Toniolo and Linder, Equation (13)

Percentage Accurate: 50.7% → 60.7%

Time: 9.1s

Alternatives: 14

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.7% 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: 60.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - n \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\\ t_2 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;{\left(n + n\right)}^{0.5} \cdot {\left(U \cdot t\_1\right)}^{0.5}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+289}:\\ \;\;\;\;\sqrt{t\_2}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot U} \cdot \sqrt{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{U* \cdot U}{Om} \cdot \frac{\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)}{Om}\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (-
          (fma -2.0 (* l (/ l Om)) t)
          (* n (* (* (/ l Om) (/ l Om)) (- U U*)))))
        (t_2
         (*
          (* (* 2.0 n) U)
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_2 0.0)
     (* (pow (+ n n) 0.5) (pow (* U t_1) 0.5))
     (if (<= t_2 5e+289)
       (sqrt t_2)
       (if (<= t_2 INFINITY)
         (* (sqrt (* (+ n n) U)) (sqrt t_1))
         (sqrt (* (* (/ (* U* U) Om) (/ (* (* n l) (* n l)) Om)) 2.0)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = fma(-2.0, (l * (l / Om)), t) - (n * (((l / Om) * (l / Om)) * (U - U_42_)));
	double t_2 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = pow((n + n), 0.5) * pow((U * t_1), 0.5);
	} else if (t_2 <= 5e+289) {
		tmp = sqrt(t_2);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((n + n) * U)) * sqrt(t_1);
	} else {
		tmp = sqrt(((((U_42_ * U) / Om) * (((n * l) * (n * l)) / Om)) * 2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) - Float64(n * Float64(Float64(Float64(l / Om) * Float64(l / Om)) * Float64(U - U_42_))))
	t_2 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64((Float64(n + n) ^ 0.5) * (Float64(U * t_1) ^ 0.5));
	elseif (t_2 <= 5e+289)
		tmp = sqrt(t_2);
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(n + n) * U)) * sqrt(t_1));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(U_42_ * U) / Om) * Float64(Float64(Float64(n * l) * Float64(n * l)) / Om)) * 2.0));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] - N[(n * N[(N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, 0.0], N[(N[Power[N[(n + n), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(U * t$95$1), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+289], N[Sqrt[t$95$2], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[(N[(U$42$ * U), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(N[(n * l), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 10.5%

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

   3. Applied rewrites 35.9%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.00000000000000031e289

   1. Initial program 97.3%

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

   if 5.00000000000000031e289 < (*.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 34.3%

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

   3. Applied rewrites 30.8%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 27.0

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

   4. Applied rewrites 27.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unswap-sqr N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 34.2

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

   6. Applied rewrites 34.2%

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

Alternative 2: 59.9% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (fma -2.0 (* l (/ l Om)) t))
        (t_2
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_2 0.0)
     (*
      (sqrt (+ n n))
      (sqrt (* U (- t_1 (* (* n (/ (* l l) (* Om Om))) (- U U*))))))
     (if (<= t_2 1e+145)
       t_2
       (if (<= t_2 INFINITY)
         (*
          (sqrt (* (+ n n) U))
          (sqrt (- t_1 (* n (* (* (/ l Om) (/ l Om)) (- U U*))))))
         (sqrt (* (* (/ (* U* U) Om) (/ (* (* n l) (* n l)) Om)) 2.0)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = fma(-2.0, (l * (l / Om)), t);
	double t_2 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = sqrt((n + n)) * sqrt((U * (t_1 - ((n * ((l * l) / (Om * Om))) * (U - U_42_)))));
	} else if (t_2 <= 1e+145) {
		tmp = t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((n + n) * U)) * sqrt((t_1 - (n * (((l / Om) * (l / Om)) * (U - U_42_)))));
	} else {
		tmp = sqrt(((((U_42_ * U) / Om) * (((n * l) * (n * l)) / Om)) * 2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = fma(-2.0, Float64(l * Float64(l / Om)), t)
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(Float64(U * Float64(t_1 - Float64(Float64(n * Float64(Float64(l * l) / Float64(Om * Om))) * Float64(U - U_42_))))));
	elseif (t_2 <= 1e+145)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(n + n) * U)) * sqrt(Float64(t_1 - Float64(n * Float64(Float64(Float64(l / Om) * Float64(l / Om)) * Float64(U - U_42_))))));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(U_42_ * U) / Om) * Float64(Float64(Float64(n * l) * Float64(n * l)) / Om)) * 2.0));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(t$95$1 - N[(N[(n * N[(N[(l * l), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+145], t$95$2, If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$1 - N[(n * N[(N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[(N[(U$42$ * U), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(N[(n * l), $MachinePrecision] * N[(n * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 11.9%

