (U * ((t - ((l * (l / Om)) * 2.0)) - ((U - U_42_) * (t_1 * n))))));
	} else {
		tmp = Math.sqrt(((-2.0 * U) * (((l * l) * n) * ((t_2 / (Om * Om)) + (2.0 / Om)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (l / Om) * (l / Om)
	t_2 = (U - U_42_) * n
	t_3 = (2.0 * n) * U
	t_4 = t - (2.0 * ((l * l) / Om))
	t_5 = t_3 * (t_4 - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))
	tmp = 0
	if t_5 <= 0.0:
		tmp = math.sqrt(((n * 2.0) * ((((-U * ((l * l) * (t_2 / Om))) - (((l * l) * U) * 2.0)) / Om) + (t * U))))
	elif t_5 <= 1e+289:
		tmp = math.sqrt((t_3 * (t_4 - ((n * t_1) * (U - U_42_)))))
	elif t_5 <= math.inf:
		tmp = math.sqrt(((n * 2.0) * (U * ((t - ((l * (l / Om)) * 2.0)) - ((U - U_42_) * (t_1 * n))))))
	else:
		tmp = math.sqrt(((-2.0 * U) * (((l * l) * n) * ((t_2 / (Om * Om)) + (2.0 / Om)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l / Om) * Float64(l / Om))
	t_2 = Float64(Float64(U - U_42_) * n)
	t_3 = Float64(Float64(2.0 * n) * U)
	t_4 = Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om)))
	t_5 = Float64(t_3 * Float64(t_4 - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_5 <= 0.0)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(Float64(Float64(Float64(Float64(-U) * Float64(Float64(l * l) * Float64(t_2 / Om))) - Float64(Float64(Float64(l * l) * U) * 2.0)) / Om) + Float64(t * U))));
	elseif (t_5 <= 1e+289)
		tmp = sqrt(Float64(t_3 * Float64(t_4 - Float64(Float64(n * t_1) * Float64(U - U_42_)))));
	elseif (t_5 <= 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_1 * n))))));
	else
		tmp = sqrt(Float64(Float64(-2.0 * U) * Float64(Float64(Float64(l * l) * n) * Float64(Float64(t_2 / Float64(Om * Om)) + Float64(2.0 / Om)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (l / Om) * (l / Om);
	t_2 = (U - U_42_) * n;
	t_3 = (2.0 * n) * U;
	t_4 = t - (2.0 * ((l * l) / Om));
	t_5 = t_3 * (t_4 - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)));
	tmp = 0.0;
	if (t_5 <= 0.0)
		tmp = sqrt(((n * 2.0) * ((((-U * ((l * l) * (t_2 / Om))) - (((l * l) * U) * 2.0)) / Om) + (t * U))));
	elseif (t_5 <= 1e+289)
		tmp = sqrt((t_3 * (t_4 - ((n * t_1) * (U - U_42_)))));
	elseif (t_5 <= Inf)
		tmp = sqrt(((n * 2.0) * (U * ((t - ((l * (l / Om)) * 2.0)) - ((U - U_42_) * (t_1 * n))))));
	else
		tmp = sqrt(((-2.0 * U) * (((l * l) * n) * ((t_2 / (Om * Om)) + (2.0 / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(U - U$42$), $MachinePrecision] * n), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$4 = N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 * N[(t$95$4 - 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$5, 0.0], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(N[(N[(N[((-U) * N[(N[(l * l), $MachinePrecision] * N[(t$95$2 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(l * l), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(t * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$5, 1e+289], N[Sqrt[N[(t$95$3 * N[(t$95$4 - N[(N[(n * t$95$1), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$5, 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$1 * 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[(t$95$2 / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $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 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 46.7%

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

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 98.7

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

   4. Applied rewrites 98.7%

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

   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%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 45.5

