Toniolo and Linder, Equation (13)

Percentage Accurate: 50.3% → 62.5%

Time: 9.2s

Alternatives: 15

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.3% 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: 62.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-156}:\\ \;\;\;\;-1 \cdot \left(n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}}\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(-2, l\_m \cdot \frac{l\_m}{Om}, t\right) - n \cdot \left(\left(\frac{l\_m}{Om} \cdot \frac{l\_m}{Om}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right) \cdot n\right) \cdot U\right) \cdot -2} \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))))
   (if (<= t_1 2e-156)
     (* -1.0 (* n (sqrt (* 2.0 (/ (* U t) n)))))
     (if (<= t_1 INFINITY)
       (sqrt
        (*
         (-
          (fma -2.0 (* l_m (/ l_m Om)) t)
          (* n (* (* (/ l_m Om) (/ l_m Om)) (- U U*))))
         (* (+ n n) U)))
       (*
        (sqrt (* (* (* (fma n (/ (- U U*) (* Om Om)) (/ 2.0 Om)) n) U) -2.0))
        l_m)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_1 <= 2e-156) {
		tmp = -1.0 * (n * sqrt((2.0 * ((U * t) / n))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt(((fma(-2.0, (l_m * (l_m / Om)), t) - (n * (((l_m / Om) * (l_m / Om)) * (U - U_42_)))) * ((n + n) * U)));
	} else {
		tmp = sqrt((((fma(n, ((U - U_42_) / (Om * Om)), (2.0 / Om)) * n) * U) * -2.0)) * l_m;
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_1 <= 2e-156)
		tmp = Float64(-1.0 * Float64(n * sqrt(Float64(2.0 * Float64(Float64(U * t) / n)))));
	elseif (t_1 <= Inf)
		tmp = sqrt(Float64(Float64(fma(-2.0, Float64(l_m * Float64(l_m / Om)), t) - Float64(n * Float64(Float64(Float64(l_m / Om) * Float64(l_m / Om)) * Float64(U - U_42_)))) * Float64(Float64(n + n) * U)));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(fma(n, Float64(Float64(U - U_42_) / Float64(Om * Om)), Float64(2.0 / Om)) * n) * U) * -2.0)) * l_m);
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 2e-156], N[(-1.0 * N[(n * N[Sqrt[N[(2.0 * N[(N[(U * t), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[Sqrt[N[(N[(N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] - N[(n * N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(n * N[(N[(U - U$42$), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] * l$95$m), $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*))))) < 2.00000000000000008e-156

   1. Initial program 50.3%

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

   3. Applied rewrites 31.2%

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

   4. Taylor expanded in l around 0

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 36.1

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

   6. Applied rewrites 36.1%

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

   7. Taylor expanded in n around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 18.3

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

   9. Applied rewrites 18.3%

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

   if 2.00000000000000008e-156 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 50.3%