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

   3. Applied rewrites 37.0%

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

   5. Applied rewrites 36.3%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 9.9999999999999999e144

   1. Initial program 97.3%

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

   if 9.9999999999999999e144 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 34.2%

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

   3. Applied rewrites 30.8%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 24.2

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

   4. Applied rewrites 24.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unswap-sqr N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 31.9

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

   6. Applied rewrites 31.9%

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

Alternative 3: 58.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 11.9%

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

   3. Applied rewrites 37.0%

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

   5. Applied rewrites 36.3%

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

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

   1. Initial program 69.0%

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

   3. Applied rewrites 71.6%

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

   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. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 24.2

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

   4. Applied rewrites 24.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unswap-sqr N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 31.9

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

   6. Applied rewrites 31.9%

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

Alternative 4: 57.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 10.5%

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

   2. Taylor expanded in Om around inf

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 29.9%

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

   5. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 16.0%

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

   8. Taylor expanded in t around inf

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

      1. sqrt-prod N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sqrt.f64 27.9

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

   10. Applied rewrites 27.9%

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

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

   1. Initial program 69.0%

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

   3. Applied rewrites 71.6%

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

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

   1. Initial program 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. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 27.0

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

   4. Applied rewrites 27.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unswap-sqr N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 34.2

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

   6. Applied rewrites 34.2%

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

Alternative 5: 52.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{Om} \cdot \frac{\ell}{Om}\\ \mathbf{if}\;t \leq -4.35 \cdot 10^{-156}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(n \cdot t\_1\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;t \leq 1.76 \cdot 10^{-208}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot U} \cdot \sqrt{t - n \cdot \left(t\_1 \cdot \left(U - U*\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (/ l Om) (/ l Om))))
   (if (<= t -4.35e-156)
     (sqrt (* (* (* 2.0 n) U) (- t (* (* n t_1) (- U U*)))))
     (if (<= t 1.76e-208)
       (sqrt (* (+ n n) (* U (- t (* 2.0 (/ (* l l) Om))))))
       (* (sqrt (* (+ n n) U)) (sqrt (- t (* n (* t_1 (- U U*))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l / Om) * (l / Om);
	double tmp;
	if (t <= -4.35e-156) {
		tmp = sqrt((((2.0 * n) * U) * (t - ((n * t_1) * (U - U_42_)))));
	} else if (t <= 1.76e-208) {
		tmp = sqrt(((n + n) * (U * (t - (2.0 * ((l * l) / Om))))));
	} else {
		tmp = sqrt(((n + n) * U)) * sqrt((t - (n * (t_1 * (U - U_42_)))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (l / om) * (l / om)
    if (t <= (-4.35d-156)) then
        tmp = sqrt((((2.0d0 * n) * u) * (t - ((n * t_1) * (u - u_42)))))
    else if (t <= 1.76d-208) then
        tmp = sqrt(((n + n) * (u * (t - (2.0d0 * ((l * l) / om))))))
    else
        tmp = sqrt(((n + n) * u)) * sqrt((t - (n * (t_1 * (u - u_42)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l / Om) * (l / Om);
	double tmp;
	if (t <= -4.35e-156) {
		tmp = Math.sqrt((((2.0 * n) * U) * (t - ((n * t_1) * (U - U_42_)))));
	} else if (t <= 1.76e-208) {
		tmp = Math.sqrt(((n + n) * (U * (t - (2.0 * ((l * l) / Om))))));
	} else {
		tmp = Math.sqrt(((n + n) * U)) * Math.sqrt((t - (n * (t_1 * (U - U_42_)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (l / Om) * (l / Om)
	tmp = 0
	if t <= -4.35e-156:
		tmp = math.sqrt((((2.0 * n) * U) * (t - ((n * t_1) * (U - U_42_)))))
	elif t <= 1.76e-208:
		tmp = math.sqrt(((n + n) * (U * (t - (2.0 * ((l * l) / Om))))))
	else:
		tmp = math.sqrt(((n + n) * U)) * math.sqrt((t - (n * (t_1 * (U - U_42_)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l / Om) * Float64(l / Om))
	tmp = 0.0
	if (t <= -4.35e-156)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(t - Float64(Float64(n * t_1) * Float64(U - U_42_)))));
	elseif (t <= 1.76e-208)
		tmp = sqrt(Float64(Float64(n + n) * Float64(U * Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))))));
	else
		tmp = Float64(sqrt(Float64(Float64(n + n) * U)) * sqrt(Float64(t - Float64(n * Float64(t_1 * Float64(U - U_42_))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (l / Om) * (l / Om);
	tmp = 0.0;
	if (t <= -4.35e-156)
		tmp = sqrt((((2.0 * n) * U) * (t - ((n * t_1) * (U - U_42_)))));
	elseif (t <= 1.76e-208)
		tmp = sqrt(((n + n) * (U * (t - (2.0 * ((l * l) / Om))))));
	else
		tmp = sqrt(((n + n) * U)) * sqrt((t - (n * (t_1 * (U - U_42_)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.35e-156], 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[t, 1.76e-208], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(U * N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t - N[(n * N[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.35000000000000005e-156