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

   6. Applied rewrites 45.5%

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

   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

   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. lower-+.f64 N/A

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

   5. Applied rewrites 39.4%

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

4. Final simplification 68.4%

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

Alternative 2: 60.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l / Om) * (l / Om);
	double t_2 = (2.0 * n) * U;
	double t_3 = sqrt(((n * 2.0) * (U * ((t - ((l * (l / Om)) * 2.0)) - ((U - U_42_) * (t_1 * n))))));
	double t_4 = t - (2.0 * ((l * l) / Om));
	double t_5 = t_2 * (t_4 - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_5 <= 0.0) {
		tmp = t_3;
	} else if (t_5 <= 1e+289) {
		tmp = sqrt((t_2 * (t_4 - ((n * t_1) * (U - U_42_)))));
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = sqrt(((-2.0 * U) * (((l * l) * n) * ((((U - U_42_) * n) / (Om * Om)) + (2.0 / Om)))));
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l / Om) * (l / Om);
	double t_2 = (2.0 * n) * U;
	double t_3 = Math.sqrt(((n * 2.0) * (U * ((t - ((l * (l / Om)) * 2.0)) - ((U - U_42_) * (t_1 * n))))));
	double t_4 = t - (2.0 * ((l * l) / Om));
	double t_5 = t_2 * (t_4 - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_5 <= 0.0) {
		tmp = t_3;
	} else if (t_5 <= 1e+289) {
		tmp = Math.sqrt((t_2 * (t_4 - ((n * t_1) * (U - U_42_)))));
	} else if (t_5 <= Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else {
		tmp = Math.sqrt(((-2.0 * U) * (((l * l) * n) * ((((U - U_42_) * n) / (Om * Om)) + (2.0 / Om)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (l / Om) * (l / Om)
	t_2 = (2.0 * n) * U
	t_3 = math.sqrt(((n * 2.0) * (U * ((t - ((l * (l / Om)) * 2.0)) - ((U - U_42_) * (t_1 * n))))))
	t_4 = t - (2.0 * ((l * l) / Om))
	t_5 = t_2 * (t_4 - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))
	tmp = 0
	if t_5 <= 0.0:
		tmp = t_3
	elif t_5 <= 1e+289:
		tmp = math.sqrt((t_2 * (t_4 - ((n * t_1) * (U - U_42_)))))
	elif t_5 <= math.inf:
		tmp = t_3
	else:
		tmp = math.sqrt(((-2.0 * U) * (((l * l) * n) * ((((U - U_42_) * n) / (Om * Om)) + (2.0 / Om)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l / Om) * Float64(l / Om))
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = 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_1 * n))))))
	t_4 = Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om)))
	t_5 = Float64(t_2 * Float64(t_4 - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_5 <= 0.0)
		tmp = t_3;
	elseif (t_5 <= 1e+289)
		tmp = sqrt(Float64(t_2 * Float64(t_4 - Float64(Float64(n * t_1) * Float64(U - U_42_)))));
	elseif (t_5 <= Inf)
		tmp = t_3;
	else
		tmp = sqrt(Float64(Float64(-2.0 * U) * Float64(Float64(Float64(l * l) * n) * Float64(Float64(Float64(Float64(U - U_42_) * n) / Float64(Om * Om)) + Float64(2.0 / Om)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (l / Om) * (l / Om);
	t_2 = (2.0 * n) * U;
	t_3 = sqrt(((n * 2.0) * (U * ((t - ((l * (l / Om)) * 2.0)) - ((U - U_42_) * (t_1 * n))))));
	t_4 = t - (2.0 * ((l * l) / Om));
	t_5 = t_2 * (t_4 - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)));
	tmp = 0.0;
	if (t_5 <= 0.0)
		tmp = t_3;
	elseif (t_5 <= 1e+289)
		tmp = sqrt((t_2 * (t_4 - ((n * t_1) * (U - U_42_)))));
	elseif (t_5 <= Inf)
		tmp = t_3;
	else
		tmp = sqrt(((-2.0 * U) * (((l * l) * n) * ((((U - U_42_) * n) / (Om * Om)) + (2.0 / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = 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$1 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 * N[(t$95$4 - 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$5, 0.0], t$95$3, If[LessEqual[t$95$5, 1e+289], N[Sqrt[N[(t$95$2 * N[(t$95$4 - N[(N[(n * t$95$1), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$5, Infinity], t$95$3, 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), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 / 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 or 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 26.7%

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

   4. Applied rewrites 45.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 45.1

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

   6. Applied rewrites 45.1%

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

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 98.7

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

   4. Applied rewrites 98.7%

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

   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

   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. lower-+.f64 N/A

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

   5. Applied rewrites 39.4%

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

Alternative 3: 53.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t - 2 \cdot \frac{\ell \cdot \ell}{Om}\\ t_3 := t\_1 \cdot \left(t\_2 - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\_2\right)\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \frac{\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U*\right) \cdot U}{Om \cdot Om}}\\ \end{array} \end{array} \]

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))))
        (t_3 (* t_1 (- t_2 (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_3 2e-303)
     (sqrt (* 2.0 (* U (* n t_2))))
     (if (<= t_3 INFINITY)
       (sqrt (* t_1 (- t (* (* l (/ l Om)) 2.0))))
       (sqrt (* (* n 2.0) (/ (* (* (* (* l l) n) U*) U) (* Om Om))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t - (2.0 * ((l * l) / Om));
	double t_3 = t_1 * (t_2 - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_3 <= 2e-303) {
		tmp = sqrt((2.0 * (U * (n * t_2))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * (t - ((l * (l / Om)) * 2.0))));
	} else {
		tmp = sqrt(((n * 2.0) * (((((l * l) * n) * U_42_) * U) / (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 = (2.0 * n) * U;
	double t_2 = t - (2.0 * ((l * l) / Om));
	double t_3 = t_1 * (t_2 - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_3 <= 2e-303) {
		tmp = Math.sqrt((2.0 * (U * (n * t_2))));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_1 * (t - ((l * (l / Om)) * 2.0))));
	} else {
		tmp = Math.sqrt(((n * 2.0) * (((((l * l) * n) * U_42_) * U) / (Om * Om))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (2.0 * n) * U
	t_2 = t - (2.0 * ((l * l) / Om))
	t_3 = t_1 * (t_2 - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))
	tmp = 0
	if t_3 <= 2e-303:
		tmp = math.sqrt((2.0 * (U * (n * t_2))))
	elif t_3 <= math.inf:
		tmp = math.sqrt((t_1 * (t - ((l * (l / Om)) * 2.0))))
	else:
		tmp = math.sqrt(((n * 2.0) * (((((l * l) * n) * U_42_) * U) / (Om * Om))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om)))
	t_3 = Float64(t_1 * Float64(t_2 - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_3 <= 2e-303)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t_2))));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_1 * Float64(t - Float64(Float64(l * Float64(l / Om)) * 2.0))));
	else
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(Float64(Float64(Float64(Float64(l * l) * n) * U_42_) * U) / Float64(Om * 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 = t - (2.0 * ((l * l) / Om));
	t_3 = t_1 * (t_2 - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)));
	tmp = 0.0;
	if (t_3 <= 2e-303)
		tmp = sqrt((2.0 * (U * (n * t_2))));
	elseif (t_3 <= Inf)
		tmp = sqrt((t_1 * (t - ((l * (l / Om)) * 2.0))));
	else
		tmp = sqrt(((n * 2.0) * (((((l * l) * n) * U_42_) * U) / (Om * 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[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(t$95$2 - 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, 2e-303], N[Sqrt[N[(2.0 * N[(U * N[(n * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$1 * N[(t - N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(N[(N[(N[(N[(l * l), $MachinePrecision] * n), $MachinePrecision] * U$42$), $MachinePrecision] * U), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1.99999999999999986e-303

