Toniolo and Linder, Equation (13)

Percentage Accurate: 48.8% → 60.5%

Time: 14.4s

Alternatives: 18

Speedup: N/A×

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: 48.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 60.5% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := {\left(\frac{l\_m}{Om}\right)}^{2}\\ t_2 := \sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(\mathsf{fma}\left(-2, l\_m \cdot \frac{l\_m}{Om}, t\right) - \left(U - U*\right) \cdot \left(t\_1 \cdot n\right)\right)\right)}\\ t_3 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot t\_1\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-321}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\sqrt{t\_3}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(-U*\right) \cdot \left(\frac{\left(-l\_m\right) \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{\frac{U}{U*}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (pow (/ l_m Om) 2.0))
        (t_2
         (sqrt
          (*
           (* n 2.0)
           (* U (- (fma -2.0 (* l_m (/ l_m Om)) t) (* (- U U*) (* t_1 n)))))))
        (t_3
         (*
          (* (* 2.0 n) U)
          (- (- t (* 2.0 (/ (* l_m l_m) Om))) (* (* n t_1) (- U U*))))))
   (if (<= t_3 2e-321)
     t_2
     (if (<= t_3 2e+303)
       (sqrt t_3)
       (if (<= t_3 INFINITY)
         t_2
         (*
          (- U*)
          (* (/ (* (- l_m) (* n (sqrt 2.0))) Om) (sqrt (/ U U*)))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = pow((l_m / Om), 2.0);
	double t_2 = sqrt(((n * 2.0) * (U * (fma(-2.0, (l_m * (l_m / Om)), t) - ((U - U_42_) * (t_1 * n))))));
	double t_3 = ((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * t_1) * (U - U_42_)));
	double tmp;
	if (t_3 <= 2e-321) {
		tmp = t_2;
	} else if (t_3 <= 2e+303) {
		tmp = sqrt(t_3);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = -U_42_ * (((-l_m * (n * sqrt(2.0))) / Om) * sqrt((U / U_42_)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(l_m / Om) ^ 2.0
	t_2 = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(fma(-2.0, Float64(l_m * Float64(l_m / Om)), t) - Float64(Float64(U - U_42_) * Float64(t_1 * n))))))
	t_3 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * t_1) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_3 <= 2e-321)
		tmp = t_2;
	elseif (t_3 <= 2e+303)
		tmp = sqrt(t_3);
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(Float64(-U_42_) * Float64(Float64(Float64(Float64(-l_m) * Float64(n * sqrt(2.0))) / Om) * sqrt(Float64(U / U_42_))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(t$95$1 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * t$95$1), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 2e-321], t$95$2, If[LessEqual[t$95$3, 2e+303], N[Sqrt[t$95$3], $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$2, N[((-U$42$) * N[(N[(N[((-l$95$m) * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[Sqrt[N[(U / U$42$), $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*)))) < 2.00097e-321 or 2e303 < (*.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 28.8%

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

   4. Applied rewrites 41.7%

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

   if 2.00097e-321 < (*.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*)))) < 2e303

   1. Initial program 98.9%

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

   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

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

   5. Applied rewrites 3.2%

      \[\leadsto \color{blue}{\left(-U*\right) \cdot \mathsf{fma}\left(\sqrt{\frac{U}{{U*}^{3}}} \cdot \frac{\left(Om \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - \frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om \cdot Om}\right)}{\ell}, 0.5, \frac{\left(\left(-1 \cdot \sqrt{2}\right) \cdot n\right) \cdot \ell}{Om} \cdot \sqrt{\frac{U}{U*}}\right)} \]

   6. Taylor expanded in U* around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-/.f64 15.8

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

   8. Applied rewrites 15.8%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 2 \cdot 10^{-321}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-U*\right) \cdot \left(\frac{\left(-\ell\right) \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{\frac{U}{U*}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 58.5% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := {\left(\frac{l\_m}{Om}\right)}^{2}\\ t_2 := U \cdot \left(\mathsf{fma}\left(-2, l\_m \cdot \frac{l\_m}{Om}, t\right) - \left(U - U*\right) \cdot \left(t\_1 \cdot n\right)\right)\\ \mathbf{if}\;n \leq -1.22 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(n \cdot t\_1\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;n \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;{\left(n \cdot 2\right)}^{0.5} \cdot {t\_2}^{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (pow (/ l_m Om) 2.0))
        (t_2 (* U (- (fma -2.0 (* l_m (/ l_m Om)) t) (* (- U U*) (* t_1 n))))))
   (if (<= n -1.22e+142)
     (sqrt (* (* (* 2.0 n) U) (- t (* (* n t_1) (- U U*)))))
     (if (<= n -2e-310)
       (sqrt (* (* n 2.0) t_2))
       (* (pow (* n 2.0) 0.5) (pow t_2 0.5))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = pow((l_m / Om), 2.0);
	double t_2 = U * (fma(-2.0, (l_m * (l_m / Om)), t) - ((U - U_42_) * (t_1 * n)));
	double tmp;
	if (n <= -1.22e+142) {
		tmp = sqrt((((2.0 * n) * U) * (t - ((n * t_1) * (U - U_42_)))));
	} else if (n <= -2e-310) {
		tmp = sqrt(((n * 2.0) * t_2));
	} else {
		tmp = pow((n * 2.0), 0.5) * pow(t_2, 0.5);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(l_m / Om) ^ 2.0
	t_2 = Float64(U * Float64(fma(-2.0, Float64(l_m * Float64(l_m / Om)), t) - Float64(Float64(U - U_42_) * Float64(t_1 * n))))
	tmp = 0.0
	if (n <= -1.22e+142)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t - Float64(Float64(n * t_1) * Float64(U - U_42_)))));
	elseif (n <= -2e-310)
		tmp = sqrt(Float64(Float64(n * 2.0) * t_2));
	else
		tmp = Float64((Float64(n * 2.0) ^ 0.5) * (t_2 ^ 0.5));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(U * N[(N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] - N[(N[(U - U$42$), $MachinePrecision] * N[(t$95$1 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.22e+142], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(t - N[(N[(n * t$95$1), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, -2e-310], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision], N[(N[Power[N[(n * 2.0), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[t$95$2, 0.5], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.21999999999999998e142

   1. Initial program 57.3%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 65.5%

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

   if -1.21999999999999998e142 < n < -1.999999999999994e-310

   1. Initial program 46.1%

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

   4. Applied rewrites 53.2%

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

   if -1.999999999999994e-310 < n

   1. Initial program 51.6%

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

   4. Applied rewrites 62.6%

      \[\leadsto \color{blue}{{\left(n \cdot 2\right)}^{0.5} \cdot {\left(U \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right)\right)}^{0.5}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 56.3% accurate, N/A× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (pow (/ l_m Om) 2.0)))
   (if (<=
        (*
         (* (* 2.0 n) U)
         (- (- t (* 2.0 (/ (* l_m l_m) Om))) (* (* n t_1) (- U U*))))
        INFINITY)
     (sqrt
      (*
       (* n 2.0)
       (* U (- (fma -2.0 (* l_m (/ l_m Om)) t) (* (- U U*) (* t_1 n))))))
     (* (- U*) (* (/ (* (- l_m) (* n (sqrt 2.0))) Om) (sqrt (/ U U*)))))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 58.6%

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

   4. Applied rewrites 60.4%

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

   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

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

   5. Applied rewrites 3.2%

      \[\leadsto \color{blue}{\left(-U*\right) \cdot \mathsf{fma}\left(\sqrt{\frac{U}{{U*}^{3}}} \cdot \frac{\left(Om \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - \frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om \cdot Om}\right)}{\ell}, 0.5, \frac{\left(\left(-1 \cdot \sqrt{2}\right) \cdot n\right) \cdot \ell}{Om} \cdot \sqrt{\frac{U}{U*}}\right)} \]