   1. Initial program 52.1%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 54.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 54.3

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

   6. Applied rewrites 54.3%

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

   if -4.35000000000000005e-156 < t < 1.76000000000000008e-208

   1. Initial program 45.6%

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

   3. Applied rewrites 48.5%

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

   4. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 36.0

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

   6. Applied rewrites 36.0%

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

   if 1.76000000000000008e-208 < t

   1. Initial program 51.8%

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

   3. Applied rewrites 52.8%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 51.3%

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

Alternative 6: 52.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;U \cdot \sqrt{\frac{n \cdot t}{U} \cdot 2}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+289}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(t - \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{U* \cdot U}{Om} \cdot \frac{\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)}{Om}\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_2 0.0)
     (* U (sqrt (* (/ (* n t) U) 2.0)))
     (if (<= t_2 5e+289)
       (sqrt (* t_1 (- t (* (* n (* (/ l Om) (/ l Om))) (- U U*)))))
       (sqrt (* (* (/ (* U* U) Om) (/ (* (* n l) (* n l)) Om)) 2.0))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = U * sqrt((((n * t) / U) * 2.0));
	} else if (t_2 <= 5e+289) {
		tmp = sqrt((t_1 * (t - ((n * ((l / Om) * (l / Om))) * (U - U_42_)))));
	} else {
		tmp = sqrt(((((U_42_ * U) / Om) * (((n * l) * (n * l)) / Om)) * 2.0));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (2.0d0 * n) * u
    t_2 = t_1 * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))
    if (t_2 <= 0.0d0) then
        tmp = u * sqrt((((n * t) / u) * 2.0d0))
    else if (t_2 <= 5d+289) then
        tmp = sqrt((t_1 * (t - ((n * ((l / om) * (l / om))) * (u - u_42)))))
    else
        tmp = sqrt(((((u_42 * u) / om) * (((n * l) * (n * l)) / om)) * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = U * Math.sqrt((((n * t) / U) * 2.0));
	} else if (t_2 <= 5e+289) {
		tmp = Math.sqrt((t_1 * (t - ((n * ((l / Om) * (l / Om))) * (U - U_42_)))));
	} else {
		tmp = Math.sqrt(((((U_42_ * U) / Om) * (((n * l) * (n * l)) / Om)) * 2.0));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (2.0 * n) * U
	t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))
	tmp = 0
	if t_2 <= 0.0:
		tmp = U * math.sqrt((((n * t) / U) * 2.0))
	elif t_2 <= 5e+289:
		tmp = math.sqrt((t_1 * (t - ((n * ((l / Om) * (l / Om))) * (U - U_42_)))))
	else:
		tmp = math.sqrt(((((U_42_ * U) / Om) * (((n * l) * (n * l)) / Om)) * 2.0))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(U * sqrt(Float64(Float64(Float64(n * t) / U) * 2.0)));
	elseif (t_2 <= 5e+289)
		tmp = sqrt(Float64(t_1 * Float64(t - Float64(Float64(n * Float64(Float64(l / Om) * Float64(l / Om))) * Float64(U - U_42_)))));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(U_42_ * U) / Om) * Float64(Float64(Float64(n * l) * Float64(n * l)) / Om)) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (2.0 * n) * U;
	t_2 = t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)));
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = U * sqrt((((n * t) / U) * 2.0));
	elseif (t_2 <= 5e+289)
		tmp = sqrt((t_1 * (t - ((n * ((l / Om) * (l / Om))) * (U - U_42_)))));
	else
		tmp = sqrt(((((U_42_ * U) / Om) * (((n * l) * (n * l)) / Om)) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 10.5%