   1. Initial program 20.5%

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

   3. Taylor expanded in U* around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow-prod-down N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 14.2

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

   5. Applied rewrites 14.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 14.2

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

   7. Applied rewrites 14.2%

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

   8. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 45.9

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

   10. Applied rewrites 45.9%

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

   if 1.99999999999999986e-303 < (*.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 73.1%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 69.9

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

   5. Applied rewrites 69.9%

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

   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 U* around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 35.1

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

   7. Applied rewrites 35.1%

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

Alternative 4: 53.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t - 2 \cdot \frac{\ell \cdot \ell}{Om}\\ t_3 := t\_1 \cdot \left(t\_2 - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\_2\right)\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \left(\left(\left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot n\right)\right) \cdot U*\right) \cdot U\right)}{Om \cdot Om}}\\ \end{array} \end{array} \]

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))))
        (t_3 (* t_1 (- t_2 (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_3 2e-303)
     (sqrt (* 2.0 (* U (* n t_2))))
     (if (<= t_3 INFINITY)
       (sqrt (* t_1 (- t (* (* l (/ l Om)) 2.0))))
       (sqrt (/ (* 2.0 (* (* (* (* l n) (* l n)) U*) U)) (* Om Om)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t - (2.0 * ((l * l) / Om));
	double t_3 = t_1 * (t_2 - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_3 <= 2e-303) {
		tmp = sqrt((2.0 * (U * (n * t_2))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * (t - ((l * (l / Om)) * 2.0))));
	} else {
		tmp = sqrt(((2.0 * ((((l * n) * (l * n)) * U_42_) * U)) / (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 = (2.0 * n) * U;
	double t_2 = t - (2.0 * ((l * l) / Om));
	double t_3 = t_1 * (t_2 - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_3 <= 2e-303) {
		tmp = Math.sqrt((2.0 * (U * (n * t_2))));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_1 * (t - ((l * (l / Om)) * 2.0))));
	} else {
		tmp = Math.sqrt(((2.0 * ((((l * n) * (l * n)) * U_42_) * U)) / (Om * Om)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (2.0 * n) * U
	t_2 = t - (2.0 * ((l * l) / Om))
	t_3 = t_1 * (t_2 - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))
	tmp = 0
	if t_3 <= 2e-303:
		tmp = math.sqrt((2.0 * (U * (n * t_2))))
	elif t_3 <= math.inf:
		tmp = math.sqrt((t_1 * (t - ((l * (l / Om)) * 2.0))))
	else:
		tmp = math.sqrt(((2.0 * ((((l * n) * (l * n)) * U_42_) * U)) / (Om * Om)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om)))
	t_3 = Float64(t_1 * Float64(t_2 - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_3 <= 2e-303)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t_2))));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_1 * Float64(t - Float64(Float64(l * Float64(l / Om)) * 2.0))));
	else
		tmp = sqrt(Float64(Float64(2.0 * Float64(Float64(Float64(Float64(l * n) * Float64(l * n)) * U_42_) * U)) / Float64(Om * 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 = t - (2.0 * ((l * l) / Om));
	t_3 = t_1 * (t_2 - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)));
	tmp = 0.0;
	if (t_3 <= 2e-303)
		tmp = sqrt((2.0 * (U * (n * t_2))));
	elseif (t_3 <= Inf)
		tmp = sqrt((t_1 * (t - ((l * (l / Om)) * 2.0))));
	else
		tmp = sqrt(((2.0 * ((((l * n) * (l * n)) * U_42_) * U)) / (Om * 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[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(t$95$2 - 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, 2e-303], N[Sqrt[N[(2.0 * N[(U * N[(n * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$1 * N[(t - N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * N[(N[(N[(N[(l * n), $MachinePrecision] * N[(l * n), $MachinePrecision]), $MachinePrecision] * U$42$), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1.99999999999999986e-303

   1. Initial program 20.5%

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

   3. Taylor expanded in U* around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow-prod-down N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 14.2

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

   5. Applied rewrites 14.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 14.2

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

   7. Applied rewrites 14.2%

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

   8. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 45.9

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

   10. Applied rewrites 45.9%

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

   if 1.99999999999999986e-303 < (*.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 73.1%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 69.9

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

   5. Applied rewrites 69.9%

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

   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 U* around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow-prod-down N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 32.8

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

   5. Applied rewrites 32.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 32.8

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

   7. Applied rewrites 32.8%

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

Alternative 5: 52.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t - 2 \cdot \frac{\ell \cdot \ell}{Om}\\ t_3 := t\_1 \cdot \left(t\_2 - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\_2\right)\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U* \cdot U} \cdot \frac{\left(\sqrt{2} \cdot n\right) \cdot \ell}{Om}\\ \end{array} \end{array} \]