   6. Taylor expanded in U* around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-/.f64 15.8

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

   8. Applied rewrites 15.8%

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

4. Final simplification 54.3%

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

Alternative 4: 51.2% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.7 \cdot 10^{-31}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot {\left(l\_m \cdot n\right)}^{2}}{Om}, 4 \cdot \left(\left(l\_m \cdot l\_m\right) \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \left(l\_m \cdot l\_m\right) \cdot \mathsf{fma}\left(2, {Om}^{-1}, \frac{n \cdot \left(U - U*\right)}{Om \cdot Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 2.7e-31)
   (sqrt
    (fma
     -1.0
     (/
      (*
       U
       (fma -2.0 (/ (* U* (pow (* l_m n) 2.0)) Om) (* 4.0 (* (* l_m l_m) n))))
      Om)
     (* 2.0 (* (* U n) t))))
   (sqrt
    (*
     (* n 2.0)
     (*
      U
      (-
       t
       (*
        (* l_m l_m)
        (fma 2.0 (pow Om -1.0) (/ (* n (- U U*)) (* Om Om))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 2.7e-31) {
		tmp = sqrt(fma(-1.0, ((U * fma(-2.0, ((U_42_ * pow((l_m * n), 2.0)) / Om), (4.0 * ((l_m * l_m) * n)))) / Om), (2.0 * ((U * n) * t))));
	} else {
		tmp = sqrt(((n * 2.0) * (U * (t - ((l_m * l_m) * fma(2.0, pow(Om, -1.0), ((n * (U - U_42_)) / (Om * Om))))))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 2.7e-31)
		tmp = sqrt(fma(-1.0, Float64(Float64(U * fma(-2.0, Float64(Float64(U_42_ * (Float64(l_m * n) ^ 2.0)) / Om), Float64(4.0 * Float64(Float64(l_m * l_m) * n)))) / Om), Float64(2.0 * Float64(Float64(U * n) * t))));
	else
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t - Float64(Float64(l_m * l_m) * fma(2.0, (Om ^ -1.0), Float64(Float64(n * Float64(U - U_42_)) / Float64(Om * Om))))))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 2.7e-31], N[Sqrt[N[(-1.0 * N[(N[(U * N[(-2.0 * N[(N[(U$42$ * N[Power[N[(l$95$m * n), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(4.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 * N[(N[(U * n), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t - N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(2.0 * N[Power[Om, -1.0], $MachinePrecision] + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.70000000000000014e-31

   1. Initial program 55.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 54.7%

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

   6. Taylor expanded in Om around -inf

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

      1. lower-fma.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(-1, \color{blue}{\frac{2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{Om} + 4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

   8. Applied rewrites 48.6%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \frac{U \cdot \left({\left(\ell \cdot n\right)}^{2} \cdot \left(U - U*\right)\right)}{Om}, 4 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)}} \]

   9. Taylor expanded in U around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \left(-2 \cdot \frac{U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)}{Om} + 4 \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)}{Om}, 4 \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       3. lower-/.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)}{Om}, 4 \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)}{Om}, 4 \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       5. unpow-prod-down N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot {\left(\ell \cdot n\right)}^{2}}{Om}, 4 \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       6. lift-pow.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot {\left(\ell \cdot n\right)}^{2}}{Om}, 4 \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       7. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot {\left(\ell \cdot n\right)}^{2}}{Om}, 4 \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot {\left(\ell \cdot n\right)}^{2}}{Om}, 4 \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       9. pow2 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot {\left(\ell \cdot n\right)}^{2}}{Om}, 4 \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       10. lift-*.f64 N/A

           \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot {\left(\ell \cdot n\right)}^{2}}{Om}, 4 \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       11. lift-*.f64 51.3

           \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot {\left(\ell \cdot n\right)}^{2}}{Om}, 4 \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

   11. Applied rewrites 51.3%

       \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot {\left(\ell \cdot n\right)}^{2}}{Om}, 4 \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

   if 2.70000000000000014e-31 < l

   1. Initial program 37.6%

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

   4. Applied rewrites 46.9%

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

   5. Taylor expanded in l around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift--.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 44.2

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

   7. Applied rewrites 44.2%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.7 \cdot 10^{-31}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot {\left(\ell \cdot n\right)}^{2}}{Om}, 4 \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(2, {Om}^{-1}, \frac{n \cdot \left(U - U*\right)}{Om \cdot Om}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.9% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\\ \mathbf{if}\;l\_m \leq 1.85 \cdot 10^{-158}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot {\left(l\_m \cdot n\right)}^{2}}{Om}, 4 \cdot \left(\left(l\_m \cdot l\_m\right) \cdot n\right)\right)}{Om}, t\_1\right)}\\ \mathbf{elif}\;l\_m \leq 6.5 \cdot 10^{-68} \lor \neg \left(l\_m \leq 2 \cdot 10^{+124}\right):\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, l\_m \cdot \frac{l\_m}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-1, \frac{n \cdot \mathsf{fma}\left(2, \frac{U \cdot \left(\left(l\_m \cdot l\_m\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{Om}, 4 \cdot \left(U \cdot \left(l\_m \cdot l\_m\right)\right)\right)}{Om}, t\_1\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* 2.0 (* (* U n) t))))
   (if (<= l_m 1.85e-158)
     (sqrt
      (fma
       -1.0
       (/
        (*
         U
         (fma
          -2.0
          (/ (* U* (pow (* l_m n) 2.0)) Om)
          (* 4.0 (* (* l_m l_m) n))))
        Om)
       t_1))
     (if (or (<= l_m 6.5e-68) (not (<= l_m 2e+124)))
       (sqrt (* (* (* (fma -2.0 (* l_m (/ l_m Om)) t) n) U) 2.0))
       (sqrt
        (fma
         -1.0
         (/
          (*
           n
           (fma
            2.0
            (/ (* U (* (* l_m l_m) (* n (- U U*)))) Om)
            (* 4.0 (* U (* l_m l_m)))))
          Om)
         t_1))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = 2.0 * ((U * n) * t);
	double tmp;
	if (l_m <= 1.85e-158) {
		tmp = sqrt(fma(-1.0, ((U * fma(-2.0, ((U_42_ * pow((l_m * n), 2.0)) / Om), (4.0 * ((l_m * l_m) * n)))) / Om), t_1));
	} else if ((l_m <= 6.5e-68) || !(l_m <= 2e+124)) {
		tmp = sqrt((((fma(-2.0, (l_m * (l_m / Om)), t) * n) * U) * 2.0));
	} else {
		tmp = sqrt(fma(-1.0, ((n * fma(2.0, ((U * ((l_m * l_m) * (n * (U - U_42_)))) / Om), (4.0 * (U * (l_m * l_m))))) / Om), t_1));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(2.0 * Float64(Float64(U * n) * t))
	tmp = 0.0
	if (l_m <= 1.85e-158)
		tmp = sqrt(fma(-1.0, Float64(Float64(U * fma(-2.0, Float64(Float64(U_42_ * (Float64(l_m * n) ^ 2.0)) / Om), Float64(4.0 * Float64(Float64(l_m * l_m) * n)))) / Om), t_1));
	elseif ((l_m <= 6.5e-68) || !(l_m <= 2e+124))
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(l_m * Float64(l_m / Om)), t) * n) * U) * 2.0));
	else
		tmp = sqrt(fma(-1.0, Float64(Float64(n * fma(2.0, Float64(Float64(U * Float64(Float64(l_m * l_m) * Float64(n * Float64(U - U_42_)))) / Om), Float64(4.0 * Float64(U * Float64(l_m * l_m))))) / Om), t_1));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(2.0 * N[(N[(U * n), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l$95$m, 1.85e-158], N[Sqrt[N[(-1.0 * N[(N[(U * N[(-2.0 * N[(N[(U$42$ * N[Power[N[(l$95$m * n), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(4.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[l$95$m, 6.5e-68], N[Not[LessEqual[l$95$m, 2e+124]], $MachinePrecision]], N[Sqrt[N[(N[(N[(N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-1.0 * N[(N[(n * N[(2.0 * N[(N[(U * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(4.0 * N[(U * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.85e-158