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

   3. Applied rewrites 53.1%

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

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

   1. Initial program 50.3%

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

   2. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 26.4%

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

Alternative 2: 59.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := \sqrt{t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-144}:\\ \;\;\;\;-1 \cdot \left(n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}}\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(\left(-\frac{\mathsf{fma}\left(l\_m \cdot l\_m, \frac{\left(U - U*\right) \cdot n}{Om}, \left(l\_m \cdot l\_m\right) \cdot 2\right)}{Om}\right) + t\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{\left(-\mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{Om \cdot Om}\right)\right) \cdot \left(2 \cdot \left(U \cdot n\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (sqrt
          (*
           t_1
           (-
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))))
   (if (<= t_2 5e-144)
     (* -1.0 (* n (sqrt (* 2.0 (/ (* U t) n)))))
     (if (<= t_2 2e+153)
       (sqrt
        (*
         t_1
         (+
          (-
           (/ (fma (* l_m l_m) (/ (* (- U U*) n) Om) (* (* l_m l_m) 2.0)) Om))
          t)))
       (*
        l_m
        (sqrt
         (*
          (- (fma 2.0 (/ 1.0 Om) (/ (* n (- U U*)) (* Om Om))))
          (* 2.0 (* U n)))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = sqrt((t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 5e-144) {
		tmp = -1.0 * (n * sqrt((2.0 * ((U * t) / n))));
	} else if (t_2 <= 2e+153) {
		tmp = sqrt((t_1 * (-(fma((l_m * l_m), (((U - U_42_) * n) / Om), ((l_m * l_m) * 2.0)) / Om) + t)));
	} else {
		tmp = l_m * sqrt((-fma(2.0, (1.0 / Om), ((n * (U - U_42_)) / (Om * Om))) * (2.0 * (U * n))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_2 <= 5e-144)
		tmp = Float64(-1.0 * Float64(n * sqrt(Float64(2.0 * Float64(Float64(U * t) / n)))));
	elseif (t_2 <= 2e+153)
		tmp = sqrt(Float64(t_1 * Float64(Float64(-Float64(fma(Float64(l_m * l_m), Float64(Float64(Float64(U - U_42_) * n) / Om), Float64(Float64(l_m * l_m) * 2.0)) / Om)) + t)));
	else
		tmp = Float64(l_m * sqrt(Float64(Float64(-fma(2.0, Float64(1.0 / Om), Float64(Float64(n * Float64(U - U_42_)) / Float64(Om * Om)))) * Float64(2.0 * Float64(U * n)))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 5e-144], N[(-1.0 * N[(n * N[Sqrt[N[(2.0 * N[(N[(U * t), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+153], N[Sqrt[N[(t$95$1 * N[((-N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(N[(U - U$42$), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]) + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[((-N[(2.0 * N[(1.0 / Om), $MachinePrecision] + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) * N[(2.0 * N[(U * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 4.9999999999999998e-144

   1. Initial program 50.3%

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

   3. Applied rewrites 31.2%

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

   4. Taylor expanded in l around 0

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 36.1

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

   6. Applied rewrites 36.1%

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

   7. Taylor expanded in n around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 18.3

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

   9. Applied rewrites 18.3%

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

   if 4.9999999999999998e-144 < (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*))))) < 2e153

   1. Initial program 50.3%

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

   2. Taylor expanded in Om around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   4. Applied rewrites 45.8%

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

   if 2e153 < (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 50.3%

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

   3. Applied rewrites 31.2%

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

   4. Taylor expanded in l around inf

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-*.f64 N/A

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

   6. Applied rewrites 28.9%

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

Alternative 3: 59.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := \sqrt{t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-144}:\\ \;\;\;\;-1 \cdot \left(n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}}\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(-2, l\_m \cdot \frac{l\_m}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \sqrt{\left(-\mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{Om \cdot Om}\right)\right) \cdot \left(2 \cdot \left(U \cdot n\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (sqrt
          (*
           t_1
           (-
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))))
   (if (<= t_2 5e-144)
     (* -1.0 (* n (sqrt (* 2.0 (/ (* U t) n)))))
     (if (<= t_2 2e+153)
       (sqrt (* t_1 (fma -2.0 (* l_m (/ l_m Om)) t)))
       (*
        l_m
        (sqrt
         (*
          (- (fma 2.0 (/ 1.0 Om) (/ (* n (- U U*)) (* Om Om))))
          (* 2.0 (* U n)))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = sqrt((t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 5e-144) {
		tmp = -1.0 * (n * sqrt((2.0 * ((U * t) / n))));
	} else if (t_2 <= 2e+153) {
		tmp = sqrt((t_1 * fma(-2.0, (l_m * (l_m / Om)), t)));
	} else {
		tmp = l_m * sqrt((-fma(2.0, (1.0 / Om), ((n * (U - U_42_)) / (Om * Om))) * (2.0 * (U * n))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_2 <= 5e-144)
		tmp = Float64(-1.0 * Float64(n * sqrt(Float64(2.0 * Float64(Float64(U * t) / n)))));
	elseif (t_2 <= 2e+153)
		tmp = sqrt(Float64(t_1 * fma(-2.0, Float64(l_m * Float64(l_m / Om)), t)));
	else
		tmp = Float64(l_m * sqrt(Float64(Float64(-fma(2.0, Float64(1.0 / Om), Float64(Float64(n * Float64(U - U_42_)) / Float64(Om * Om)))) * Float64(2.0 * Float64(U * n)))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 5e-144], N[(-1.0 * N[(n * N[Sqrt[N[(2.0 * N[(N[(U * t), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+153], N[Sqrt[N[(t$95$1 * N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m * N[Sqrt[N[((-N[(2.0 * N[(1.0 / Om), $MachinePrecision] + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) * N[(2.0 * N[(U * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 4.9999999999999998e-144