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))))
        (t_3 (* t_1 (- t_2 (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_3 2e-303)
     (sqrt (* 2.0 (* U (* n t_2))))
     (if (<= t_3 INFINITY)
       (sqrt (* t_1 (- t (* (* l (/ l Om)) 2.0))))
       (* (sqrt (* U* U)) (/ (* (* (sqrt 2.0) n) l) 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 = t - (2.0 * ((l * l) / Om));
	double t_3 = t_1 * (t_2 - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_3 <= 2e-303) {
		tmp = sqrt((2.0 * (U * (n * t_2))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * (t - ((l * (l / Om)) * 2.0))));
	} else {
		tmp = sqrt((U_42_ * U)) * (((sqrt(2.0) * n) * l) / Om);
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t - (2.0 * ((l * l) / Om));
	double t_3 = t_1 * (t_2 - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_3 <= 2e-303) {
		tmp = Math.sqrt((2.0 * (U * (n * t_2))));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_1 * (t - ((l * (l / Om)) * 2.0))));
	} else {
		tmp = Math.sqrt((U_42_ * U)) * (((Math.sqrt(2.0) * n) * l) / Om);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (2.0 * n) * U
	t_2 = t - (2.0 * ((l * l) / Om))
	t_3 = t_1 * (t_2 - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))
	tmp = 0
	if t_3 <= 2e-303:
		tmp = math.sqrt((2.0 * (U * (n * t_2))))
	elif t_3 <= math.inf:
		tmp = math.sqrt((t_1 * (t - ((l * (l / Om)) * 2.0))))
	else:
		tmp = math.sqrt((U_42_ * U)) * (((math.sqrt(2.0) * n) * l) / Om)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om)))
	t_3 = Float64(t_1 * Float64(t_2 - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_3 <= 2e-303)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t_2))));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_1 * Float64(t - Float64(Float64(l * Float64(l / Om)) * 2.0))));
	else
		tmp = Float64(sqrt(Float64(U_42_ * U)) * Float64(Float64(Float64(sqrt(2.0) * n) * l) / 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 = t - (2.0 * ((l * l) / Om));
	t_3 = t_1 * (t_2 - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)));
	tmp = 0.0;
	if (t_3 <= 2e-303)
		tmp = sqrt((2.0 * (U * (n * t_2))));
	elseif (t_3 <= Inf)
		tmp = sqrt((t_1 * (t - ((l * (l / Om)) * 2.0))));
	else
		tmp = sqrt((U_42_ * U)) * (((sqrt(2.0) * n) * l) / 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[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(t$95$2 - 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, 2e-303], N[Sqrt[N[(2.0 * N[(U * N[(n * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$1 * N[(t - N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U$42$ * U), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * n), $MachinePrecision] * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1.99999999999999986e-303

   1. Initial program 20.5%

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

   3. Taylor expanded in U* around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow-prod-down N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 14.2

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

   5. Applied rewrites 14.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 14.2

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

   7. Applied rewrites 14.2%

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

   8. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 45.9

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

   10. Applied rewrites 45.9%

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

   if 1.99999999999999986e-303 < (*.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 73.1%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 69.9

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

   5. Applied rewrites 69.9%

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

   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 U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 13.0

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

   5. Applied rewrites 13.0%

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

Alternative 6: 50.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t - 2 \cdot t\_1\\ t_4 := t\_2 \cdot \left(t\_3 - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_4 \leq 2 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\_3\right)\right)}\\ \mathbf{elif}\;t\_4 \leq 10^{+289}:\\ \;\;\;\;\sqrt{t\_2 \cdot \left(-2 \cdot t\_1 + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right)\right)}\\ \end{array} \end{array} \]

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 (* 2.0 t_1)))
        (t_4 (* t_2 (- t_3 (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_4 2e-303)
     (sqrt (* 2.0 (* U (* n t_3))))
     (if (<= t_4 1e+289)
       (sqrt (* t_2 (+ (* -2.0 t_1) t)))
       (sqrt (* (* n 2.0) (* U (- t (* (* (/ l Om) l) 2.0)))))))))

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 - (2.0 * t_1);
	double t_4 = t_2 * (t_3 - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_4 <= 2e-303) {
		tmp = sqrt((2.0 * (U * (n * t_3))));
	} else if (t_4 <= 1e+289) {
		tmp = sqrt((t_2 * ((-2.0 * t_1) + t)));
	} else {
		tmp = sqrt(((n * 2.0) * (U * (t - (((l / Om) * l) * 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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = (l * l) / om
    t_2 = (2.0d0 * n) * u
    t_3 = t - (2.0d0 * t_1)
    t_4 = t_2 * (t_3 - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))
    if (t_4 <= 2d-303) then
        tmp = sqrt((2.0d0 * (u * (n * t_3))))
    else if (t_4 <= 1d+289) then
        tmp = sqrt((t_2 * (((-2.0d0) * t_1) + t)))
    else
        tmp = sqrt(((n * 2.0d0) * (u * (t - (((l / om) * l) * 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 t_1 = (l * l) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = t - (2.0 * t_1);
	double t_4 = t_2 * (t_3 - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_4 <= 2e-303) {
		tmp = Math.sqrt((2.0 * (U * (n * t_3))));
	} else if (t_4 <= 1e+289) {
		tmp = Math.sqrt((t_2 * ((-2.0 * t_1) + t)));
	} else {
		tmp = Math.sqrt(((n * 2.0) * (U * (t - (((l / Om) * l) * 2.0)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (l * l) / Om
	t_2 = (2.0 * n) * U
	t_3 = t - (2.0 * t_1)
	t_4 = t_2 * (t_3 - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))
	tmp = 0
	if t_4 <= 2e-303:
		tmp = math.sqrt((2.0 * (U * (n * t_3))))
	elif t_4 <= 1e+289:
		tmp = math.sqrt((t_2 * ((-2.0 * t_1) + t)))
	else:
		tmp = math.sqrt(((n * 2.0) * (U * (t - (((l / Om) * l) * 2.0)))))
	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 - Float64(2.0 * t_1))
	t_4 = Float64(t_2 * Float64(t_3 - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_4 <= 2e-303)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t_3))));
	elseif (t_4 <= 1e+289)
		tmp = sqrt(Float64(t_2 * Float64(Float64(-2.0 * t_1) + t)));
	else
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t - Float64(Float64(Float64(l / Om) * l) * 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (l * l) / Om;
	t_2 = (2.0 * n) * U;
	t_3 = t - (2.0 * t_1);
	t_4 = t_2 * (t_3 - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)));
	tmp = 0.0;
	if (t_4 <= 2e-303)
		tmp = sqrt((2.0 * (U * (n * t_3))));
	elseif (t_4 <= 1e+289)
		tmp = sqrt((t_2 * ((-2.0 * t_1) + t)));
	else
		tmp = sqrt(((n * 2.0) * (U * (t - (((l / Om) * l) * 2.0)))));
	end
	tmp_2 = 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 - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * N[(t$95$3 - 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$4, 2e-303], N[Sqrt[N[(2.0 * N[(U * N[(n * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, 1e+289], N[Sqrt[N[(t$95$2 * N[(N[(-2.0 * t$95$1), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t - N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1.99999999999999986e-303