   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

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

   5. Applied rewrites 54.8%

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

   6. Taylor expanded in Om around -inf

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

      1. lower-fma.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(-1, \color{blue}{\frac{2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{Om} + 4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

   8. Applied rewrites 50.1%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \frac{U \cdot \left({\left(\ell \cdot n\right)}^{2} \cdot \left(U - U*\right)\right)}{Om}, 4 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)}} \]

   9. Taylor expanded in U around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \left(-2 \cdot \frac{U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)}{Om} + 4 \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)}{Om}, 4 \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       3. lower-/.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)}{Om}, 4 \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)}{Om}, 4 \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       5. unpow-prod-down N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot {\left(\ell \cdot n\right)}^{2}}{Om}, 4 \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       6. lift-pow.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot {\left(\ell \cdot n\right)}^{2}}{Om}, 4 \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       7. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot {\left(\ell \cdot n\right)}^{2}}{Om}, 4 \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot {\left(\ell \cdot n\right)}^{2}}{Om}, 4 \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       9. pow2 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot {\left(\ell \cdot n\right)}^{2}}{Om}, 4 \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       10. lift-*.f64 N/A

           \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot {\left(\ell \cdot n\right)}^{2}}{Om}, 4 \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       11. lift-*.f64 50.9

           \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot {\left(\ell \cdot n\right)}^{2}}{Om}, 4 \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

   11. Applied rewrites 50.9%

       \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot {\left(\ell \cdot n\right)}^{2}}{Om}, 4 \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

   if 1.85e-158 < l < 6.4999999999999997e-68 or 1.9999999999999999e124 < l

   1. Initial program 36.0%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 51.9

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

   5. Applied rewrites 51.9%

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

   if 6.4999999999999997e-68 < l < 1.9999999999999999e124

   1. Initial program 50.1%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 45.9%

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

   6. Taylor expanded in Om around -inf

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

      1. lower-fma.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(-1, \color{blue}{\frac{2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{Om} + 4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

   8. Applied rewrites 41.4%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \frac{U \cdot \left({\left(\ell \cdot n\right)}^{2} \cdot \left(U - U*\right)\right)}{Om}, 4 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)}} \]

   9. Taylor expanded in n around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{n \cdot \left(2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{Om} + 4 \cdot \left(U \cdot {\ell}^{2}\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{n \cdot \mathsf{fma}\left(2, \frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{Om}, 4 \cdot \left(U \cdot {\ell}^{2}\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       3. lower-/.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{n \cdot \mathsf{fma}\left(2, \frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{Om}, 4 \cdot \left(U \cdot {\ell}^{2}\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{n \cdot \mathsf{fma}\left(2, \frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{Om}, 4 \cdot \left(U \cdot {\ell}^{2}\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{n \cdot \mathsf{fma}\left(2, \frac{U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{Om}, 4 \cdot \left(U \cdot {\ell}^{2}\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       6. pow2 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{n \cdot \mathsf{fma}\left(2, \frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{Om}, 4 \cdot \left(U \cdot {\ell}^{2}\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       7. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{n \cdot \mathsf{fma}\left(2, \frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{Om}, 4 \cdot \left(U \cdot {\ell}^{2}\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{n \cdot \mathsf{fma}\left(2, \frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{Om}, 4 \cdot \left(U \cdot {\ell}^{2}\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       9. lift--.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{n \cdot \mathsf{fma}\left(2, \frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{Om}, 4 \cdot \left(U \cdot {\ell}^{2}\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{n \cdot \mathsf{fma}\left(2, \frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{Om}, 4 \cdot \left(U \cdot {\ell}^{2}\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       11. lower-*.f64 N/A

           \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{n \cdot \mathsf{fma}\left(2, \frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{Om}, 4 \cdot \left(U \cdot {\ell}^{2}\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       12. pow2 N/A

           \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{n \cdot \mathsf{fma}\left(2, \frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{Om}, 4 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       13. lift-*.f64 55.0

           \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{n \cdot \mathsf{fma}\left(2, \frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{Om}, 4 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

   11. Applied rewrites 55.0%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.85 \cdot 10^{-158}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot {\left(\ell \cdot n\right)}^{2}}{Om}, 4 \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)}\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{-68} \lor \neg \left(\ell \leq 2 \cdot 10^{+124}\right):\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-1, \frac{n \cdot \mathsf{fma}\left(2, \frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot \left(U - U*\right)\right)\right)}{Om}, 4 \cdot \left(U \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.4% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 8.5 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot {\left(l\_m \cdot n\right)}^{2}}{Om}, 4 \cdot \left(\left(l\_m \cdot l\_m\right) \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, l\_m \cdot \frac{l\_m}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 8.5e+19)
   (sqrt
    (fma
     -1.0
     (/
      (*
       U
       (fma -2.0 (/ (* U* (pow (* l_m n) 2.0)) Om) (* 4.0 (* (* l_m l_m) n))))
      Om)
     (* 2.0 (* (* U n) t))))
   (sqrt (* (* (* (fma -2.0 (* l_m (/ l_m Om)) t) n) U) 2.0))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 8.5e+19) {
		tmp = sqrt(fma(-1.0, ((U * fma(-2.0, ((U_42_ * pow((l_m * n), 2.0)) / Om), (4.0 * ((l_m * l_m) * n)))) / Om), (2.0 * ((U * n) * t))));
	} else {
		tmp = sqrt((((fma(-2.0, (l_m * (l_m / Om)), t) * n) * U) * 2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 8.5e+19)
		tmp = sqrt(fma(-1.0, Float64(Float64(U * fma(-2.0, Float64(Float64(U_42_ * (Float64(l_m * n) ^ 2.0)) / Om), Float64(4.0 * Float64(Float64(l_m * l_m) * n)))) / Om), Float64(2.0 * Float64(Float64(U * n) * t))));
	else
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(l_m * Float64(l_m / Om)), t) * n) * U) * 2.0));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 8.5e+19], N[Sqrt[N[(-1.0 * N[(N[(U * N[(-2.0 * N[(N[(U$42$ * N[Power[N[(l$95$m * n), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(4.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 * N[(N[(U * n), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 8.5e19

   1. Initial program 54.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 52.8%

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

   6. Taylor expanded in Om around -inf

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

      1. lower-fma.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(-1, \color{blue}{\frac{2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left({n}^{2} \cdot \left(U - U*\right)\right)\right)}{Om} + 4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}, 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)} \]

   8. Applied rewrites 47.7%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \frac{U \cdot \left({\left(\ell \cdot n\right)}^{2} \cdot \left(U - U*\right)\right)}{Om}, 4 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)}} \]

   9. Taylor expanded in U around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \left(-2 \cdot \frac{U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)}{Om} + 4 \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)}{Om}, 4 \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       3. lower-/.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)}{Om}, 4 \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot \left({\ell}^{2} \cdot {n}^{2}\right)}{Om}, 4 \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       5. unpow-prod-down N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot {\left(\ell \cdot n\right)}^{2}}{Om}, 4 \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       6. lift-pow.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot {\left(\ell \cdot n\right)}^{2}}{Om}, 4 \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       7. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot {\left(\ell \cdot n\right)}^{2}}{Om}, 4 \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot {\left(\ell \cdot n\right)}^{2}}{Om}, 4 \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       9. pow2 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot {\left(\ell \cdot n\right)}^{2}}{Om}, 4 \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       10. lift-*.f64 N/A