   1. Initial program 50.3%

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

   3. Applied rewrites 31.2%

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

   4. Taylor expanded in l around 0

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 36.1

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

   6. Applied rewrites 36.1%

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

   7. Taylor expanded in n around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 18.3

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

   9. Applied rewrites 18.3%

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

   if 4.9999999999999998e-144 < (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*))))) < 2e153

   1. Initial program 50.3%

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

   2. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 47.5

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

   4. Applied rewrites 47.5%

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

   if 2e153 < (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 50.3%

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

   3. Applied rewrites 31.2%

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

   4. Taylor expanded in l around inf

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-*.f64 N/A

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

   6. Applied rewrites 28.9%

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

Alternative 4: 56.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := \sqrt{t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-144}:\\ \;\;\;\;-1 \cdot \left(n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}}\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(-2, l\_m \cdot \frac{l\_m}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right) \cdot n\right) \cdot U\right) \cdot -2} \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (sqrt
          (*
           t_1
           (-
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))))
   (if (<= t_2 5e-144)
     (* -1.0 (* n (sqrt (* 2.0 (/ (* U t) n)))))
     (if (<= t_2 INFINITY)
       (sqrt (* t_1 (fma -2.0 (* l_m (/ l_m Om)) t)))
       (*
        (sqrt (* (* (* (fma n (/ (- U U*) (* Om Om)) (/ 2.0 Om)) n) U) -2.0))
        l_m)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = sqrt((t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 5e-144) {
		tmp = -1.0 * (n * sqrt((2.0 * ((U * t) / n))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * fma(-2.0, (l_m * (l_m / Om)), t)));
	} else {
		tmp = sqrt((((fma(n, ((U - U_42_) / (Om * Om)), (2.0 / Om)) * n) * U) * -2.0)) * l_m;
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_2 <= 5e-144)
		tmp = Float64(-1.0 * Float64(n * sqrt(Float64(2.0 * Float64(Float64(U * t) / n)))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(t_1 * fma(-2.0, Float64(l_m * Float64(l_m / Om)), t)));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(fma(n, Float64(Float64(U - U_42_) / Float64(Om * Om)), Float64(2.0 / Om)) * n) * U) * -2.0)) * l_m);
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 5e-144], N[(-1.0 * N[(n * N[Sqrt[N[(2.0 * N[(N[(U * t), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(t$95$1 * N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(n * N[(N[(U - U$42$), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] * l$95$m), $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*))))) < 4.9999999999999998e-144

   1. Initial program 50.3%

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

   3. Applied rewrites 31.2%

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

   4. Taylor expanded in l around 0

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 36.1

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

   6. Applied rewrites 36.1%

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

   7. Taylor expanded in n around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 18.3

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

   9. Applied rewrites 18.3%

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

   if 4.9999999999999998e-144 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 50.3%

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

   2. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 47.5

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

   4. Applied rewrites 47.5%

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

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

   1. Initial program 50.3%

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

   2. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 26.4%

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

Alternative 5: 54.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-312}:\\ \;\;\;\;-1 \cdot \left(n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}}\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(-2, l\_m \cdot \frac{l\_m}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{U \cdot U*}{Om} \cdot \frac{\left(l\_m \cdot n\right) \cdot \left(l\_m \cdot n\right)}{Om}\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_2 5e-312)
     (* -1.0 (* n (sqrt (* 2.0 (/ (* U t) n)))))
     (if (<= t_2 INFINITY)
       (sqrt (* t_1 (fma -2.0 (* l_m (/ l_m Om)) t)))
       (sqrt (* (* (/ (* U U*) Om) (/ (* (* l_m n) (* l_m n)) Om)) 2.0))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 5e-312) {
		tmp = -1.0 * (n * sqrt((2.0 * ((U * t) / n))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * fma(-2.0, (l_m * (l_m / Om)), t)));
	} else {
		tmp = sqrt(((((U * U_42_) / Om) * (((l_m * n) * (l_m * n)) / Om)) * 2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_2 <= 5e-312)
		tmp = Float64(-1.0 * Float64(n * sqrt(Float64(2.0 * Float64(Float64(U * t) / n)))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(t_1 * fma(-2.0, Float64(l_m * Float64(l_m / Om)), t)));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(U * U_42_) / Om) * Float64(Float64(Float64(l_m * n) * Float64(l_m * n)) / Om)) * 2.0));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-312], N[(-1.0 * N[(n * N[Sqrt[N[(2.0 * N[(N[(U * t), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(t$95$1 * N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(U * U$42$), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(N[(l$95$m * n), $MachinePrecision] * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.0000000000022e-312