   1. Initial program 20.5%

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

   3. Taylor expanded in U* around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow-prod-down N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 14.2

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

   5. Applied rewrites 14.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 14.2

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

   7. Applied rewrites 14.2%

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

   8. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 45.9

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

   10. Applied rewrites 45.9%

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

   if 1.99999999999999986e-303 < (*.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 98.8%

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

   3. Taylor expanded in Om around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 90.9

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

   8. Applied rewrites 90.9%

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

   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*))))

   1. Initial program 22.7%

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

   4. Applied rewrites 33.2%

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

   5. Taylor expanded in n around 0

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

      1. associate--l- N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. *-commutative N/A

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

      12. associate--l- N/A

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

   7. Applied rewrites 27.7%

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

Alternative 7: 49.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t - 2 \cdot \frac{\ell \cdot \ell}{Om}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 2 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right)}\\ \end{array} \end{array} \]

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*)))) 2e-303)
     (sqrt (* 2.0 (* U (* n t_2))))
     (sqrt (* t_1 (- t (* (* l (/ l Om)) 2.0)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t - (2.0 * ((l * l) / Om));
	double tmp;
	if ((t_1 * (t_2 - ((n * pow((l / Om), 2.0)) * (U - U_42_)))) <= 2e-303) {
		tmp = sqrt((2.0 * (U * (n * t_2))));
	} else {
		tmp = sqrt((t_1 * (t - ((l * (l / Om)) * 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (2.0d0 * n) * u
    t_2 = t - (2.0d0 * ((l * l) / om))
    if ((t_1 * (t_2 - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))) <= 2d-303) then
        tmp = sqrt((2.0d0 * (u * (n * t_2))))
    else
        tmp = sqrt((t_1 * (t - ((l * (l / om)) * 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 t_1 = (2.0 * n) * U;
	double t_2 = t - (2.0 * ((l * l) / Om));
	double tmp;
	if ((t_1 * (t_2 - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))) <= 2e-303) {
		tmp = Math.sqrt((2.0 * (U * (n * t_2))));
	} else {
		tmp = Math.sqrt((t_1 * (t - ((l * (l / Om)) * 2.0))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (2.0 * n) * U
	t_2 = t - (2.0 * ((l * l) / Om))
	tmp = 0
	if (t_1 * (t_2 - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))) <= 2e-303:
		tmp = math.sqrt((2.0 * (U * (n * t_2))))
	else:
		tmp = math.sqrt((t_1 * (t - ((l * (l / Om)) * 2.0))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om)))
	tmp = 0.0
	if (Float64(t_1 * Float64(t_2 - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))) <= 2e-303)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t_2))));
	else
		tmp = sqrt(Float64(t_1 * Float64(t - Float64(Float64(l * Float64(l / Om)) * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (2.0 * n) * U;
	t_2 = t - (2.0 * ((l * l) / Om));
	tmp = 0.0;
	if ((t_1 * (t_2 - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)))) <= 2e-303)
		tmp = sqrt((2.0 * (U * (n * t_2))));
	else
		tmp = sqrt((t_1 * (t - ((l * (l / Om)) * 2.0))));
	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[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(t$95$2 - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-303], N[Sqrt[N[(2.0 * N[(U * N[(n * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t$95$1 * N[(t - N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1.99999999999999986e-303

   1. Initial program 20.5%

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

   3. Taylor expanded in U* around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow-prod-down N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 14.2

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

   5. Applied rewrites 14.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 14.2

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

   7. Applied rewrites 14.2%

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

   8. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 45.9

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

   10. Applied rewrites 45.9%

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

   if 1.99999999999999986e-303 < (*.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 61.8%

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

   3. Taylor expanded in n around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 59.4

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

   5. Applied rewrites 59.4%

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

Alternative 8: 38.0% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U)))
   (if (<=
        (*
         t_1
         (-
          (- t (* 2.0 (/ (* l l) Om)))
          (* (* n (pow (/ l Om) 2.0)) (- U U*))))
        5e-272)
     (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 ((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))) <= 5e-272) {
		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 ((t_1 * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))) <= 5d-272) 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 ((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))) <= 5e-272) {
		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 (t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))) <= 5e-272:
		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 (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_)))) <= 5e-272)
		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 ((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)))) <= 5e-272)
		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[(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], 5e-272], 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 (*.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%

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

   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 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 48.8%