           \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot {\left(\ell \cdot n\right)}^{2}}{Om}, 4 \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

       11. lift-*.f64 51.0

           \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot {\left(\ell \cdot n\right)}^{2}}{Om}, 4 \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

   11. Applied rewrites 51.0%

       \[\leadsto \sqrt{\mathsf{fma}\left(-1, \frac{U \cdot \mathsf{fma}\left(-2, \frac{U* \cdot {\left(\ell \cdot n\right)}^{2}}{Om}, 4 \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right)}{Om}, 2 \cdot \left(\left(U \cdot n\right) \cdot t\right)\right)} \]

   if 8.5e19 < l

   1. Initial program 36.9%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 48.7

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

   5. Applied rewrites 48.7%

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

Alternative 7: 50.2% accurate, N/A× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<=
      (*
       (* (* 2.0 n) U)
       (-
        (- t (* 2.0 (/ (* l_m l_m) Om)))
        (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))
      INFINITY)
   (sqrt (* (* (* (fma -2.0 (* l_m (/ l_m Om)) t) n) U) 2.0))
   (* (- U*) (* (/ (* (- l_m) (* n (sqrt 2.0))) Om) (sqrt (/ U U*))))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 58.6%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 52.9

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

   5. Applied rewrites 52.9%

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

   if +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

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

   5. Applied rewrites 3.2%

      \[\leadsto \color{blue}{\left(-U*\right) \cdot \mathsf{fma}\left(\sqrt{\frac{U}{{U*}^{3}}} \cdot \frac{\left(Om \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - \frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om \cdot Om}\right)}{\ell}, 0.5, \frac{\left(\left(-1 \cdot \sqrt{2}\right) \cdot n\right) \cdot \ell}{Om} \cdot \sqrt{\frac{U}{U*}}\right)} \]

   6. Taylor expanded in U* around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-/.f64 15.8

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

   8. Applied rewrites 15.8%

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

4. Final simplification 47.8%

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

Alternative 8: 47.2% accurate, N/A× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<=
      (*
       (* (* 2.0 n) U)
       (-
        (- t (* 2.0 (/ (* l_m l_m) Om)))
        (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))
      INFINITY)
   (sqrt
    (*
     (* 2.0 U)
     (*
      (-
       t
       (fma (* l_m (/ l_m Om)) 2.0 (/ (* (* (* l_m l_m) n) (- U*)) (* Om Om))))
      n)))
   (* (- U*) (* (/ (* (- l_m) (* n (sqrt 2.0))) Om) (sqrt (/ U U*))))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 58.6%

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

   3. Taylor expanded in U around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 47.6%

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

   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

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

   5. Applied rewrites 3.2%

      \[\leadsto \color{blue}{\left(-U*\right) \cdot \mathsf{fma}\left(\sqrt{\frac{U}{{U*}^{3}}} \cdot \frac{\left(Om \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - \frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om \cdot Om}\right)}{\ell}, 0.5, \frac{\left(\left(-1 \cdot \sqrt{2}\right) \cdot n\right) \cdot \ell}{Om} \cdot \sqrt{\frac{U}{U*}}\right)} \]

   6. Taylor expanded in U* around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-/.f64 15.8

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

   8. Applied rewrites 15.8%

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

4. Final simplification 43.2%

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

Alternative 9: 40.5% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(l\_m \cdot l\_m\right) \cdot \frac{\sqrt{2}}{Om}\right) \cdot \sqrt{\frac{U \cdot n}{t}}, -1, \sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(-t\right) \cdot \mathsf{fma}\left(-2, \frac{U}{n}, \frac{\mathsf{fma}\left(-4, \frac{U}{Om} \cdot \frac{l\_m \cdot l\_m}{n}, -2 \cdot \frac{U \cdot \left(\left(l\_m \cdot l\_m\right) \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)}{-t}\right)\right) \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-U*\right) \cdot \left(\frac{\left(-l\_m\right) \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{\frac{U}{U*}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))))
   (if (<= t_1 5e+151)
     (fma
      (* (* (* l_m l_m) (/ (sqrt 2.0) Om)) (sqrt (/ (* U n) t)))
      -1.0
      (sqrt (* (* (* t n) U) 2.0)))
     (if (<= t_1 INFINITY)
       (sqrt
        (*
         (*
          (- t)
          (fma
           -2.0
           (/ U n)
           (/
            (fma
             -4.0
             (* (/ U Om) (/ (* l_m l_m) n))
             (* -2.0 (/ (* U (* (* l_m l_m) (- U U*))) (* Om Om))))
            (- t))))
         (* n n)))
       (* (- U*) (* (/ (* (- l_m) (* n (sqrt 2.0))) Om) (sqrt (/ U U*))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_1 <= 5e+151) {
		tmp = fma((((l_m * l_m) * (sqrt(2.0) / Om)) * sqrt(((U * n) / t))), -1.0, sqrt((((t * n) * U) * 2.0)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt(((-t * fma(-2.0, (U / n), (fma(-4.0, ((U / Om) * ((l_m * l_m) / n)), (-2.0 * ((U * ((l_m * l_m) * (U - U_42_))) / (Om * Om)))) / -t))) * (n * n)));
	} else {
		tmp = -U_42_ * (((-l_m * (n * sqrt(2.0))) / Om) * sqrt((U / U_42_)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_1 <= 5e+151)
		tmp = fma(Float64(Float64(Float64(l_m * l_m) * Float64(sqrt(2.0) / Om)) * sqrt(Float64(Float64(U * n) / t))), -1.0, sqrt(Float64(Float64(Float64(t * n) * U) * 2.0)));
	elseif (t_1 <= Inf)
		tmp = sqrt(Float64(Float64(Float64(-t) * fma(-2.0, Float64(U / n), Float64(fma(-4.0, Float64(Float64(U / Om) * Float64(Float64(l_m * l_m) / n)), Float64(-2.0 * Float64(Float64(U * Float64(Float64(l_m * l_m) * Float64(U - U_42_))) / Float64(Om * Om)))) / Float64(-t)))) * Float64(n * n)));
	else
		tmp = Float64(Float64(-U_42_) * Float64(Float64(Float64(Float64(-l_m) * Float64(n * sqrt(2.0))) / Om) * sqrt(Float64(U / U_42_))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 5e+151], N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(U * n), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -1.0 + N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[Sqrt[N[(N[((-t) * N[(-2.0 * N[(U / n), $MachinePrecision] + N[(N[(-4.0 * N[(N[(U / Om), $MachinePrecision] * N[(N[(l$95$m * l$95$m), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(U * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[((-U$42$) * N[(N[(N[((-l$95$m) * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[Sqrt[N[(U / U$42$), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5.0000000000000002e151

   1. Initial program 75.1%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 54.0%

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

   if 5.0000000000000002e151 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 36.1%

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

   3. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 32.4%

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

   6. Taylor expanded in t around -inf

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

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(-1 \cdot \left(t \cdot \left(-2 \cdot \frac{U}{n} + -1 \cdot \frac{-4 \cdot \frac{U \cdot {\ell}^{2}}{Om \cdot n} + -2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}{t}\right)\right)\right) \cdot \left(n \cdot n\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(-1 \cdot \left(t \cdot \left(-2 \cdot \frac{U}{n} + -1 \cdot \frac{-4 \cdot \frac{U \cdot {\ell}^{2}}{Om \cdot n} + -2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}{t}\right)\right)\right) \cdot \left(n \cdot n\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \sqrt{\left(-1 \cdot \left(t \cdot \mathsf{fma}\left(-2, \frac{U}{n}, -1 \cdot \frac{-4 \cdot \frac{U \cdot {\ell}^{2}}{Om \cdot n} + -2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}{t}\right)\right)\right) \cdot \left(n \cdot n\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(-1 \cdot \left(t \cdot \mathsf{fma}\left(-2, \frac{U}{n}, -1 \cdot \frac{-4 \cdot \frac{U \cdot {\ell}^{2}}{Om \cdot n} + -2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}{t}\right)\right)\right) \cdot \left(n \cdot n\right)} \]