   1. Initial program 50.3%

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

   3. Applied rewrites 31.2%

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

   4. Taylor expanded in l around 0

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 36.1

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

   6. Applied rewrites 36.1%

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

   7. Taylor expanded in n around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 18.3

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

   9. Applied rewrites 18.3%

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

   if 5.0000000000022e-312 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 50.3%

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

   2. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 47.5

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

   4. Applied rewrites 47.5%

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

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

   1. Initial program 50.3%

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

   2. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow-prod-down N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 17.5

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

   4. Applied rewrites 17.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. pow-prod-down N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow-prod-down N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 19.9

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

   6. Applied rewrites 19.9%

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

Alternative 6: 53.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-312}:\\ \;\;\;\;-1 \cdot \left(n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}}\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(-2, l\_m \cdot \frac{l\_m}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(U \cdot U*\right) \cdot \left(\left(l\_m \cdot n\right) \cdot \left(l\_m \cdot n\right)\right)}{Om \cdot Om} \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_2 5e-312)
     (* -1.0 (* n (sqrt (* 2.0 (/ (* U t) n)))))
     (if (<= t_2 INFINITY)
       (sqrt (* t_1 (fma -2.0 (* l_m (/ l_m Om)) t)))
       (sqrt (* (/ (* (* U U*) (* (* l_m n) (* l_m n))) (* Om Om)) 2.0))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 5e-312) {
		tmp = -1.0 * (n * sqrt((2.0 * ((U * t) / n))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * fma(-2.0, (l_m * (l_m / Om)), t)));
	} else {
		tmp = sqrt(((((U * U_42_) * ((l_m * n) * (l_m * n))) / (Om * Om)) * 2.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_2 <= 5e-312)
		tmp = Float64(-1.0 * Float64(n * sqrt(Float64(2.0 * Float64(Float64(U * t) / n)))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(t_1 * fma(-2.0, Float64(l_m * Float64(l_m / Om)), t)));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(U * U_42_) * Float64(Float64(l_m * n) * Float64(l_m * n))) / Float64(Om * Om)) * 2.0));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-312], N[(-1.0 * N[(n * N[Sqrt[N[(2.0 * N[(N[(U * t), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(t$95$1 * N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(U * U$42$), $MachinePrecision] * N[(N[(l$95$m * n), $MachinePrecision] * N[(l$95$m * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.0000000000022e-312

   1. Initial program 50.3%

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

   3. Applied rewrites 31.2%

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

   4. Taylor expanded in l around 0

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 36.1

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

   6. Applied rewrites 36.1%

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

   7. Taylor expanded in n around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 18.3

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

   9. Applied rewrites 18.3%

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

   if 5.0000000000022e-312 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 50.3%

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

   2. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 47.5

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

   4. Applied rewrites 47.5%

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

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

   1. Initial program 50.3%

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

   2. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow-prod-down N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 17.5

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

   4. Applied rewrites 17.5%

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

Alternative 7: 53.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-312}:\\ \;\;\;\;-1 \cdot \left(n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}}\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(-2, l\_m \cdot \frac{l\_m}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(U \cdot \frac{U* \cdot \left(\left(l\_m \cdot l\_m\right) \cdot n\right)}{Om \cdot Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_2 5e-312)
     (* -1.0 (* n (sqrt (* 2.0 (/ (* U t) n)))))
     (if (<= t_2 INFINITY)
       (sqrt (* t_1 (fma -2.0 (* l_m (/ l_m Om)) t)))
       (sqrt (* (+ n n) (* U (/ (* U* (* (* l_m l_m) n)) (* Om Om)))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 5e-312) {
		tmp = -1.0 * (n * sqrt((2.0 * ((U * t) / n))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * fma(-2.0, (l_m * (l_m / Om)), t)));
	} else {
		tmp = sqrt(((n + n) * (U * ((U_42_ * ((l_m * l_m) * n)) / (Om * Om)))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_2 <= 5e-312)
		tmp = Float64(-1.0 * Float64(n * sqrt(Float64(2.0 * Float64(Float64(U * t) / n)))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(t_1 * fma(-2.0, Float64(l_m * Float64(l_m / Om)), t)));
	else
		tmp = sqrt(Float64(Float64(n + n) * Float64(U * Float64(Float64(U_42_ * Float64(Float64(l_m * l_m) * n)) / Float64(Om * Om)))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-312], N[(-1.0 * N[(n * N[Sqrt[N[(2.0 * N[(N[(U * t), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(t$95$1 * N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(U * N[(N[(U$42$ * N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.0000000000022e-312

