Toniolo and Linder, Equation (13)

Percentage Accurate: 49.6% → 61.4%

Time: 10.9s

Alternatives: 10

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 49.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 61.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\\ t_2 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;{2}^{0.5} \cdot \left({t\_1}^{0.5} \cdot {U}^{0.5}\right)\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+138}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;{2}^{0.5} \cdot {\left(t\_1 \cdot U\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (fma -2.0 (* l (/ l Om)) t) n))
        (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)
     (* (pow 2.0 0.5) (* (pow t_1 0.5) (pow U 0.5)))
     (if (<= t_2 4e+138) t_2 (* (pow 2.0 0.5) (pow (* t_1 U) 0.5))))))

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;
	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 = pow(2.0, 0.5) * (pow(t_1, 0.5) * pow(U, 0.5));
	} else if (t_2 <= 4e+138) {
		tmp = t_2;
	} else {
		tmp = pow(2.0, 0.5) * pow((t_1 * U), 0.5);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) * n)
	t_2 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64((2.0 ^ 0.5) * Float64((t_1 ^ 0.5) * (U ^ 0.5)));
	elseif (t_2 <= 4e+138)
		tmp = t_2;
	else
		tmp = Float64((2.0 ^ 0.5) * (Float64(t_1 * U) ^ 0.5));
	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), $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[Power[2.0, 0.5], $MachinePrecision] * N[(N[Power[t$95$1, 0.5], $MachinePrecision] * N[Power[U, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+138], t$95$2, N[(N[Power[2.0, 0.5], $MachinePrecision] * N[Power[N[(t$95$1 * U), $MachinePrecision], 0.5], $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 13.1%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/2 N/A

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

      4. lower-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lower-pow.f64 N/A

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

   5. Applied rewrites 32.0%

      \[\leadsto \color{blue}{{2}^{0.5} \cdot {\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right)}^{0.5}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 40.0%

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

   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*))))) < 4.0000000000000001e138

   1. Initial program 97.8%

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

   if 4.0000000000000001e138 < (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 22.8%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/2 N/A

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

      4. lower-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lower-pow.f64 N/A

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

   5. Applied rewrites 41.5%

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

Alternative 2: 58.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6 \cdot 10^{-154}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right)\right)}\\ \mathbf{elif}\;\ell \leq 4.6 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot \left(t - \mathsf{fma}\left(-1, \frac{U*}{Om} \cdot \frac{\left(\ell \cdot \ell\right) \cdot n}{Om}, 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;{2}^{0.5} \cdot {\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 6e-154)
   (sqrt (* (* n 2.0) (* U (- t (* (- U U*) (* (pow (/ l Om) 2.0) n))))))
   (if (<= l 4.6e+49)
     (sqrt
      (*
       (*
        2.0
        (*
         n
         (-
          t
          (fma
           -1.0
           (* (/ U* Om) (/ (* (* l l) n) Om))
           (* 2.0 (/ (* l l) Om))))))
       U))
     (* (pow 2.0 0.5) (pow (* (* (fma -2.0 (* l (/ l Om)) t) n) U) 0.5)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 6e-154) {
		tmp = sqrt(((n * 2.0) * (U * (t - ((U - U_42_) * (pow((l / Om), 2.0) * n))))));
	} else if (l <= 4.6e+49) {
		tmp = sqrt(((2.0 * (n * (t - fma(-1.0, ((U_42_ / Om) * (((l * l) * n) / Om)), (2.0 * ((l * l) / Om)))))) * U));
	} else {
		tmp = pow(2.0, 0.5) * pow(((fma(-2.0, (l * (l / Om)), t) * n) * U), 0.5);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 6e-154)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t - Float64(Float64(U - U_42_) * Float64((Float64(l / Om) ^ 2.0) * n))))));
	elseif (l <= 4.6e+49)
		tmp = sqrt(Float64(Float64(2.0 * Float64(n * Float64(t - fma(-1.0, Float64(Float64(U_42_ / Om) * Float64(Float64(Float64(l * l) * n) / Om)), Float64(2.0 * Float64(Float64(l * l) / Om)))))) * U));
	else
		tmp = Float64((2.0 ^ 0.5) * (Float64(Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) * n) * U) ^ 0.5));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 6e-154], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t - N[(N[(U - U$42$), $MachinePrecision] * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 4.6e+49], N[Sqrt[N[(N[(2.0 * N[(n * N[(t - N[(-1.0 * N[(N[(U$42$ / Om), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], N[(N[Power[2.0, 0.5], $MachinePrecision] * N[Power[N[(N[(N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 6.0000000000000005e-154