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

   2. Taylor expanded in Om around inf

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 29.9%

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

   5. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 16.0%

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

   8. Taylor expanded in t around inf

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

      1. sqrt-prod N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sqrt.f64 27.9

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

   10. Applied rewrites 27.9%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.00000000000000031e289

   1. Initial program 97.3%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 85.7%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 85.7

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

   6. Applied rewrites 85.7%

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

   if 5.00000000000000031e289 < (*.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 23.7%

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

   2. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 25.9

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

   4. Applied rewrites 25.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unswap-sqr N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 32.0

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

   6. Applied rewrites 32.0%

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

Alternative 7: 49.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;U \cdot \sqrt{\frac{n \cdot t}{U} \cdot 2}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+289}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{U* \cdot U}{Om} \cdot \frac{\left(n \cdot \ell\right) \cdot \left(n \cdot \ell\right)}{Om}\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* (* 2.0 n) U)
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_1 0.0)
     (* U (sqrt (* (/ (* n t) U) 2.0)))
     (if (<= t_1 5e+289)
       (sqrt (* (* (+ n n) U) (fma -2.0 (* l (/ l Om)) t)))
       (sqrt (* (* (/ (* U* U) Om) (/ (* (* n l) (* n l)) Om)) 2.0))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = U * sqrt((((n * t) / U) * 2.0));
	} else if (t_1 <= 5e+289) {
		tmp = sqrt((((n + n) * U) * fma(-2.0, (l * (l / Om)), t)));
	} else {
		tmp = sqrt(((((U_42_ * U) / Om) * (((n * l) * (n * l)) / Om)) * 2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(U * sqrt(Float64(Float64(Float64(n * t) / U) * 2.0)));
	elseif (t_1 <= 5e+289)
		tmp = sqrt(Float64(Float64(Float64(n + n) * U) * fma(-2.0, Float64(l * Float64(l / Om)), t)));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(U_42_ * U) / Om) * Float64(Float64(Float64(n * l) * Float64(n * l)) / Om)) * 2.0));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 10.5%

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

   2. Taylor expanded in Om around inf

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 29.9%

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

   5. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 16.0%

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

   8. Taylor expanded in t around inf

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

      1. sqrt-prod N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sqrt.f64 27.9

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

   10. Applied rewrites 27.9%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.00000000000000031e289

   1. Initial program 97.3%

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

   2. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 86.2

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

   4. Applied rewrites 86.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 86.2

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

   6. Applied rewrites 86.2%

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

   if 5.00000000000000031e289 < (*.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 23.7%