   1. Initial program 50.3%

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

   3. Applied rewrites 31.2%

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

   4. Taylor expanded in l around 0

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 36.1

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

   6. Applied rewrites 36.1%

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

   7. Taylor expanded in n around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 18.3

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

   9. Applied rewrites 18.3%

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

   if 5.0000000000022e-312 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 50.3%

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

   2. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 47.5

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

   4. Applied rewrites 47.5%

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

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

   1. Initial program 50.3%

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

   3. Applied rewrites 53.1%

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

   4. Taylor expanded in U* around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 16.5

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

   6. Applied rewrites 16.5%

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

Alternative 8: 53.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-312}:\\ \;\;\;\;-1 \cdot \left(n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}}\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(-2, l\_m \cdot \frac{l\_m}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{U* \cdot \left(\left(l\_m \cdot l\_m\right) \cdot n\right)}{Om \cdot Om} \cdot \left(\left(n + n\right) \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_2 5e-312)
     (* -1.0 (* n (sqrt (* 2.0 (/ (* U t) n)))))
     (if (<= t_2 INFINITY)
       (sqrt (* t_1 (fma -2.0 (* l_m (/ l_m Om)) t)))
       (sqrt (* (/ (* U* (* (* l_m l_m) n)) (* Om Om)) (* (+ n n) U)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 5e-312) {
		tmp = -1.0 * (n * sqrt((2.0 * ((U * t) / n))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * fma(-2.0, (l_m * (l_m / Om)), t)));
	} else {
		tmp = sqrt((((U_42_ * ((l_m * l_m) * n)) / (Om * Om)) * ((n + n) * U)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_2 <= 5e-312)
		tmp = Float64(-1.0 * Float64(n * sqrt(Float64(2.0 * Float64(Float64(U * t) / n)))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(t_1 * fma(-2.0, Float64(l_m * Float64(l_m / Om)), t)));
	else
		tmp = sqrt(Float64(Float64(Float64(U_42_ * Float64(Float64(l_m * l_m) * n)) / Float64(Om * Om)) * Float64(Float64(n + n) * U)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-312], N[(-1.0 * N[(n * N[Sqrt[N[(2.0 * N[(N[(U * t), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(t$95$1 * N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(U$42$ * N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.0000000000022e-312

   1. Initial program 50.3%

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

   3. Applied rewrites 31.2%

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

   4. Taylor expanded in l around 0

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 36.1

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

   6. Applied rewrites 36.1%

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

   7. Taylor expanded in n around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 18.3

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

   9. Applied rewrites 18.3%

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

   if 5.0000000000022e-312 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 50.3%