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

Alternative 9: 53.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\\ t_2 := \left(U - U*\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right)\\ \mathbf{if}\;Om \leq -1.36 \cdot 10^{+175}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t\_1 - t\_2\right)\right)}\\ \mathbf{elif}\;Om \leq -1.46 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{\frac{\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot U}{Om} \cdot -4 - -2 \cdot \left(\left(t \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;Om \leq 2.6 \cdot 10^{-26}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - t\_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (- t (* (* l (/ l Om)) 2.0)))
        (t_2 (* (- U U*) (* (* (/ l Om) (/ l Om)) n))))
   (if (<= Om -1.36e+175)
     (sqrt (* (* n 2.0) (* U (- t_1 t_2))))
     (if (<= Om -1.46e-12)
       (sqrt (- (* (/ (* (* l (* l n)) U) Om) -4.0) (* -2.0 (* (* t n) U))))
       (if (<= Om 2.6e-26)
         (sqrt (* (* n 2.0) (* U (- t t_2))))
         (sqrt (* (* (* 2.0 n) U) t_1)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = t - ((l * (l / Om)) * 2.0);
	double t_2 = (U - U_42_) * (((l / Om) * (l / Om)) * n);
	double tmp;
	if (Om <= -1.36e+175) {
		tmp = sqrt(((n * 2.0) * (U * (t_1 - t_2))));
	} else if (Om <= -1.46e-12) {
		tmp = sqrt((((((l * (l * n)) * U) / Om) * -4.0) - (-2.0 * ((t * n) * U))));
	} else if (Om <= 2.6e-26) {
		tmp = sqrt(((n * 2.0) * (U * (t - t_2))));
	} else {
		tmp = sqrt((((2.0 * n) * U) * 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) :: t_2
    real(8) :: tmp
    t_1 = t - ((l * (l / om)) * 2.0d0)
    t_2 = (u - u_42) * (((l / om) * (l / om)) * n)
    if (om <= (-1.36d+175)) then
        tmp = sqrt(((n * 2.0d0) * (u * (t_1 - t_2))))
    else if (om <= (-1.46d-12)) then
        tmp = sqrt((((((l * (l * n)) * u) / om) * (-4.0d0)) - ((-2.0d0) * ((t * n) * u))))
    else if (om <= 2.6d-26) then
        tmp = sqrt(((n * 2.0d0) * (u * (t - t_2))))
    else
        tmp = sqrt((((2.0d0 * n) * u) * 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 = t - ((l * (l / Om)) * 2.0);
	double t_2 = (U - U_42_) * (((l / Om) * (l / Om)) * n);
	double tmp;
	if (Om <= -1.36e+175) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t_1 - t_2))));
	} else if (Om <= -1.46e-12) {
		tmp = Math.sqrt((((((l * (l * n)) * U) / Om) * -4.0) - (-2.0 * ((t * n) * U))));
	} else if (Om <= 2.6e-26) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t - t_2))));
	} else {
		tmp = Math.sqrt((((2.0 * n) * U) * t_1));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = t - ((l * (l / Om)) * 2.0)
	t_2 = (U - U_42_) * (((l / Om) * (l / Om)) * n)
	tmp = 0
	if Om <= -1.36e+175:
		tmp = math.sqrt(((n * 2.0) * (U * (t_1 - t_2))))
	elif Om <= -1.46e-12:
		tmp = math.sqrt((((((l * (l * n)) * U) / Om) * -4.0) - (-2.0 * ((t * n) * U))))
	elif Om <= 2.6e-26:
		tmp = math.sqrt(((n * 2.0) * (U * (t - t_2))))
	else:
		tmp = math.sqrt((((2.0 * n) * U) * t_1))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(t - Float64(Float64(l * Float64(l / Om)) * 2.0))
	t_2 = Float64(Float64(U - U_42_) * Float64(Float64(Float64(l / Om) * Float64(l / Om)) * n))
	tmp = 0.0
	if (Om <= -1.36e+175)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t_1 - t_2))));
	elseif (Om <= -1.46e-12)
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(l * Float64(l * n)) * U) / Om) * -4.0) - Float64(-2.0 * Float64(Float64(t * n) * U))));
	elseif (Om <= 2.6e-26)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t - t_2))));
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = t - ((l * (l / Om)) * 2.0);
	t_2 = (U - U_42_) * (((l / Om) * (l / Om)) * n);
	tmp = 0.0;
	if (Om <= -1.36e+175)
		tmp = sqrt(((n * 2.0) * (U * (t_1 - t_2))));
	elseif (Om <= -1.46e-12)
		tmp = sqrt((((((l * (l * n)) * U) / Om) * -4.0) - (-2.0 * ((t * n) * U))));
	elseif (Om <= 2.6e-26)
		tmp = sqrt(((n * 2.0) * (U * (t - t_2))));
	else
		tmp = sqrt((((2.0 * n) * U) * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(t - N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(U - U$42$), $MachinePrecision] * N[(N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Om, -1.36e+175], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, -1.46e-12], N[Sqrt[N[(N[(N[(N[(N[(l * N[(l * n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision] / Om), $MachinePrecision] * -4.0), $MachinePrecision] - N[(-2.0 * N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 2.6e-26], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if Om < -1.36e175

   1. Initial program 61.8%

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

   4. Applied rewrites 72.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 72.9

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

   6. Applied rewrites 72.9%

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

   if -1.36e175 < Om < -1.46000000000000004e-12

   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 Om around inf

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

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

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 61.9

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

   5. Applied rewrites 61.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 72.4

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

   7. Applied rewrites 72.4%

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

   if -1.46000000000000004e-12 < Om < 2.6000000000000001e-26

   1. Initial program 47.4%

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

   4. Applied rewrites 53.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 53.4

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

   6. Applied rewrites 53.4%

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

   7. Taylor expanded in t around inf

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

   9. Applied rewrites 63.2%

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

   if 2.6000000000000001e-26 < Om

   1. Initial program 57.4%

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

   3. Taylor expanded in 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. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 62.3