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

   2. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 25.9

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

   4. Applied rewrites 25.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unswap-sqr N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 32.0

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

   6. Applied rewrites 32.0%

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

Alternative 8: 48.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\\ t_2 := \left(n + n\right) \cdot U\\ \mathbf{if}\;t \leq 6 \cdot 10^{-286}:\\ \;\;\;\;\sqrt{t\_2 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_2} \cdot \sqrt{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (fma -2.0 (* l (/ l Om)) t)) (t_2 (* (+ n n) U)))
   (if (<= t 6e-286) (sqrt (* t_2 t_1)) (* (sqrt t_2) (sqrt t_1)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = fma(-2.0, (l * (l / Om)), t);
	double t_2 = (n + n) * U;
	double tmp;
	if (t <= 6e-286) {
		tmp = sqrt((t_2 * t_1));
	} else {
		tmp = sqrt(t_2) * sqrt(t_1);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = fma(-2.0, Float64(l * Float64(l / Om)), t)
	t_2 = Float64(Float64(n + n) * U)
	tmp = 0.0
	if (t <= 6e-286)
		tmp = sqrt(Float64(t_2 * t_1));
	else
		tmp = Float64(sqrt(t_2) * sqrt(t_1));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]}, If[LessEqual[t, 6e-286], N[Sqrt[N[(t$95$2 * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[t$95$2], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if t < 6.0000000000000001e-286

   1. Initial program 50.6%

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

   2. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 48.1

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

   4. Applied rewrites 48.1%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 48.1

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

   6. Applied rewrites 48.1%

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

   if 6.0000000000000001e-286 < t

   1. Initial program 50.9%

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

   2. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 47.7

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

   4. Applied rewrites 47.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 N/A

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

      9. lower-sqrt.f64 48.6

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

   6. Applied rewrites 48.6%

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

Alternative 9: 48.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(n + n\right) \cdot U\\ \mathbf{if}\;t \leq 1.15 \cdot 10^{+188}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (+ n n) U)))
   (if (<= t 1.15e+188)
     (sqrt (* t_1 (fma -2.0 (* l (/ l Om)) t)))
     (* (sqrt t_1) (sqrt t)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n + n) * U;
	double tmp;
	if (t <= 1.15e+188) {
		tmp = sqrt((t_1 * fma(-2.0, (l * (l / Om)), t)));
	} else {
		tmp = sqrt(t_1) * sqrt(t);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(n + n) * U)
	tmp = 0.0
	if (t <= 1.15e+188)
		tmp = sqrt(Float64(t_1 * fma(-2.0, Float64(l * Float64(l / Om)), t)));
	else
		tmp = Float64(sqrt(t_1) * sqrt(t));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]}, If[LessEqual[t, 1.15e+188], N[Sqrt[N[(t$95$1 * N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.15000000000000006e188