   1. Initial program 51.1%

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 51.6%

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

   if 6.0000000000000005e-154 < l < 4.60000000000000004e49

   1. Initial program 58.8%

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

   3. Taylor expanded in U around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 42.3%

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

   6. Taylor expanded in Om around -inf

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

      1. lower-fma.f64 N/A

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

   8. Applied rewrites 44.4%

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

   9. Taylor expanded in U around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. lower-fma.f64 N/A

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

   11. Applied rewrites 74.0%

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

   if 4.60000000000000004e49 < l

   1. Initial program 23.6%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/2 N/A

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

      4. lower-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lower-pow.f64 N/A

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

   5. Applied rewrites 50.7%

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

Alternative 3: 58.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\\ t_2 := {\left(\frac{\ell}{Om}\right)}^{2}\\ \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot t\_2\right) \cdot \left(U - U*\right)\right)} \leq 10^{+116}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t\_1 - \left(U - U*\right) \cdot \left(t\_2 \cdot n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{2}^{0.5} \cdot {\left(\left(t\_1 \cdot n\right) \cdot U\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (fma -2.0 (* l (/ l Om)) t)) (t_2 (pow (/ l Om) 2.0)))
   (if (<=
        (sqrt
         (*
          (* (* 2.0 n) U)
          (- (- t (* 2.0 (/ (* l l) Om))) (* (* n t_2) (- U U*)))))
        1e+116)
     (sqrt (* (* n 2.0) (* U (- t_1 (* (- U U*) (* t_2 n))))))
     (* (pow 2.0 0.5) (pow (* (* t_1 n) U) 0.5)))))

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 = pow((l / Om), 2.0);
	double tmp;
	if (sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * t_2) * (U - U_42_))))) <= 1e+116) {
		tmp = sqrt(((n * 2.0) * (U * (t_1 - ((U - U_42_) * (t_2 * n))))));
	} else {
		tmp = pow(2.0, 0.5) * pow(((t_1 * n) * U), 0.5);
	}
	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(l / Om) ^ 2.0
	tmp = 0.0
	if (sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * t_2) * Float64(U - U_42_))))) <= 1e+116)
		tmp = sqrt(Float64(Float64(n * 2.0) * Float64(U * Float64(t_1 - Float64(Float64(U - U_42_) * Float64(t_2 * n))))));
	else
		tmp = Float64((2.0 ^ 0.5) * (Float64(Float64(t_1 * n) * U) ^ 0.5));
	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[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * t$95$2), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1e+116], N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * N[(U * N[(t$95$1 - N[(N[(U - U$42$), $MachinePrecision] * N[(t$95$2 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Power[2.0, 0.5], $MachinePrecision] * N[Power[N[(N[(t$95$1 * n), $MachinePrecision] * U), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1.00000000000000002e116

   1. Initial program 71.3%

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

   4. Applied rewrites 75.5%

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

   if 1.00000000000000002e116 < (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 25.8%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/2 N/A