   8. Applied rewrites 31.2%

      \[\leadsto \sqrt{\left(-1 \cdot \left(t \cdot \mathsf{fma}\left(-2, \frac{U}{n}, -1 \cdot \frac{\mathsf{fma}\left(-4, \frac{U}{Om} \cdot \frac{\ell \cdot \ell}{n}, -2 \cdot \frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)}{t}\right)\right)\right) \cdot \left(\color{blue}{n} \cdot n\right)} \]

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

   1. Initial program 0.0%

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

   3. Taylor expanded in U* around -inf

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

   5. Applied rewrites 2.9%

      \[\leadsto \color{blue}{\left(-U*\right) \cdot \mathsf{fma}\left(\sqrt{\frac{U}{{U*}^{3}}} \cdot \frac{\left(Om \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - \frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om \cdot Om}\right)}{\ell}, 0.5, \frac{\left(\left(-1 \cdot \sqrt{2}\right) \cdot n\right) \cdot \ell}{Om} \cdot \sqrt{\frac{U}{U*}}\right)} \]

   6. Taylor expanded in U* around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-/.f64 14.4

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

   8. Applied rewrites 14.4%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\ell \cdot \ell\right) \cdot \frac{\sqrt{2}}{Om}\right) \cdot \sqrt{\frac{U \cdot n}{t}}, -1, \sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\right)\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(-t\right) \cdot \mathsf{fma}\left(-2, \frac{U}{n}, \frac{\mathsf{fma}\left(-4, \frac{U}{Om} \cdot \frac{\ell \cdot \ell}{n}, -2 \cdot \frac{U \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(U - U*\right)\right)}{Om \cdot Om}\right)}{-t}\right)\right) \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-U*\right) \cdot \left(\frac{\left(-\ell\right) \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{\frac{U}{U*}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 38.8% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(l\_m \cdot l\_m\right) \cdot \frac{\sqrt{2}}{Om}\right) \cdot \sqrt{\frac{U \cdot n}{t}}, -1, \sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(l\_m \cdot n\right) \cdot \sqrt{2}}{Om} \cdot \sqrt{U \cdot U*}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<=
      (sqrt
       (*
        (* (* 2.0 n) U)
        (-
         (- t (* 2.0 (/ (* l_m l_m) Om)))
         (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))
      5e+151)
   (fma
    (* (* (* l_m l_m) (/ (sqrt 2.0) Om)) (sqrt (/ (* U n) t)))
    -1.0
    (sqrt (* (* (* t n) U) 2.0)))
   (* (/ (* (* l_m n) (sqrt 2.0)) Om) (sqrt (* U U*)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_))))) <= 5e+151) {
		tmp = fma((((l_m * l_m) * (sqrt(2.0) / Om)) * sqrt(((U * n) / t))), -1.0, sqrt((((t * n) * U) * 2.0)));
	} else {
		tmp = (((l_m * n) * sqrt(2.0)) / Om) * sqrt((U * U_42_));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))) <= 5e+151)
		tmp = fma(Float64(Float64(Float64(l_m * l_m) * Float64(sqrt(2.0) / Om)) * sqrt(Float64(Float64(U * n) / t))), -1.0, sqrt(Float64(Float64(Float64(t * n) * U) * 2.0)));
	else
		tmp = Float64(Float64(Float64(Float64(l_m * n) * sqrt(2.0)) / Om) * sqrt(Float64(U * U_42_)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 5e+151], N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(U * n), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -1.0 + N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l$95$m * n), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 5.0000000000000002e151

   1. Initial program 75.1%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 54.0%

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

   if 5.0000000000000002e151 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 24.9%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 30.1%

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

   6. Taylor expanded in U* around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   8. Applied rewrites 24.2%

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

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\ell \cdot \ell\right) \cdot \frac{\sqrt{2}}{Om}\right) \cdot \sqrt{\frac{U \cdot n}{t}}, -1, \sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot n\right) \cdot \sqrt{2}}{Om} \cdot \sqrt{U \cdot U*}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 34.8% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\frac{n \cdot t}{U} \cdot 2}\\ \mathbf{if}\;U \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-U\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot \left(-t\_1\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (* (/ (* n t) U) 2.0))))
   (if (<= U -5e-310) (* (- U) t_1) (* (- U) (- t_1)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt((((n * t) / U) * 2.0));
	double tmp;
	if (U <= -5e-310) {
		tmp = -U * t_1;
	} else {
		tmp = -U * -t_1;
	}
	return tmp;
}

Fortran

l_m =     private
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_m, 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_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((((n * t) / u) * 2.0d0))
    if (u <= (-5d-310)) then
        tmp = -u * t_1
    else
        tmp = -u * -t_1
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = Math.sqrt((((n * t) / U) * 2.0));
	double tmp;
	if (U <= -5e-310) {
		tmp = -U * t_1;
	} else {
		tmp = -U * -t_1;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = math.sqrt((((n * t) / U) * 2.0))
	tmp = 0
	if U <= -5e-310:
		tmp = -U * t_1
	else:
		tmp = -U * -t_1
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(n * t) / U) * 2.0))
	tmp = 0.0
	if (U <= -5e-310)
		tmp = Float64(Float64(-U) * t_1);
	else
		tmp = Float64(Float64(-U) * Float64(-t_1));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt((((n * t) / U) * 2.0));
	tmp = 0.0;
	if (U <= -5e-310)
		tmp = -U * t_1;
	else
		tmp = -U * -t_1;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(n * t), $MachinePrecision] / U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[U, -5e-310], N[((-U) * t$95$1), $MachinePrecision], N[((-U) * (-t$95$1)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if U < -4.999999999999985e-310

   1. Initial program 50.1%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 36.1%

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

   6. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

   8. Applied rewrites 29.9%

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

   9. Taylor expanded in t around inf

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

       1. sqrt-prod N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-sqrt.f64 38.7