   1. Initial program 51.0%

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

   2. Taylor expanded in n around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 47.8

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

   4. Applied rewrites 47.8%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 47.8

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

   6. Applied rewrites 47.8%

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

   if 1.15000000000000006e188 < t

   1. Initial program 48.2%

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

   3. Applied rewrites 59.6%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 57.6%

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

Alternative 10: 44.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;U \cdot \sqrt{\frac{n \cdot t}{U} \cdot 2}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+289}:\\ \;\;\;\;\sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U* \cdot U} \cdot \frac{\left(\sqrt{2} \cdot n\right) \cdot \ell}{Om}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* (* 2.0 n) U)
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_1 0.0)
     (* U (sqrt (* (/ (* n t) U) 2.0)))
     (if (<= t_1 5e+289)
       (sqrt (* t (* (+ n n) U)))
       (* (sqrt (* U* U)) (/ (* (* (sqrt 2.0) n) l) Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = U * sqrt((((n * t) / U) * 2.0));
	} else if (t_1 <= 5e+289) {
		tmp = sqrt((t * ((n + n) * U)));
	} else {
		tmp = sqrt((U_42_ * U)) * (((sqrt(2.0) * n) * l) / Om);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))
    if (t_1 <= 0.0d0) then
        tmp = u * sqrt((((n * t) / u) * 2.0d0))
    else if (t_1 <= 5d+289) then
        tmp = sqrt((t * ((n + n) * u)))
    else
        tmp = sqrt((u_42 * u)) * (((sqrt(2.0d0) * n) * l) / om)
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = U * Math.sqrt((((n * t) / U) * 2.0));
	} else if (t_1 <= 5e+289) {
		tmp = Math.sqrt((t * ((n + n) * U)));
	} else {
		tmp = Math.sqrt((U_42_ * U)) * (((Math.sqrt(2.0) * n) * l) / Om);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))
	tmp = 0
	if t_1 <= 0.0:
		tmp = U * math.sqrt((((n * t) / U) * 2.0))
	elif t_1 <= 5e+289:
		tmp = math.sqrt((t * ((n + n) * U)))
	else:
		tmp = math.sqrt((U_42_ * U)) * (((math.sqrt(2.0) * n) * l) / Om)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(U * sqrt(Float64(Float64(Float64(n * t) / U) * 2.0)));
	elseif (t_1 <= 5e+289)
		tmp = sqrt(Float64(t * Float64(Float64(n + n) * U)));
	else
		tmp = Float64(sqrt(Float64(U_42_ * U)) * Float64(Float64(Float64(sqrt(2.0) * n) * l) / Om));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = U * sqrt((((n * t) / U) * 2.0));
	elseif (t_1 <= 5e+289)
		tmp = sqrt((t * ((n + n) * U)));
	else
		tmp = sqrt((U_42_ * U)) * (((sqrt(2.0) * n) * l) / Om);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 10.5%

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

   2. Taylor expanded in Om around inf

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 29.9%

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

   5. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 16.0%

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

   8. Taylor expanded in t around inf

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

      1. sqrt-prod N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sqrt.f64 27.9

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

   10. Applied rewrites 27.9%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.00000000000000031e289

   1. Initial program 97.3%

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

   3. Applied rewrites 94.0%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 75.1%

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

   if 5.00000000000000031e289 < (*.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 23.7%

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

   2. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 22.8

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

   4. Applied rewrites 22.8%

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

Alternative 11: 39.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(n + n\right) \cdot U\\ \mathbf{if}\;t \leq 6.5 \cdot 10^{-294}:\\ \;\;\;\;\sqrt{t \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1} \cdot \sqrt{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (+ n n) U)))
   (if (<= t 6.5e-294) (sqrt (* t t_1)) (* (sqrt t_1) (sqrt t)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n + n) * U;
	double tmp;
	if (t <= 6.5e-294) {
		tmp = sqrt((t * t_1));
	} else {
		tmp = sqrt(t_1) * sqrt(t);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (n + n) * u
    if (t <= 6.5d-294) then
        tmp = sqrt((t * t_1))
    else
        tmp = sqrt(t_1) * sqrt(t)
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (n + n) * U;
	double tmp;
	if (t <= 6.5e-294) {
		tmp = Math.sqrt((t * t_1));
	} else {
		tmp = Math.sqrt(t_1) * Math.sqrt(t);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (n + n) * U
	tmp = 0
	if t <= 6.5e-294:
		tmp = math.sqrt((t * t_1))
	else:
		tmp = math.sqrt(t_1) * math.sqrt(t)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(n + n) * U)
	tmp = 0.0
	if (t <= 6.5e-294)
		tmp = sqrt(Float64(t * t_1));
	else
		tmp = Float64(sqrt(t_1) * sqrt(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (n + n) * U;
	tmp = 0.0;
	if (t <= 6.5e-294)
		tmp = sqrt((t * t_1));
	else
		tmp = sqrt(t_1) * sqrt(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]}, If[LessEqual[t, 6.5e-294], N[Sqrt[N[(t * t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 6.4999999999999995e-294