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

      4. lower-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lower-pow.f64 N/A

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

   5. Applied rewrites 43.0%

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

Alternative 4: 58.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\\ \mathbf{if}\;U \leq -5 \cdot 10^{-310}:\\ \;\;\;\;{2}^{0.5} \cdot {\left(t\_1 \cdot U\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{2}^{0.5} \cdot \left({t\_1}^{0.5} \cdot {U}^{0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (fma -2.0 (* l (/ l Om)) t) n)))
   (if (<= U -5e-310)
     (* (pow 2.0 0.5) (pow (* t_1 U) 0.5))
     (* (pow 2.0 0.5) (* (pow t_1 0.5) (pow U 0.5))))))

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;
	double tmp;
	if (U <= -5e-310) {
		tmp = pow(2.0, 0.5) * pow((t_1 * U), 0.5);
	} else {
		tmp = pow(2.0, 0.5) * (pow(t_1, 0.5) * pow(U, 0.5));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) * n)
	tmp = 0.0
	if (U <= -5e-310)
		tmp = Float64((2.0 ^ 0.5) * (Float64(t_1 * U) ^ 0.5));
	else
		tmp = Float64((2.0 ^ 0.5) * Float64((t_1 ^ 0.5) * (U ^ 0.5)));
	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), $MachinePrecision]}, If[LessEqual[U, -5e-310], N[(N[Power[2.0, 0.5], $MachinePrecision] * N[Power[N[(t$95$1 * U), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], N[(N[Power[2.0, 0.5], $MachinePrecision] * N[(N[Power[t$95$1, 0.5], $MachinePrecision] * N[Power[U, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if U < -4.999999999999985e-310

   1. Initial program 46.3%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/2 N/A

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

      4. lower-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lower-pow.f64 N/A

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

   5. Applied rewrites 55.0%

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

   if -4.999999999999985e-310 < U

   1. Initial program 50.8%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/2 N/A

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

      4. lower-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lower-pow.f64 N/A

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

   5. Applied rewrites 51.4%

      \[\leadsto \color{blue}{{2}^{0.5} \cdot {\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right)}^{0.5}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 62.0%

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

Alternative 5: 57.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\\ \mathbf{if}\;U \leq -5 \cdot 10^{-310}:\\ \;\;\;\;e^{\log 2 \cdot 0.5} \cdot {\left(t\_1 \cdot U\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{2}^{0.5} \cdot \left({t\_1}^{0.5} \cdot {U}^{0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (fma -2.0 (* l (/ l Om)) t) n)))
   (if (<= U -5e-310)
     (* (exp (* (log 2.0) 0.5)) (pow (* t_1 U) 0.5))
     (* (pow 2.0 0.5) (* (pow t_1 0.5) (pow U 0.5))))))

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;
	double tmp;
	if (U <= -5e-310) {
		tmp = exp((log(2.0) * 0.5)) * pow((t_1 * U), 0.5);
	} else {
		tmp = pow(2.0, 0.5) * (pow(t_1, 0.5) * pow(U, 0.5));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) * n)
	tmp = 0.0
	if (U <= -5e-310)
		tmp = Float64(exp(Float64(log(2.0) * 0.5)) * (Float64(t_1 * U) ^ 0.5));
	else
		tmp = Float64((2.0 ^ 0.5) * Float64((t_1 ^ 0.5) * (U ^ 0.5)));
	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), $MachinePrecision]}, If[LessEqual[U, -5e-310], N[(N[Exp[N[(N[Log[2.0], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[Power[N[(t$95$1 * U), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], N[(N[Power[2.0, 0.5], $MachinePrecision] * N[(N[Power[t$95$1, 0.5], $MachinePrecision] * N[Power[U, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if U < -4.999999999999985e-310

   1. Initial program 46.3%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/2 N/A

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

      4. lower-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lower-pow.f64 N/A

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

   5. Applied rewrites 55.0%

      \[\leadsto \color{blue}{{2}^{0.5} \cdot {\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right)}^{0.5}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 54.8