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

   2. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 47.5

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

   4. Applied rewrites 47.5%

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

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

   1. Initial program 50.3%

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

   3. Applied rewrites 53.1%

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

   4. Taylor expanded in U* around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 16.2

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

   6. Applied rewrites 16.2%

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

Alternative 9: 51.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-312}:\\ \;\;\;\;-1 \cdot \left(n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}}\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(-2, l\_m \cdot \frac{l\_m}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m \cdot \sqrt{-n \cdot \left(U - U*\right)}}{Om} \cdot \sqrt{\left(n + n\right) \cdot U}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_2 5e-312)
     (* -1.0 (* n (sqrt (* 2.0 (/ (* U t) n)))))
     (if (<= t_2 INFINITY)
       (sqrt (* t_1 (fma -2.0 (* l_m (/ l_m Om)) t)))
       (* (/ (* l_m (sqrt (- (* n (- U U*))))) Om) (sqrt (* (+ n n) U)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 5e-312) {
		tmp = -1.0 * (n * sqrt((2.0 * ((U * t) / n))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * fma(-2.0, (l_m * (l_m / Om)), t)));
	} else {
		tmp = ((l_m * sqrt(-(n * (U - U_42_)))) / Om) * sqrt(((n + n) * U));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_2 <= 5e-312)
		tmp = Float64(-1.0 * Float64(n * sqrt(Float64(2.0 * Float64(Float64(U * t) / n)))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(t_1 * fma(-2.0, Float64(l_m * Float64(l_m / Om)), t)));
	else
		tmp = Float64(Float64(Float64(l_m * sqrt(Float64(-Float64(n * Float64(U - U_42_))))) / Om) * sqrt(Float64(Float64(n + n) * U)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-312], N[(-1.0 * N[(n * N[Sqrt[N[(2.0 * N[(N[(U * t), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(t$95$1 * N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[(l$95$m * N[Sqrt[(-N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[Sqrt[N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.0000000000022e-312

   1. Initial program 50.3%

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

   3. Applied rewrites 31.2%

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

   4. Taylor expanded in l around 0

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 36.1

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

   6. Applied rewrites 36.1%

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

   7. Taylor expanded in n around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 18.3

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

   9. Applied rewrites 18.3%

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

   if 5.0000000000022e-312 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 50.3%

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

   2. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 47.5

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

   4. Applied rewrites 47.5%

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

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

   1. Initial program 50.3%

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

   3. Applied rewrites 31.2%

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

   4. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 5.8

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

   6. Applied rewrites 5.8%

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

   7. Taylor expanded in l around 0

      \[\leadsto \frac{\ell \cdot \sqrt{\mathsf{neg}\left(n \cdot \left(U - U*\right)\right)}}{Om} \cdot \sqrt{\left(n + n\right) \cdot U} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 6.3

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

   9. Applied rewrites 6.3%

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

Alternative 10: 51.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-312}:\\ \;\;\;\;-1 \cdot \left(n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}}\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(-2, l\_m \cdot \frac{l\_m}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{U* \cdot \left(\left(l\_m \cdot l\_m\right) \cdot n\right)}}{Om} \cdot \sqrt{\left(n + n\right) \cdot U}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))))
   (if (<= t_2 5e-312)
     (* -1.0 (* n (sqrt (* 2.0 (/ (* U t) n)))))
     (if (<= t_2 INFINITY)
       (sqrt (* t_1 (fma -2.0 (* l_m (/ l_m Om)) t)))
       (* (/ (sqrt (* U* (* (* l_m l_m) n))) Om) (sqrt (* (+ n n) U)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_2 <= 5e-312) {
		tmp = -1.0 * (n * sqrt((2.0 * ((U * t) / n))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * fma(-2.0, (l_m * (l_m / Om)), t)));
	} else {
		tmp = (sqrt((U_42_ * ((l_m * l_m) * n))) / Om) * sqrt(((n + n) * U));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_2 <= 5e-312)
		tmp = Float64(-1.0 * Float64(n * sqrt(Float64(2.0 * Float64(Float64(U * t) / n)))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(t_1 * fma(-2.0, Float64(l_m * Float64(l_m / Om)), t)));
	else
		tmp = Float64(Float64(sqrt(Float64(U_42_ * Float64(Float64(l_m * l_m) * n))) / Om) * sqrt(Float64(Float64(n + n) * U)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-312], N[(-1.0 * N[(n * N[Sqrt[N[(2.0 * N[(N[(U * t), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(t$95$1 * N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[N[(U$42$ * N[(N[(l$95$m * l$95$m), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Om), $MachinePrecision] * N[Sqrt[N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 5.0000000000022e-312

   1. Initial program 50.3%

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

   3. Applied rewrites 31.2%

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

   4. Taylor expanded in l around 0

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 36.1

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

   6. Applied rewrites 36.1%

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

   7. Taylor expanded in n around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 18.3

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

   9. Applied rewrites 18.3%

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

   if 5.0000000000022e-312 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 50.3%