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

   5. Applied rewrites 62.3%

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

Alternative 10: 52.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq -1.05 \cdot 10^{+179}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right)\right)}\\ \mathbf{elif}\;Om \leq -1.46 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{\frac{\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot U}{Om} \cdot -4 - -2 \cdot \left(\left(t \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;Om \leq 2.6 \cdot 10^{-26}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \left(U - U*\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om -1.05e+179)
   (sqrt (* (* n 2.0) (* U (- t (* (* (/ l Om) l) 2.0)))))
   (if (<= Om -1.46e-12)
     (sqrt (- (* (/ (* (* l (* l n)) U) Om) -4.0) (* -2.0 (* (* t n) U))))
     (if (<= Om 2.6e-26)
       (sqrt
        (* (* n 2.0) (* U (- t (* (- U U*) (* (* (/ l Om) (/ l Om)) n))))))
       (sqrt (* (* (* 2.0 n) U) (- t (* (* l (/ l Om)) 2.0))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= -1.05e+179) {
		tmp = sqrt(((n * 2.0) * (U * (t - (((l / Om) * l) * 2.0)))));
	} else if (Om <= -1.46e-12) {
		tmp = sqrt((((((l * (l * n)) * U) / Om) * -4.0) - (-2.0 * ((t * n) * U))));
	} else if (Om <= 2.6e-26) {
		tmp = sqrt(((n * 2.0) * (U * (t - ((U - U_42_) * (((l / Om) * (l / Om)) * n))))));
	} else {
		tmp = sqrt((((2.0 * n) * U) * (t - ((l * (l / Om)) * 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 (om <= (-1.05d+179)) then
        tmp = sqrt(((n * 2.0d0) * (u * (t - (((l / om) * l) * 2.0d0)))))
    else if (om <= (-1.46d-12)) then
        tmp = sqrt((((((l * (l * n)) * u) / om) * (-4.0d0)) - ((-2.0d0) * ((t * n) * u))))
    else if (om <= 2.6d-26) then
        tmp = sqrt(((n * 2.0d0) * (u * (t - ((u - u_42) * (((l / om) * (l / om)) * n))))))
    else
        tmp = sqrt((((2.0d0 * n) * u) * (t - ((l * (l / om)) * 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 (Om <= -1.05e+179) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t - (((l / Om) * l) * 2.0)))));
	} else if (Om <= -1.46e-12) {
		tmp = Math.sqrt((((((l * (l * n)) * U) / Om) * -4.0) - (-2.0 * ((t * n) * U))));
	} else if (Om <= 2.6e-26) {
		tmp = Math.sqrt(((n * 2.0) * (U * (t - ((U - U_42_) * (((l / Om) * (l / Om)) * n))))));
	} else {
		tmp = Math.sqrt((((2.0 * n) * U) * (t - ((l * (l / Om)) * 2.0))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if Om <= -1.05e+179:
		tmp = math.sqrt(((n * 2.0) * (U * (t - (((l / Om) * l) * 2.0)))))
	elif Om <= -1.46e-12:
		tmp = math.sqrt((((((l * (l * n)) * U) / Om) * -4.0) - (-2.0 * ((t * n) * U))))
	elif Om <= 2.6e-26:
		tmp = math.sqrt(((n * 2.0) * (U * (t - ((U - U_42_) * (((l / Om) * (l / Om)) * n))))))
	else:
		tmp = math.sqrt((((2.0 * n) * U) * (t - ((l * (l / Om)) * 2.0))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (Om <= -1.05e+179)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t - Float64(Float64(Float64(l / Om) * l) * 2.0)))));
	elseif (Om <= -1.46e-12)
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(l * Float64(l * n)) * U) / Om) * -4.0) - Float64(-2.0 * Float64(Float64(t * n) * U))));
	elseif (Om <= 2.6e-26)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t - Float64(Float64(U - U_42_) * Float64(Float64(Float64(l / Om) * Float64(l / Om)) * n))))));
	else
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t - Float64(Float64(l * Float64(l / Om)) * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (Om <= -1.05e+179)
		tmp = sqrt(((n * 2.0) * (U * (t - (((l / Om) * l) * 2.0)))));
	elseif (Om <= -1.46e-12)
		tmp = sqrt((((((l * (l * n)) * U) / Om) * -4.0) - (-2.0 * ((t * n) * U))));
	elseif (Om <= 2.6e-26)
		tmp = sqrt(((n * 2.0) * (U * (t - ((U - U_42_) * (((l / Om) * (l / Om)) * n))))));
	else
		tmp = sqrt((((2.0 * n) * U) * (t - ((l * (l / Om)) * 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[Om, -1.05e+179], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t - N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, -1.46e-12], N[Sqrt[N[(N[(N[(N[(N[(l * N[(l * n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision] / Om), $MachinePrecision] * -4.0), $MachinePrecision] - N[(-2.0 * N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 2.6e-26], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t - 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[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t - N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if Om < -1.0499999999999999e179

   1. Initial program 61.8%

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

   4. Applied rewrites 72.9%

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

   5. Taylor expanded in n around 0

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

      1. associate--l- N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. *-commutative N/A

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

      12. associate--l- N/A

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

   7. Applied rewrites 72.7%

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

   if -1.0499999999999999e179 < Om < -1.46000000000000004e-12

   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 Om around inf

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

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

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 61.9

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

   5. Applied rewrites 61.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 72.4

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

   7. Applied rewrites 72.4%

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

   if -1.46000000000000004e-12 < Om < 2.6000000000000001e-26

   1. Initial program 47.4%

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

   4. Applied rewrites 53.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 53.4

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

   6. Applied rewrites 53.4%

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

   7. Taylor expanded in t around inf

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

   9. Applied rewrites 63.2%

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

   if 2.6000000000000001e-26 < Om

   1. Initial program 57.4%

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

   3. Taylor expanded in 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. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 62.3