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

   7. Applied rewrites 54.8%

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

   if -4.999999999999985e-310 < U

   1. Initial program 50.8%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/2 N/A

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

      4. lower-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lower-pow.f64 N/A

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

   5. Applied rewrites 51.4%

      \[\leadsto \color{blue}{{2}^{0.5} \cdot {\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right)}^{0.5}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 62.0%

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

Alternative 6: 54.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\\ t_2 := e^{\log 2 \cdot 0.5}\\ \mathbf{if}\;U \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_2 \cdot {\left(t\_1 \cdot U\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left({t\_1}^{0.5} \cdot {U}^{0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (fma -2.0 (* l (/ l Om)) t) n)) (t_2 (exp (* (log 2.0) 0.5))))
   (if (<= U -5e-310)
     (* t_2 (pow (* t_1 U) 0.5))
     (* t_2 (* (pow t_1 0.5) (pow U 0.5))))))

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;
	double t_2 = exp((log(2.0) * 0.5));
	double tmp;
	if (U <= -5e-310) {
		tmp = t_2 * pow((t_1 * U), 0.5);
	} else {
		tmp = t_2 * (pow(t_1, 0.5) * pow(U, 0.5));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) * n)
	t_2 = exp(Float64(log(2.0) * 0.5))
	tmp = 0.0
	if (U <= -5e-310)
		tmp = Float64(t_2 * (Float64(t_1 * U) ^ 0.5));
	else
		tmp = Float64(t_2 * Float64((t_1 ^ 0.5) * (U ^ 0.5)));
	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), $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[Log[2.0], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[U, -5e-310], N[(t$95$2 * N[Power[N[(t$95$1 * U), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(N[Power[t$95$1, 0.5], $MachinePrecision] * N[Power[U, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if U < -4.999999999999985e-310

   1. Initial program 46.3%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/2 N/A

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

      4. lower-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lower-pow.f64 N/A

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

   5. Applied rewrites 55.0%

      \[\leadsto \color{blue}{{2}^{0.5} \cdot {\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right)}^{0.5}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 54.8

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

   7. Applied rewrites 54.8%

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

   if -4.999999999999985e-310 < U

   1. Initial program 50.8%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/2 N/A

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

      4. lower-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lower-pow.f64 N/A

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

   5. Applied rewrites 51.4%

      \[\leadsto \color{blue}{{2}^{0.5} \cdot {\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right)}^{0.5}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 51.2

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

   7. Applied rewrites 51.2%

      \[\leadsto e^{\log 2 \cdot 0.5} \cdot {\color{blue}{\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right)}}^{0.5} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-pow.f64 61.8

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

   9. Applied rewrites 61.8%

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

Alternative 7: 54.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\log 2 \cdot 0.5}\\ \mathbf{if}\;U \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_1 \cdot {\left(t \cdot \mathsf{fma}\left(-2, \frac{U}{Om} \cdot \frac{\left(\ell \cdot \ell\right) \cdot n}{t}, U \cdot n\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left({\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right)}^{0.5} \cdot {U}^{0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (exp (* (log 2.0) 0.5))))
   (if (<= U -5e-310)
     (*
      t_1
      (pow (* t (fma -2.0 (* (/ U Om) (/ (* (* l l) n) t)) (* U n))) 0.5))
     (* t_1 (* (pow (* (fma -2.0 (* l (/ l Om)) t) n) 0.5) (pow U 0.5))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = exp((log(2.0) * 0.5));
	double tmp;
	if (U <= -5e-310) {
		tmp = t_1 * pow((t * fma(-2.0, ((U / Om) * (((l * l) * n) / t)), (U * n))), 0.5);
	} else {
		tmp = t_1 * (pow((fma(-2.0, (l * (l / Om)), t) * n), 0.5) * pow(U, 0.5));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = exp(Float64(log(2.0) * 0.5))
	tmp = 0.0
	if (U <= -5e-310)
		tmp = Float64(t_1 * (Float64(t * fma(-2.0, Float64(Float64(U / Om) * Float64(Float64(Float64(l * l) * n) / t)), Float64(U * n))) ^ 0.5));
	else
		tmp = Float64(t_1 * Float64((Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) * n) ^ 0.5) * (U ^ 0.5)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Exp[N[(N[Log[2.0], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[U, -5e-310], N[(t$95$1 * N[Power[N[(t * N[(-2.0 * N[(N[(U / Om), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * n), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + N[(U * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[Power[N[(N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[U, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if U < -4.999999999999985e-310