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

   2. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 47.5

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

   4. Applied rewrites 47.5%

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

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

   1. Initial program 50.3%

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

   3. Applied rewrites 31.2%

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

   4. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 5.8

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

   6. Applied rewrites 5.8%

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

   7. Taylor expanded in U around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 6.1

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

   9. Applied rewrites 6.1%

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

Alternative 11: 49.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ \mathbf{if}\;\sqrt{t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \leq 2 \cdot 10^{-156}:\\ \;\;\;\;-1 \cdot \left(n \cdot \sqrt{2 \cdot \frac{U \cdot t}{n}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(-2, l\_m \cdot \frac{l\_m}{Om}, t\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U)))
   (if (<=
        (sqrt
         (*
          t_1
          (-
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U U*)))))
        2e-156)
     (* -1.0 (* n (sqrt (* 2.0 (/ (* U t) n)))))
     (sqrt (* t_1 (fma -2.0 (* l_m (/ l_m Om)) t))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double tmp;
	if (sqrt((t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_))))) <= 2e-156) {
		tmp = -1.0 * (n * sqrt((2.0 * ((U * t) / n))));
	} else {
		tmp = sqrt((t_1 * fma(-2.0, (l_m * (l_m / Om)), t)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	tmp = 0.0
	if (sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_))))) <= 2e-156)
		tmp = Float64(-1.0 * Float64(n * sqrt(Float64(2.0 * Float64(Float64(U * t) / n)))));
	else
		tmp = sqrt(Float64(t_1 * fma(-2.0, Float64(l_m * Float64(l_m / Om)), t)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2e-156], N[(-1.0 * N[(n * N[Sqrt[N[(2.0 * N[(N[(U * t), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(t$95$1 * N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2.00000000000000008e-156

   1. Initial program 50.3%

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

   3. Applied rewrites 31.2%

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

   4. Taylor expanded in l around 0

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 36.1

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

   6. Applied rewrites 36.1%

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

   7. Taylor expanded in n around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 18.3

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

   9. Applied rewrites 18.3%

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

   if 2.00000000000000008e-156 < (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 50.3%

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

   2. Taylor expanded in n around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 47.5

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

   4. Applied rewrites 47.5%

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

Alternative 12: 49.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{+207}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, l\_m \cdot \frac{l\_m}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \frac{1}{\sqrt{t}}\right) \cdot \sqrt{\left(n + n\right) \cdot U}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t 9.5e+207)
   (sqrt (* (* (* (fma -2.0 (* l_m (/ l_m Om)) t) n) U) 2.0))
   (* (* t (/ 1.0 (sqrt t))) (sqrt (* (+ n n) U)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= 9.5e+207) {
		tmp = sqrt((((fma(-2.0, (l_m * (l_m / Om)), t) * n) * U) * 2.0));
	} else {
		tmp = (t * (1.0 / sqrt(t))) * sqrt(((n + n) * U));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= 9.5e+207)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(l_m * Float64(l_m / Om)), t) * n) * U) * 2.0));
	else
		tmp = Float64(Float64(t * Float64(1.0 / sqrt(t))) * sqrt(Float64(Float64(n + n) * U)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[t, 9.5e+207], N[Sqrt[N[(N[(N[(N[(-2.0 * N[(l$95$m * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[(t * N[(1.0 / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.5000000000000005e207

   1. Initial program 50.3%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 48.1

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

   4. Applied rewrites 48.1%

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

   if 9.5000000000000005e207 < t

   1. Initial program 50.3%

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

   3. Applied rewrites 31.2%

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

   4. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 21.1

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

   6. Applied rewrites 21.1%

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

Alternative 13: 39.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq -1.72 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(U \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t} \cdot \sqrt{\left(n + n\right) \cdot U}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= t -1.72e-307)
   (sqrt (* t (* 2.0 (* U n))))
   (* (sqrt t) (sqrt (* (+ n n) U)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= -1.72e-307) {
		tmp = sqrt((t * (2.0 * (U * n))));
	} else {
		tmp = sqrt(t) * sqrt(((n + n) * U));
	}
	return tmp;
}