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

   5. Applied rewrites 62.3%

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

Alternative 11: 41.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.3 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 5.3e-156)
   (sqrt (* (* (* 2.0 n) U) t))
   (if (<= l 3.4e+140)
     (sqrt (* 2.0 (* U (* n (- t (* 2.0 (/ (* l l) Om)))))))
     (sqrt (* (* n 2.0) (* U (- t (* (* (/ l Om) l) 2.0))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 5.3e-156) {
		tmp = sqrt((((2.0 * n) * U) * t));
	} else if (l <= 3.4e+140) {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * ((l * l) / Om)))))));
	} else {
		tmp = sqrt(((n * 2.0) * (U * (t - (((l / Om) * l) * 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 (l <= 5.3d-156) then
        tmp = sqrt((((2.0d0 * n) * u) * t))
    else if (l <= 3.4d+140) then
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * ((l * l) / om)))))))
    else
        tmp = sqrt(((n * 2.0d0) * (u * (t - (((l / om) * l) * 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 (l <= 5.3e-156) {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	} else if (l <= 3.4e+140) {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * ((l * l) / Om)))))));
	} else {
		tmp = Math.sqrt(((n * 2.0) * (U * (t - (((l / Om) * l) * 2.0)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 5.3e-156:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	elif l <= 3.4e+140:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * ((l * l) / Om)))))))
	else:
		tmp = math.sqrt(((n * 2.0) * (U * (t - (((l / Om) * l) * 2.0)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 5.3e-156)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	elseif (l <= 3.4e+140)
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om)))))));
	else
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t - Float64(Float64(Float64(l / Om) * l) * 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 5.3e-156)
		tmp = sqrt((((2.0 * n) * U) * t));
	elseif (l <= 3.4e+140)
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * ((l * l) / Om)))))));
	else
		tmp = sqrt(((n * 2.0) * (U * (t - (((l / Om) * l) * 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 5.3e-156], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.4e+140], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t - N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 5.29999999999999968e-156

   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 51.1%

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

   if 5.29999999999999968e-156 < l < 3.4e140

   1. Initial program 57.0%

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

   3. Taylor expanded in U* around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow-prod-down N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 22.1

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

   5. Applied rewrites 22.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 22.1

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

   7. Applied rewrites 22.1%

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

   8. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 55.8

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

   10. Applied rewrites 55.8%

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

   if 3.4e140 < l

   1. Initial program 19.6%

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

   4. Applied rewrites 48.1%

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

   5. Taylor expanded in n around 0

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

      1. associate--l- N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. *-commutative N/A

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

      12. associate--l- N/A

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

   7. Applied rewrites 48.8%

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

Alternative 12: 41.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.3 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(t - \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 5.3e-156)
   (sqrt (* (* (* 2.0 n) U) t))
   (sqrt (* (* (* (- t (* (* l (/ l Om)) 2.0)) n) U) 2.0))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 5.3e-156) {
		tmp = sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = sqrt(((((t - ((l * (l / Om)) * 2.0)) * n) * U) * 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 (l <= 5.3d-156) then
        tmp = sqrt((((2.0d0 * n) * u) * t))
    else
        tmp = sqrt(((((t - ((l * (l / om)) * 2.0d0)) * n) * u) * 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 (l <= 5.3e-156) {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = Math.sqrt(((((t - ((l * (l / Om)) * 2.0)) * n) * U) * 2.0));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 5.3e-156:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	else:
		tmp = math.sqrt(((((t - ((l * (l / Om)) * 2.0)) * n) * U) * 2.0))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 5.3e-156)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(t - Float64(Float64(l * Float64(l / Om)) * 2.0)) * n) * U) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 5.3e-156)
		tmp = sqrt((((2.0 * n) * U) * t));
	else
		tmp = sqrt(((((t - ((l * (l / Om)) * 2.0)) * n) * U) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 5.3e-156], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(t - N[(N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.29999999999999968e-156

   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 51.1%

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

   if 5.29999999999999968e-156 < l

   1. Initial program 45.7%

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

   3. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   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 48.4

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

   5. Applied rewrites 48.4%

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

Alternative 13: 39.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.3 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 5.3e-156)
   (sqrt (* (* (* 2.0 n) U) t))
   (sqrt (* 2.0 (* U (* n (- t (* 2.0 (/ (* l l) Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 5.3e-156) {
		tmp = sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * ((l * l) / 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) :: tmp
    if (l <= 5.3d-156) then
        tmp = sqrt((((2.0d0 * n) * u) * t))
    else
        tmp = sqrt((2.0d0 * (u * (n * (t - (2.0d0 * ((l * l) / 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 tmp;
	if (l <= 5.3e-156) {
		tmp = Math.sqrt((((2.0 * n) * U) * t));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * (t - (2.0 * ((l * l) / Om)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 5.3e-156:
		tmp = math.sqrt((((2.0 * n) * U) * t))
	else:
		tmp = math.sqrt((2.0 * (U * (n * (t - (2.0 * ((l * l) / Om)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 5.3e-156)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * t));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 5.3e-156)
		tmp = sqrt((((2.0 * n) * U) * t));
	else
		tmp = sqrt((2.0 * (U * (n * (t - (2.0 * ((l * l) / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 5.3e-156], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.29999999999999968e-156

   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 51.1%

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

   if 5.29999999999999968e-156 < l

   1. Initial program 45.7%

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

   3. Taylor expanded in U* around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow-prod-down N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 21.6

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

   5. Applied rewrites 21.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 21.6

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

   7. Applied rewrites 21.6%

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

   8. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 44.3

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

   10. Applied rewrites 44.3%

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

Alternative 14: 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*))))))