   1. Initial program 46.3%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/2 N/A

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

      4. lower-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lower-pow.f64 N/A

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

   5. Applied rewrites 55.0%

      \[\leadsto \color{blue}{{2}^{0.5} \cdot {\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right)}^{0.5}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 54.8

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

   7. Applied rewrites 54.8%

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

   8. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 46.0

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

   10. Applied rewrites 46.0%

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

   if -4.999999999999985e-310 < U

   1. Initial program 50.8%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/2 N/A

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

      4. lower-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lower-pow.f64 N/A

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

   5. Applied rewrites 51.4%

      \[\leadsto \color{blue}{{2}^{0.5} \cdot {\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right)}^{0.5}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 51.2

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

   7. Applied rewrites 51.2%

      \[\leadsto e^{\log 2 \cdot 0.5} \cdot {\color{blue}{\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right)}}^{0.5} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-pow.f64 61.8

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

   9. Applied rewrites 61.8%

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

Alternative 8: 42.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq -8.5 \cdot 10^{-88}:\\ \;\;\;\;t \cdot \left(\left(-1 \cdot {\left(\frac{U \cdot n}{t}\right)}^{0.5}\right) \cdot {2}^{0.5}\right)\\ \mathbf{elif}\;U \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t \cdot \left(U \cdot \left({\left(\frac{n}{U \cdot t}\right)}^{0.5} \cdot {2}^{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log 2 \cdot 0.5} \cdot \left({\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right)}^{0.5} \cdot {U}^{0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U -8.5e-88)
   (* t (* (* -1.0 (pow (/ (* U n) t) 0.5)) (pow 2.0 0.5)))
   (if (<= U -5e-310)
     (* t (* U (* (pow (/ n (* U t)) 0.5) (pow 2.0 0.5))))
     (*
      (exp (* (log 2.0) 0.5))
      (* (pow (* (fma -2.0 (* l (/ l Om)) t) n) 0.5) (pow U 0.5))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -8.5e-88) {
		tmp = t * ((-1.0 * pow(((U * n) / t), 0.5)) * pow(2.0, 0.5));
	} else if (U <= -5e-310) {
		tmp = t * (U * (pow((n / (U * t)), 0.5) * pow(2.0, 0.5)));
	} else {
		tmp = exp((log(2.0) * 0.5)) * (pow((fma(-2.0, (l * (l / Om)), t) * n), 0.5) * pow(U, 0.5));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= -8.5e-88)
		tmp = Float64(t * Float64(Float64(-1.0 * (Float64(Float64(U * n) / t) ^ 0.5)) * (2.0 ^ 0.5)));
	elseif (U <= -5e-310)
		tmp = Float64(t * Float64(U * Float64((Float64(n / Float64(U * t)) ^ 0.5) * (2.0 ^ 0.5))));
	else
		tmp = Float64(exp(Float64(log(2.0) * 0.5)) * Float64((Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) * n) ^ 0.5) * (U ^ 0.5)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, -8.5e-88], N[(t * N[(N[(-1.0 * N[Power[N[(N[(U * n), $MachinePrecision] / t), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[Power[2.0, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[U, -5e-310], N[(t * N[(U * N[(N[Power[N[(n / N[(U * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[2.0, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[2.0], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[U, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if U < -8.4999999999999996e-88