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

   11. Applied rewrites 38.7%

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

   if -4.999999999999985e-310 < U

   1. Initial program 51.1%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 29.1%

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

   6. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

   8. Applied rewrites 2.4%

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

   9. Taylor expanded in t around -inf

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

       1. mul-1-neg N/A

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

       2. sqrt-unprod N/A

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

       3. metadata-eval N/A

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

       4. sqrt-prod N/A

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

       5. lower-neg.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-sqrt.f64 39.1

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

   11. Applied rewrites 39.1%

       \[\leadsto -1 \cdot \left(U \cdot \left(-\sqrt{\frac{n \cdot t}{U} \cdot 2}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-U\right) \cdot \sqrt{\frac{n \cdot t}{U} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot \left(-\sqrt{\frac{n \cdot t}{U} \cdot 2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 24.3% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;U \leq -2.9 \cdot 10^{-243}:\\ \;\;\;\;\left(-U\right) \cdot \left(n \cdot \mathsf{fma}\left(-1, \frac{\left(l\_m \cdot l\_m\right) \cdot \sqrt{2}}{-Om} \cdot \frac{1}{\sqrt{U \cdot \left(n \cdot t\right)}}, \sqrt{\frac{t}{U \cdot n} \cdot 2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot \left(-\sqrt{\frac{n \cdot t}{U} \cdot 2}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= U -2.9e-243)
   (*
    (- U)
    (*
     n
     (fma
      -1.0
      (* (/ (* (* l_m l_m) (sqrt 2.0)) (- Om)) (/ 1.0 (sqrt (* U (* n t)))))
      (sqrt (* (/ t (* U n)) 2.0)))))
   (* (- U) (- (sqrt (* (/ (* n t) U) 2.0))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (U <= -2.9e-243) {
		tmp = -U * (n * fma(-1.0, ((((l_m * l_m) * sqrt(2.0)) / -Om) * (1.0 / sqrt((U * (n * t))))), sqrt(((t / (U * n)) * 2.0))));
	} else {
		tmp = -U * -sqrt((((n * t) / U) * 2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (U <= -2.9e-243)
		tmp = Float64(Float64(-U) * Float64(n * fma(-1.0, Float64(Float64(Float64(Float64(l_m * l_m) * sqrt(2.0)) / Float64(-Om)) * Float64(1.0 / sqrt(Float64(U * Float64(n * t))))), sqrt(Float64(Float64(t / Float64(U * n)) * 2.0)))));
	else
		tmp = Float64(Float64(-U) * Float64(-sqrt(Float64(Float64(Float64(n * t) / U) * 2.0))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[U, -2.9e-243], N[((-U) * N[(n * N[(-1.0 * N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / (-Om)), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[(t / N[(U * n), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-U) * (-N[Sqrt[N[(N[(N[(n * t), $MachinePrecision] / U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -2.89999999999999977e-243

   1. Initial program 54.0%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 36.5%

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

   6. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

   8. Applied rewrites 33.6%

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

   9. Taylor expanded in n around inf

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

       1. lower-*.f64 N/A

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

       2. associate-*r* N/A

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

       3. mul-1-neg N/A

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

       4. lower-fma.f64 N/A

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

   11. Applied rewrites 20.3%

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

   12. Taylor expanded in U around -inf

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

   14. Applied rewrites 25.2%

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

   if -2.89999999999999977e-243 < U

   1. Initial program 47.6%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 29.9%

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

   6. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

   8. Applied rewrites 3.5%

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

   9. Taylor expanded in t around -inf

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

       1. mul-1-neg N/A

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

       2. sqrt-unprod N/A

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

       3. metadata-eval N/A

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

       4. sqrt-prod N/A

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

       5. lower-neg.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-sqrt.f64 34.6

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

   11. Applied rewrites 34.6%

       \[\leadsto -1 \cdot \left(U \cdot \left(-\sqrt{\frac{n \cdot t}{U} \cdot 2}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -2.9 \cdot 10^{-243}:\\ \;\;\;\;\left(-U\right) \cdot \left(n \cdot \mathsf{fma}\left(-1, \frac{\left(\ell \cdot \ell\right) \cdot \sqrt{2}}{-Om} \cdot \frac{1}{\sqrt{U \cdot \left(n \cdot t\right)}}, \sqrt{\frac{t}{U \cdot n} \cdot 2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot \left(-\sqrt{\frac{n \cdot t}{U} \cdot 2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 20.4% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-159}:\\ \;\;\;\;\left(-U\right) \cdot \left(n \cdot \mathsf{fma}\left(-1, \frac{\left(l\_m \cdot l\_m\right) \cdot \sqrt{2}}{-Om} \cdot \frac{1}{\sqrt{U \cdot \left(n \cdot t\right)}}, \sqrt{\frac{t}{U \cdot n} \cdot 2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-l\_m\right) \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{U \cdot U*}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t -3.8e-159)
   (*
    (- U)
    (*
     n
     (fma
      -1.0
      (* (/ (* (* l_m l_m) (sqrt 2.0)) (- Om)) (/ 1.0 (sqrt (* U (* n t)))))
      (sqrt (* (/ t (* U n)) 2.0)))))
   (* (/ (* (- l_m) (* n (sqrt 2.0))) Om) (sqrt (* U U*)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= -3.8e-159) {
		tmp = -U * (n * fma(-1.0, ((((l_m * l_m) * sqrt(2.0)) / -Om) * (1.0 / sqrt((U * (n * t))))), sqrt(((t / (U * n)) * 2.0))));
	} else {
		tmp = ((-l_m * (n * sqrt(2.0))) / Om) * sqrt((U * U_42_));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= -3.8e-159)
		tmp = Float64(Float64(-U) * Float64(n * fma(-1.0, Float64(Float64(Float64(Float64(l_m * l_m) * sqrt(2.0)) / Float64(-Om)) * Float64(1.0 / sqrt(Float64(U * Float64(n * t))))), sqrt(Float64(Float64(t / Float64(U * n)) * 2.0)))));
	else
		tmp = Float64(Float64(Float64(Float64(-l_m) * Float64(n * sqrt(2.0))) / Om) * sqrt(Float64(U * U_42_)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[t, -3.8e-159], N[((-U) * N[(n * N[(-1.0 * N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / (-Om)), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[(t / N[(U * n), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-l$95$m) * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.8000000000000001e-159

   1. Initial program 48.8%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 37.5%

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

   6. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

   8. Applied rewrites 22.0%

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

   9. Taylor expanded in n around inf

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

       1. lower-*.f64 N/A

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

       2. associate-*r* N/A

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

       3. mul-1-neg N/A

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

       4. lower-fma.f64 N/A

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

   11. Applied rewrites 24.5%

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

   12. Taylor expanded in U around -inf

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

   14. Applied rewrites 37.8%

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

   if -3.8000000000000001e-159 < t

   1. Initial program 51.8%

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

   3. Taylor expanded in U* around -inf

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

   5. Applied rewrites 1.6%

      \[\leadsto \color{blue}{\left(-U*\right) \cdot \mathsf{fma}\left(\sqrt{\frac{U}{{U*}^{3}}} \cdot \frac{\left(Om \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - \frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om \cdot Om}\right)}{\ell}, 0.5, \frac{\left(\left(-1 \cdot \sqrt{2}\right) \cdot n\right) \cdot \ell}{Om} \cdot \sqrt{\frac{U}{U*}}\right)} \]