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(n, u, t, l_m, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= (-1.72d-307)) then
        tmp = sqrt((t * (2.0d0 * (u * n))))
    else
        tmp = sqrt(t) * sqrt(((n + n) * u))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (t <= -1.72e-307) {
		tmp = Math.sqrt((t * (2.0 * (U * n))));
	} else {
		tmp = Math.sqrt(t) * Math.sqrt(((n + n) * U));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if t <= -1.72e-307:
		tmp = math.sqrt((t * (2.0 * (U * n))))
	else:
		tmp = math.sqrt(t) * math.sqrt(((n + n) * U))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (t <= -1.72e-307)
		tmp = sqrt(Float64(t * Float64(2.0 * Float64(U * n))));
	else
		tmp = Float64(sqrt(t) * sqrt(Float64(Float64(n + n) * U)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if (t <= -1.72e-307)
		tmp = sqrt((t * (2.0 * (U * n))));
	else
		tmp = sqrt(t) * sqrt(((n + n) * U));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[t, -1.72e-307], N[Sqrt[N[(t * N[(2.0 * N[(U * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[t], $MachinePrecision] * N[Sqrt[N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.72000000000000008e-307

   1. Initial program 50.3%

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

   3. Applied rewrites 31.2%

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

   4. Taylor expanded in l around 0

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 36.1

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

   6. Applied rewrites 36.1%

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

   if -1.72000000000000008e-307 < t

   1. Initial program 50.3%

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

   3. Applied rewrites 31.2%

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

   4. Taylor expanded in l around 0

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

      1. lower-sqrt.f64 21.1

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

   6. Applied rewrites 21.1%

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

Alternative 14: 38.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) - \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \leq 2 \cdot 10^{-322}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(2 \cdot \left(U \cdot n\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<=
      (*
       (* (* 2.0 n) U)
       (-
        (- t (* 2.0 (/ (* l_m l_m) Om)))
        (* (* n (pow (/ l_m Om) 2.0)) (- U U*))))
      2e-322)
   (sqrt (* (+ n n) (* U t)))
   (sqrt (* t (* 2.0 (* U n))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * pow((l_m / Om), 2.0)) * (U - U_42_)))) <= 2e-322) {
		tmp = sqrt(((n + n) * (U * t)));
	} else {
		tmp = sqrt((t * (2.0 * (U * n))));
	}
	return tmp;
}

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(n, u, t, l_m, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l_m * l_m) / om))) - ((n * ((l_m / om) ** 2.0d0)) * (u - u_42)))) <= 2d-322) then
        tmp = sqrt(((n + n) * (u * t)))
    else
        tmp = sqrt((t * (2.0d0 * (u * n))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if ((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * Math.pow((l_m / Om), 2.0)) * (U - U_42_)))) <= 2e-322) {
		tmp = Math.sqrt(((n + n) * (U * t)));
	} else {
		tmp = Math.sqrt((t * (2.0 * (U * n))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	tmp = 0
	if (((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * math.pow((l_m / Om), 2.0)) * (U - U_42_)))) <= 2e-322:
		tmp = math.sqrt(((n + n) * (U * t)))
	else:
		tmp = math.sqrt((t * (2.0 * (U * n))))
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) - Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U - U_42_)))) <= 2e-322)
		tmp = sqrt(Float64(Float64(n + n) * Float64(U * t)));
	else
		tmp = sqrt(Float64(t * Float64(2.0 * Float64(U * n))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0;
	if ((((2.0 * n) * U) * ((t - (2.0 * ((l_m * l_m) / Om))) - ((n * ((l_m / Om) ^ 2.0)) * (U - U_42_)))) <= 2e-322)
		tmp = sqrt(((n + n) * (U * t)));
	else
		tmp = sqrt((t * (2.0 * (U * n))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-322], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t * N[(2.0 * N[(U * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1.97626e-322

   1. Initial program 50.3%

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

   3. Applied rewrites 53.1%

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

   4. Taylor expanded in t around inf

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

      1. lower-*.f64 35.4

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

   6. Applied rewrites 35.4%

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

   if 1.97626e-322 < (*.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 50.3%

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

   3. Applied rewrites 31.2%

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

   4. Taylor expanded in l around 0

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 36.1

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

   6. Applied rewrites 36.1%

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

Alternative 15: 35.4% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Fortran

l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.3%

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

3. Applied rewrites 53.1%

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

4. Taylor expanded in t around inf

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

   1. lower-*.f64 35.4

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

6. Applied rewrites 35.4%

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

Reproduce

herbie shell --seed 2025142 
(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*))))))