   1. Initial program 58.6%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/2 N/A

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

      4. lower-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lower-pow.f64 N/A

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

   5. Applied rewrites 65.1%

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

   6. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 16.2%

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

   9. Taylor expanded in t around -inf

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

       1. lower-*.f64 N/A

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

       2. pow1/2 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-pow.f64 N/A

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

       6. sqrt-pow2 N/A

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

       7. metadata-eval N/A

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

       8. metadata-eval N/A

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

       9. lower-*.f64 N/A

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

       10. pow1/2 N/A

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

       11. lift-pow.f64 23.7

           \[\leadsto t \cdot \left({\left(\frac{U \cdot n}{t}\right)}^{0.5} \cdot \left(-1 \cdot {2}^{0.5}\right)\right) \]

   11. Applied rewrites 23.7%

       \[\leadsto t \cdot \left({\left(\frac{U \cdot n}{t}\right)}^{0.5} \cdot \left(-1 \cdot \color{blue}{{2}^{0.5}}\right)\right) \]

   if -8.4999999999999996e-88 < U < -4.999999999999985e-310

   1. Initial program 28.6%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/2 N/A

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

      4. lower-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lower-pow.f64 N/A

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

   5. Applied rewrites 40.6%

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

   6. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 3.5%

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

   9. Taylor expanded in U around inf

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

       1. lower-*.f64 N/A

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

       2. lower-fma.f64 N/A

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

   11. Applied rewrites 13.8%

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

   12. Taylor expanded in t around inf

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

       1. pow1/2 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-pow.f64 N/A

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

       5. pow1/2 N/A

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

       6. lift-pow.f64 N/A

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

       7. lift-*.f64 21.4

          \[\leadsto t \cdot \left(U \cdot \left({\left(\frac{n}{U \cdot t}\right)}^{0.5} \cdot {2}^{0.5}\right)\right) \]

   14. Applied rewrites 21.4%

       \[\leadsto t \cdot \left(U \cdot \left({\left(\frac{n}{U \cdot t}\right)}^{0.5} \cdot {2}^{0.5}\right)\right) \]

   if -4.999999999999985e-310 < U

   1. Initial program 50.8%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/2 N/A

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

      4. lower-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lower-pow.f64 N/A

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

   5. Applied rewrites 51.4%

      \[\leadsto \color{blue}{{2}^{0.5} \cdot {\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right)}^{0.5}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 51.2

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

   7. Applied rewrites 51.2%

      \[\leadsto e^{\log 2 \cdot 0.5} \cdot {\color{blue}{\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right)}}^{0.5} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-pow.f64 61.8