   6. Taylor expanded in U* around -inf

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

      1. lower-*.f64 N/A

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

   8. Applied rewrites 14.3%

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

4. Final simplification 23.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-159}:\\ \;\;\;\;\left(-U\right) \cdot \left(n \cdot \mathsf{fma}\left(-1, \frac{\left(\ell \cdot \ell\right) \cdot \sqrt{2}}{-Om} \cdot \frac{1}{\sqrt{U \cdot \left(n \cdot t\right)}}, \sqrt{\frac{t}{U \cdot n} \cdot 2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\ell\right) \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{U \cdot U*}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 13.4% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\frac{U}{U*}}\\ t_2 := n \cdot \sqrt{2}\\ t_3 := l\_m \cdot t\_2\\ \mathbf{if}\;t \leq -3.4 \cdot 10^{+123}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, U* \cdot \mathsf{fma}\left(-1, t\_1 \cdot t\_3, -0.5 \cdot \left(\sqrt{{\left(\frac{U}{U*}\right)}^{3}} \cdot t\_3\right)\right), Om \cdot \mathsf{fma}\left(-0.5, \frac{Om \cdot \left(t \cdot \sqrt{2}\right)}{l\_m} \cdot t\_1, t\_1 \cdot \left(l\_m \cdot \sqrt{2}\right)\right)\right)}{Om}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-l\_m\right) \cdot t\_2}{Om} \cdot \sqrt{U \cdot U*}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (/ U U*))) (t_2 (* n (sqrt 2.0))) (t_3 (* l_m t_2)))
   (if (<= t -3.4e+123)
     (/
      (fma
       -1.0
       (* U* (fma -1.0 (* t_1 t_3) (* -0.5 (* (sqrt (pow (/ U U*) 3.0)) t_3))))
       (*
        Om
        (fma
         -0.5
         (* (/ (* Om (* t (sqrt 2.0))) l_m) t_1)
         (* t_1 (* l_m (sqrt 2.0))))))
      Om)
     (* (/ (* (- l_m) t_2) Om) (sqrt (* U U*))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt((U / U_42_));
	double t_2 = n * sqrt(2.0);
	double t_3 = l_m * t_2;
	double tmp;
	if (t <= -3.4e+123) {
		tmp = fma(-1.0, (U_42_ * fma(-1.0, (t_1 * t_3), (-0.5 * (sqrt(pow((U / U_42_), 3.0)) * t_3)))), (Om * fma(-0.5, (((Om * (t * sqrt(2.0))) / l_m) * t_1), (t_1 * (l_m * sqrt(2.0)))))) / Om;
	} else {
		tmp = ((-l_m * t_2) / Om) * sqrt((U * U_42_));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(U / U_42_))
	t_2 = Float64(n * sqrt(2.0))
	t_3 = Float64(l_m * t_2)
	tmp = 0.0
	if (t <= -3.4e+123)
		tmp = Float64(fma(-1.0, Float64(U_42_ * fma(-1.0, Float64(t_1 * t_3), Float64(-0.5 * Float64(sqrt((Float64(U / U_42_) ^ 3.0)) * t_3)))), Float64(Om * fma(-0.5, Float64(Float64(Float64(Om * Float64(t * sqrt(2.0))) / l_m) * t_1), Float64(t_1 * Float64(l_m * sqrt(2.0)))))) / Om);
	else
		tmp = Float64(Float64(Float64(Float64(-l_m) * t_2) / Om) * sqrt(Float64(U * U_42_)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(U / U$42$), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(l$95$m * t$95$2), $MachinePrecision]}, If[LessEqual[t, -3.4e+123], N[(N[(-1.0 * N[(U$42$ * N[(-1.0 * N[(t$95$1 * t$95$3), $MachinePrecision] + N[(-0.5 * N[(N[Sqrt[N[Power[N[(U / U$42$), $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Om * N[(-0.5 * N[(N[(N[(Om * N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$1 * N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision], N[(N[(N[((-l$95$m) * t$95$2), $MachinePrecision] / Om), $MachinePrecision] * N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.40000000000000001e123

   1. Initial program 50.6%

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

   3. Taylor expanded in U* around -inf

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

   5. Applied rewrites 3.4%

      \[\leadsto \color{blue}{\left(-U*\right) \cdot \mathsf{fma}\left(\sqrt{\frac{U}{{U*}^{3}}} \cdot \frac{\left(Om \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - \frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om \cdot Om}\right)}{\ell}, 0.5, \frac{\left(\left(-1 \cdot \sqrt{2}\right) \cdot n\right) \cdot \ell}{Om} \cdot \sqrt{\frac{U}{U*}}\right)} \]

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 14.4%

      \[\leadsto \frac{\mathsf{fma}\left(-1, U* \cdot \mathsf{fma}\left(-1, \sqrt{\frac{U}{U*}} \cdot \left(\ell \cdot \left(n \cdot \sqrt{2}\right)\right), -0.5 \cdot \left(\sqrt{{\left(\frac{U}{U*}\right)}^{3}} \cdot \left(\ell \cdot \left(n \cdot \sqrt{2}\right)\right)\right)\right), Om \cdot \mathsf{fma}\left(-0.5, \frac{Om \cdot \left(t \cdot \sqrt{2}\right)}{\ell} \cdot \sqrt{\frac{U}{U*}}, \sqrt{\frac{U}{U*}} \cdot \left(\ell \cdot \sqrt{2}\right)\right)\right)}{\color{blue}{Om}} \]

   if -3.40000000000000001e123 < t

   1. Initial program 50.6%

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

   3. Taylor expanded in U* around -inf

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

   5. Applied rewrites 2.8%

      \[\leadsto \color{blue}{\left(-U*\right) \cdot \mathsf{fma}\left(\sqrt{\frac{U}{{U*}^{3}}} \cdot \frac{\left(Om \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - \frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om \cdot Om}\right)}{\ell}, 0.5, \frac{\left(\left(-1 \cdot \sqrt{2}\right) \cdot n\right) \cdot \ell}{Om} \cdot \sqrt{\frac{U}{U*}}\right)} \]

   6. Taylor expanded in U* around -inf

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

      1. lower-*.f64 N/A

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

   8. Applied rewrites 12.9%

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

4. Final simplification 13.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{+123}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, U* \cdot \mathsf{fma}\left(-1, \sqrt{\frac{U}{U*}} \cdot \left(\ell \cdot \left(n \cdot \sqrt{2}\right)\right), -0.5 \cdot \left(\sqrt{{\left(\frac{U}{U*}\right)}^{3}} \cdot \left(\ell \cdot \left(n \cdot \sqrt{2}\right)\right)\right)\right), Om \cdot \mathsf{fma}\left(-0.5, \frac{Om \cdot \left(t \cdot \sqrt{2}\right)}{\ell} \cdot \sqrt{\frac{U}{U*}}, \sqrt{\frac{U}{U*}} \cdot \left(\ell \cdot \sqrt{2}\right)\right)\right)}{Om}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\ell\right) \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{U \cdot U*}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 13.9% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \frac{\left(-l\_m\right) \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{U \cdot U*} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (* (/ (* (- l_m) (* n (sqrt 2.0))) Om) (sqrt (* U U*))))

C

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

Fortran

l_m =     private
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_m, 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_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = ((-l_m * (n * sqrt(2.0d0))) / om) * sqrt((u * u_42))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[(N[(N[((-l$95$m) * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.6%

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

3. Taylor expanded in U* around -inf

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

5. Applied rewrites 2.9%

   \[\leadsto \color{blue}{\left(-U*\right) \cdot \mathsf{fma}\left(\sqrt{\frac{U}{{U*}^{3}}} \cdot \frac{\left(Om \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - \frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om \cdot Om}\right)}{\ell}, 0.5, \frac{\left(\left(-1 \cdot \sqrt{2}\right) \cdot n\right) \cdot \ell}{Om} \cdot \sqrt{\frac{U}{U*}}\right)} \]

6. Taylor expanded in U* around -inf

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

   1. lower-*.f64 N/A

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

8. Applied rewrites 13.2%

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

9. Final simplification 13.2%

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

Alternative 16: 8.3% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\frac{U}{U*}}\\ \mathbf{if}\;l\_m \leq 1.6 \cdot 10^{+105}:\\ \;\;\;\;0.5 \cdot \left(\frac{l\_m \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{\frac{{U}^{3}}{U*}}\right)\\ \mathbf{else}:\\ \;\;\;\;Om \cdot \mathsf{fma}\left(-0.5, t\_1 \cdot \frac{t \cdot \sqrt{2}}{l\_m}, \frac{l\_m \cdot \sqrt{2}}{Om} \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (sqrt (/ U U*))))
   (if (<= l_m 1.6e+105)
     (* 0.5 (* (/ (* l_m (* n (sqrt 2.0))) Om) (sqrt (/ (pow U 3.0) U*))))
     (*
      Om
      (fma
       -0.5
       (* t_1 (/ (* t (sqrt 2.0)) l_m))
       (* (/ (* l_m (sqrt 2.0)) Om) t_1))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt((U / U_42_));
	double tmp;
	if (l_m <= 1.6e+105) {
		tmp = 0.5 * (((l_m * (n * sqrt(2.0))) / Om) * sqrt((pow(U, 3.0) / U_42_)));
	} else {
		tmp = Om * fma(-0.5, (t_1 * ((t * sqrt(2.0)) / l_m)), (((l_m * sqrt(2.0)) / Om) * t_1));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(U / U_42_))
	tmp = 0.0
	if (l_m <= 1.6e+105)
		tmp = Float64(0.5 * Float64(Float64(Float64(l_m * Float64(n * sqrt(2.0))) / Om) * sqrt(Float64((U ^ 3.0) / U_42_))));
	else
		tmp = Float64(Om * fma(-0.5, Float64(t_1 * Float64(Float64(t * sqrt(2.0)) / l_m)), Float64(Float64(Float64(l_m * sqrt(2.0)) / Om) * t_1)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(U / U$42$), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l$95$m, 1.6e+105], N[(0.5 * N[(N[(N[(l$95$m * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[Sqrt[N[(N[Power[U, 3.0], $MachinePrecision] / U$42$), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Om * N[(-0.5 * N[(t$95$1 * N[(N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.6e105