   1. Initial program 50.6%

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

   3. Applied rewrites 53.6%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 35.8%

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

   if 6.4999999999999995e-294 < t

   1. Initial program 50.8%

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

   3. Applied rewrites 52.1%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 43.2%

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

Alternative 12: 39.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq 9.2 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(n + n\right)} \cdot \sqrt{U}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U 9.2e-297)
   (sqrt (* t (* (+ n n) U)))
   (* (sqrt (* t (+ n n))) (sqrt U))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 9.2e-297) {
		tmp = sqrt((t * ((n + n) * U)));
	} else {
		tmp = sqrt((t * (n + n))) * sqrt(U);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= 9.2d-297) then
        tmp = sqrt((t * ((n + n) * u)))
    else
        tmp = sqrt((t * (n + n))) * sqrt(u)
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= 9.2e-297) {
		tmp = Math.sqrt((t * ((n + n) * U)));
	} else {
		tmp = Math.sqrt((t * (n + n))) * Math.sqrt(U);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= 9.2e-297:
		tmp = math.sqrt((t * ((n + n) * U)))
	else:
		tmp = math.sqrt((t * (n + n))) * math.sqrt(U)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= 9.2e-297)
		tmp = sqrt(Float64(t * Float64(Float64(n + n) * U)));
	else
		tmp = Float64(sqrt(Float64(t * Float64(n + n))) * sqrt(U));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= 9.2e-297)
		tmp = sqrt((t * ((n + n) * U)));
	else
		tmp = sqrt((t * (n + n))) * sqrt(U);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, 9.2e-297], N[Sqrt[N[(t * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(t * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 9.1999999999999996e-297

   1. Initial program 49.9%

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

   3. Applied rewrites 52.4%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 35.0%

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

   if 9.1999999999999996e-297 < U

   1. Initial program 51.5%

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

   3. Applied rewrites 54.2%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 37.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 37.3

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

   8. Applied rewrites 37.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 43.1

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

   10. Applied rewrites 43.1%

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

Alternative 13: 36.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 4 \cdot 10^{+263}:\\ \;\;\;\;\sqrt{t \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t \cdot \left(n + n\right)\right) \cdot U}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<=
      (*
       (* (* 2.0 n) U)
       (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))
      4e+263)
   (sqrt (* t (* (+ n n) U)))
   (sqrt (* (* t (+ n n)) U))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))) <= 4e+263) {
		tmp = sqrt((t * ((n + n) * U)));
	} else {
		tmp = sqrt(((t * (n + n)) * U));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))) <= 4d+263) then
        tmp = sqrt((t * ((n + n) * u)))
    else
        tmp = sqrt(((t * (n + n)) * u))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))) <= 4e+263) {
		tmp = Math.sqrt((t * ((n + n) * U)));
	} else {
		tmp = Math.sqrt(((t * (n + n)) * U));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))) <= 4e+263:
		tmp = math.sqrt((t * ((n + n) * U)))
	else:
		tmp = math.sqrt(((t * (n + n)) * U))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))) <= 4e+263)
		tmp = sqrt(Float64(t * Float64(Float64(n + n) * U)));
	else
		tmp = sqrt(Float64(Float64(t * Float64(n + n)) * U));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)))) <= 4e+263)
		tmp = sqrt((t * ((n + n) * U)));
	else
		tmp = sqrt(((t * (n + n)) * U));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+263], N[Sqrt[N[(t * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(t * N[(n + n), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.00000000000000006e263

   1. Initial program 73.2%

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

   3. Applied rewrites 71.7%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 57.9%

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

   if 4.00000000000000006e263 < (*.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 25.8%

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

   3. Applied rewrites 32.8%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 12.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 13.2

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

   8. Applied rewrites 13.2%

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

Alternative 14: 36.1% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.7%

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

3. Applied rewrites 53.2%

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

4. Taylor expanded in t around inf

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

6. Applied rewrites 36.1%

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

Reproduce

herbie shell --seed 2025117 
(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*))))))