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

   9. Applied rewrites 61.8%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -8.5 \cdot 10^{-88}:\\ \;\;\;\;t \cdot \left(\left(-1 \cdot {\left(\frac{U \cdot n}{t}\right)}^{0.5}\right) \cdot {2}^{0.5}\right)\\ \mathbf{elif}\;U \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t \cdot \left(U \cdot \left({\left(\frac{n}{U \cdot t}\right)}^{0.5} \cdot {2}^{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log 2 \cdot 0.5} \cdot \left({\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right)}^{0.5} \cdot {U}^{0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 30.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{n}{U \cdot t}\right)}^{0.5}\\ \mathbf{if}\;n \leq 4.4 \cdot 10^{-269}:\\ \;\;\;\;t \cdot \left(U \cdot \left(\left(-1 \cdot t\_1\right) \cdot {2}^{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(U \cdot \left(t\_1 \cdot {2}^{0.5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (pow (/ n (* U t)) 0.5)))
   (if (<= n 4.4e-269)
     (* t (* U (* (* -1.0 t_1) (pow 2.0 0.5))))
     (* t (* U (* t_1 (pow 2.0 0.5)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = pow((n / (U * t)), 0.5);
	double tmp;
	if (n <= 4.4e-269) {
		tmp = t * (U * ((-1.0 * t_1) * pow(2.0, 0.5)));
	} else {
		tmp = t * (U * (t_1 * pow(2.0, 0.5)));
	}
	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 / (u * t)) ** 0.5d0
    if (n <= 4.4d-269) then
        tmp = t * (u * (((-1.0d0) * t_1) * (2.0d0 ** 0.5d0)))
    else
        tmp = t * (u * (t_1 * (2.0d0 ** 0.5d0)))
    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 = Math.pow((n / (U * t)), 0.5);
	double tmp;
	if (n <= 4.4e-269) {
		tmp = t * (U * ((-1.0 * t_1) * Math.pow(2.0, 0.5)));
	} else {
		tmp = t * (U * (t_1 * Math.pow(2.0, 0.5)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = math.pow((n / (U * t)), 0.5)
	tmp = 0
	if n <= 4.4e-269:
		tmp = t * (U * ((-1.0 * t_1) * math.pow(2.0, 0.5)))
	else:
		tmp = t * (U * (t_1 * math.pow(2.0, 0.5)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n / Float64(U * t)) ^ 0.5
	tmp = 0.0
	if (n <= 4.4e-269)
		tmp = Float64(t * Float64(U * Float64(Float64(-1.0 * t_1) * (2.0 ^ 0.5))));
	else
		tmp = Float64(t * Float64(U * Float64(t_1 * (2.0 ^ 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (n / (U * t)) ^ 0.5;
	tmp = 0.0;
	if (n <= 4.4e-269)
		tmp = t * (U * ((-1.0 * t_1) * (2.0 ^ 0.5)));
	else
		tmp = t * (U * (t_1 * (2.0 ^ 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Power[N[(n / N[(U * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]}, If[LessEqual[n, 4.4e-269], N[(t * N[(U * N[(N[(-1.0 * t$95$1), $MachinePrecision] * N[Power[2.0, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(U * N[(t$95$1 * N[Power[2.0, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < 4.39999999999999968e-269

   1. Initial program 49.0%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/2 N/A

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

      4. lower-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lower-pow.f64 N/A

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

   5. Applied rewrites 52.8%

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

   6. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 11.6%

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

   9. Taylor expanded in U around inf

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

       1. lower-*.f64 N/A

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

       2. lower-fma.f64 N/A

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

   11. Applied rewrites 3.3%

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

   12. Taylor expanded in t around -inf

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

       1. lower-*.f64 N/A

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

       2. pow1/2 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-pow.f64 N/A

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

       6. sqrt-pow2 N/A

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

       7. metadata-eval N/A

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

       8. metadata-eval N/A

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

       9. lower-*.f64 N/A

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

       10. pow1/2 N/A

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

       11. lift-pow.f64 31.7

           \[\leadsto t \cdot \left(U \cdot \left({\left(\frac{n}{U \cdot t}\right)}^{0.5} \cdot \left(-1 \cdot {2}^{0.5}\right)\right)\right) \]

   14. Applied rewrites 31.7%

       \[\leadsto t \cdot \left(U \cdot \left({\left(\frac{n}{U \cdot t}\right)}^{0.5} \cdot \left(-1 \cdot {2}^{\color{blue}{0.5}}\right)\right)\right) \]

   if 4.39999999999999968e-269 < n

   1. Initial program 47.9%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/2 N/A

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

      4. lower-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. lower-pow.f64 N/A

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 10.3%

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

   9. Taylor expanded in U around inf

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

       1. lower-*.f64 N/A

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

       2. lower-fma.f64 N/A

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

   11. Applied rewrites 23.1%

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

   12. Taylor expanded in t around inf

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

       1. pow1/2 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-pow.f64 N/A

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

       5. pow1/2 N/A

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

       6. lift-pow.f64 N/A

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

       7. lift-*.f64 32.5

          \[\leadsto t \cdot \left(U \cdot \left({\left(\frac{n}{U \cdot t}\right)}^{0.5} \cdot {2}^{0.5}\right)\right) \]