   1. Initial program 54.2%

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

   3. Taylor expanded in U* around -inf

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

   5. Applied rewrites 2.9%

      \[\leadsto \color{blue}{\left(-U*\right) \cdot \mathsf{fma}\left(\sqrt{\frac{U}{{U*}^{3}}} \cdot \frac{\left(Om \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - \frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om \cdot Om}\right)}{\ell}, 0.5, \frac{\left(\left(-1 \cdot \sqrt{2}\right) \cdot n\right) \cdot \ell}{Om} \cdot \sqrt{\frac{U}{U*}}\right)} \]

   6. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \color{blue}{\sqrt{\frac{{U}^{3}}{U*}}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{\frac{{U}^{3}}{U*}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{\frac{{U}^{3}}{U*}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{\frac{{U}^{3}}{U*}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{\frac{{U}^{3}}{U*}}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{\frac{{U}^{3}}{U*}}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{\frac{{U}^{3}}{U*}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{\frac{{U}^{3}}{U*}}\right) \]

      9. lower-pow.f64 6.9

         \[\leadsto 0.5 \cdot \left(\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{\frac{{U}^{3}}{U*}}\right) \]

   8. Applied rewrites 6.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{\frac{{U}^{3}}{U*}}\right)} \]

   if 1.6e105 < l

   1. Initial program 27.5%

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

   3. Taylor expanded in U* around -inf

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

   5. Applied rewrites 3.0%

      \[\leadsto \color{blue}{\left(-U*\right) \cdot \mathsf{fma}\left(\sqrt{\frac{U}{{U*}^{3}}} \cdot \frac{\left(Om \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - \frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om \cdot Om}\right)}{\ell}, 0.5, \frac{\left(\left(-1 \cdot \sqrt{2}\right) \cdot n\right) \cdot \ell}{Om} \cdot \sqrt{\frac{U}{U*}}\right)} \]

   6. Taylor expanded in Om around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 14.6%

      \[\leadsto Om \cdot \color{blue}{\mathsf{fma}\left(-0.5, \sqrt{\frac{U}{U*}} \cdot \frac{t \cdot \sqrt{2}}{\ell}, \frac{\ell \cdot \sqrt{2}}{Om} \cdot \sqrt{\frac{U}{U*}}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 6.1% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ 0.5 \cdot \left(\frac{l\_m \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{\frac{{U}^{3}}{U*}}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (* 0.5 (* (/ (* l_m (* n (sqrt 2.0))) Om) (sqrt (/ (pow U 3.0) U*)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return 0.5 * (((l_m * (n * sqrt(2.0))) / Om) * sqrt((pow(U, 3.0) / U_42_)));
}

Fortran

l_m =     private
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_m, 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_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = 0.5d0 * (((l_m * (n * sqrt(2.0d0))) / om) * sqrt(((u ** 3.0d0) / u_42)))
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return 0.5 * (((l_m * (n * Math.sqrt(2.0))) / Om) * Math.sqrt((Math.pow(U, 3.0) / U_42_)));
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return 0.5 * (((l_m * (n * math.sqrt(2.0))) / Om) * math.sqrt((math.pow(U, 3.0) / U_42_)))

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return Float64(0.5 * Float64(Float64(Float64(l_m * Float64(n * sqrt(2.0))) / Om) * sqrt(Float64((U ^ 3.0) / U_42_))))
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.5 * (((l_m * (n * sqrt(2.0))) / Om) * sqrt(((U ^ 3.0) / U_42_)));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[(0.5 * N[(N[(N[(l$95$m * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[Sqrt[N[(N[Power[U, 3.0], $MachinePrecision] / U$42$), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.6%

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

3. Taylor expanded in U* around -inf

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

5. Applied rewrites 2.9%

   \[\leadsto \color{blue}{\left(-U*\right) \cdot \mathsf{fma}\left(\sqrt{\frac{U}{{U*}^{3}}} \cdot \frac{\left(Om \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - \frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om \cdot Om}\right)}{\ell}, 0.5, \frac{\left(\left(-1 \cdot \sqrt{2}\right) \cdot n\right) \cdot \ell}{Om} \cdot \sqrt{\frac{U}{U*}}\right)} \]

6. Taylor expanded in U around inf

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

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \color{blue}{\sqrt{\frac{{U}^{3}}{U*}}}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{\frac{{U}^{3}}{U*}}\right) \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{\frac{{U}^{3}}{U*}}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{\frac{{U}^{3}}{U*}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{\frac{{U}^{3}}{U*}}\right) \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{\frac{{U}^{3}}{U*}}\right) \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{\frac{{U}^{3}}{U*}}\right) \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{\frac{{U}^{3}}{U*}}\right) \]

   9. lower-pow.f64 6.6

      \[\leadsto 0.5 \cdot \left(\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{\frac{{U}^{3}}{U*}}\right) \]

8. Applied rewrites 6.6%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{\frac{{U}^{3}}{U*}}\right)} \]
9. Add Preprocessing

Alternative 18: 2.6% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ -0.5 \cdot \left(\frac{Om \cdot \left(t \cdot \sqrt{2}\right)}{l\_m} \cdot \sqrt{\frac{U}{U*}}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (* -0.5 (* (/ (* Om (* t (sqrt 2.0))) l_m) (sqrt (/ U U*)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return -0.5 * (((Om * (t * sqrt(2.0))) / l_m) * sqrt((U / U_42_)));
}

Fortran

l_m =     private
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_m, 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_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = (-0.5d0) * (((om * (t * sqrt(2.0d0))) / l_m) * sqrt((u / u_42)))
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return -0.5 * (((Om * (t * Math.sqrt(2.0))) / l_m) * Math.sqrt((U / U_42_)));
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return -0.5 * (((Om * (t * math.sqrt(2.0))) / l_m) * math.sqrt((U / U_42_)))

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return Float64(-0.5 * Float64(Float64(Float64(Om * Float64(t * sqrt(2.0))) / l_m) * sqrt(Float64(U / U_42_))))
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = -0.5 * (((Om * (t * sqrt(2.0))) / l_m) * sqrt((U / U_42_)));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[(-0.5 * N[(N[(N[(Om * N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] * N[Sqrt[N[(U / U$42$), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.6%

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

3. Taylor expanded in U* around -inf

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

5. Applied rewrites 2.9%

   \[\leadsto \color{blue}{\left(-U*\right) \cdot \mathsf{fma}\left(\sqrt{\frac{U}{{U*}^{3}}} \cdot \frac{\left(Om \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - \frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om \cdot Om}\right)}{\ell}, 0.5, \frac{\left(\left(-1 \cdot \sqrt{2}\right) \cdot n\right) \cdot \ell}{Om} \cdot \sqrt{\frac{U}{U*}}\right)} \]

6. Taylor expanded in t around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-/.f64 2.8

      \[\leadsto -0.5 \cdot \left(\frac{Om \cdot \left(t \cdot \sqrt{2}\right)}{\ell} \cdot \sqrt{\frac{U}{U*}}\right) \]

8. Applied rewrites 2.8%

   \[\leadsto -0.5 \cdot \color{blue}{\left(\frac{Om \cdot \left(t \cdot \sqrt{2}\right)}{\ell} \cdot \sqrt{\frac{U}{U*}}\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(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*))))))