   14. Applied rewrites 32.5%

       \[\leadsto t \cdot \left(U \cdot \left({\left(\frac{n}{U \cdot t}\right)}^{0.5} \cdot {2}^{0.5}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 4.4 \cdot 10^{-269}:\\ \;\;\;\;t \cdot \left(U \cdot \left(\left(-1 \cdot {\left(\frac{n}{U \cdot t}\right)}^{0.5}\right) \cdot {2}^{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(U \cdot \left({\left(\frac{n}{U \cdot t}\right)}^{0.5} \cdot {2}^{0.5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 17.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ t \cdot \left(U \cdot \left(\left(-1 \cdot {\left(\frac{n}{U \cdot t}\right)}^{0.5}\right) \cdot {2}^{0.5}\right)\right) \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (* t (* U (* (* -1.0 (pow (/ n (* U t)) 0.5)) (pow 2.0 0.5)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return t * (U * ((-1.0 * pow((n / (U * t)), 0.5)) * pow(2.0, 0.5)));
}

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 = t * (u * (((-1.0d0) * ((n / (u * t)) ** 0.5d0)) * (2.0d0 ** 0.5d0)))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return t * (U * ((-1.0 * Math.pow((n / (U * t)), 0.5)) * Math.pow(2.0, 0.5)));
}

Python

def code(n, U, t, l, Om, U_42_):
	return t * (U * ((-1.0 * math.pow((n / (U * t)), 0.5)) * math.pow(2.0, 0.5)))

Julia

function code(n, U, t, l, Om, U_42_)
	return Float64(t * Float64(U * Float64(Float64(-1.0 * (Float64(n / Float64(U * t)) ^ 0.5)) * (2.0 ^ 0.5))))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = t * (U * ((-1.0 * ((n / (U * t)) ^ 0.5)) * (2.0 ^ 0.5)));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[(t * N[(U * N[(N[(-1.0 * N[Power[N[(n / N[(U * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[Power[2.0, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.5%

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

3. Taylor expanded in n around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow1/2 N/A

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

   4. lower-pow.f64 N/A

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

   5. pow1/2 N/A

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

   6. lower-pow.f64 N/A

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

5. Applied rewrites 53.2%

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

6. Taylor expanded in t around inf

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

   1. lower-*.f64 N/A

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

   2. lower-fma.f64 N/A

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

8. Applied rewrites 11.0%

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

9. Taylor expanded in U around inf

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

    1. lower-*.f64 N/A

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

    2. lower-fma.f64 N/A

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

11. Applied rewrites 12.7%

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

12. Taylor expanded in t around -inf

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

    1. lower-*.f64 N/A

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

    2. pow1/2 N/A

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

    3. lift-/.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-pow.f64 N/A

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

    6. sqrt-pow2 N/A

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

    7. metadata-eval N/A

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

    8. metadata-eval N/A

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

    9. lower-*.f64 N/A

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

    10. pow1/2 N/A

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

    11. lift-pow.f64 17.7

        \[\leadsto t \cdot \left(U \cdot \left({\left(\frac{n}{U \cdot t}\right)}^{0.5} \cdot \left(-1 \cdot {2}^{0.5}\right)\right)\right) \]

14. Applied rewrites 17.7%

    \[\leadsto t \cdot \left(U \cdot \left({\left(\frac{n}{U \cdot t}\right)}^{0.5} \cdot \left(-1 \cdot {2}^{\color{blue}{0.5}}\right)\right)\right) \]

15. Final simplification 17.7%

    \[\leadsto t \cdot \left(U \cdot \left(\left(-1 \cdot {\left(\frac{n}{U \cdot t}\right)}^{0.5}\right) \cdot {2}^{0.5}\right)\right) \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(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*))))))