Toniolo and Linder, Equation (13)

Percentage Accurate: 49.9% → 64.3%

Time: 8.9s

Alternatives: 10

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 49.9% accurate, 1.0× speedup

Math

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

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: 64.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|\ell\right| \cdot \left|\ell\right|}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \frac{\left|\ell\right|}{Om}\\ t_4 := t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {t\_3}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \left(U \cdot \left(t + -2 \cdot \frac{{\left(\left|\ell\right|\right)}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;t\_4 \leq 10^{+302}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(\left(U* - U\right) \cdot t\_3, t\_3 \cdot n, \mathsf{fma}\left(-2, t\_1, t\right)\right)}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{\left(t - \left|\ell\right| \cdot \mathsf{fma}\left(\frac{\left|\ell\right|}{Om \cdot Om} \cdot n, U - U*, \frac{\left|\ell\right| + \left|\ell\right|}{Om}\right)\right) \cdot \left(U \cdot \left(n + n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* (fabs l) (fabs l)) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3 (/ (fabs l) Om))
        (t_4 (* t_2 (- (- t (* 2.0 t_1)) (* (* n (pow t_3 2.0)) (- U U*))))))
   (if (<= t_4 0.0)
     (* (sqrt n) (sqrt (* 2.0 (* U (+ t (* -2.0 (/ (pow (fabs l) 2.0) Om)))))))
     (if (<= t_4 1e+302)
       (sqrt (* t_2 (fma (* (- U* U) t_3) (* t_3 n) (fma -2.0 t_1 t))))
       (if (<= t_4 INFINITY)
         (sqrt
          (*
           (-
            t
            (*
             (fabs l)
             (fma
              (* (/ (fabs l) (* Om Om)) n)
              (- U U*)
              (/ (+ (fabs l) (fabs l)) Om))))
           (* U (+ n n))))
         (*
          (fabs l)
          (sqrt
           (*
            -2.0
            (*
             U
             (*
              n
              (fma 2.0 (/ 1.0 Om) (/ (* n (- U U*)) (pow Om 2.0)))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (fabs(l) * fabs(l)) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = fabs(l) / Om;
	double t_4 = t_2 * ((t - (2.0 * t_1)) - ((n * pow(t_3, 2.0)) * (U - U_42_)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt(n) * sqrt((2.0 * (U * (t + (-2.0 * (pow(fabs(l), 2.0) / Om))))));
	} else if (t_4 <= 1e+302) {
		tmp = sqrt((t_2 * fma(((U_42_ - U) * t_3), (t_3 * n), fma(-2.0, t_1, t))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt(((t - (fabs(l) * fma(((fabs(l) / (Om * Om)) * n), (U - U_42_), ((fabs(l) + fabs(l)) / Om)))) * (U * (n + n))));
	} else {
		tmp = fabs(l) * sqrt((-2.0 * (U * (n * fma(2.0, (1.0 / Om), ((n * (U - U_42_)) / pow(Om, 2.0)))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(abs(l) * abs(l)) / Om)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(abs(l) / Om)
	t_4 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (t_3 ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = Float64(sqrt(n) * sqrt(Float64(2.0 * Float64(U * Float64(t + Float64(-2.0 * Float64((abs(l) ^ 2.0) / Om)))))));
	elseif (t_4 <= 1e+302)
		tmp = sqrt(Float64(t_2 * fma(Float64(Float64(U_42_ - U) * t_3), Float64(t_3 * n), fma(-2.0, t_1, t))));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(Float64(t - Float64(abs(l) * fma(Float64(Float64(abs(l) / Float64(Om * Om)) * n), Float64(U - U_42_), Float64(Float64(abs(l) + abs(l)) / Om)))) * Float64(U * Float64(n + n))));
	else
		tmp = Float64(abs(l) * sqrt(Float64(-2.0 * Float64(U * Float64(n * fma(2.0, Float64(1.0 / Om), Float64(Float64(n * Float64(U - U_42_)) / (Om ^ 2.0))))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[l], $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(U * N[(t + N[(-2.0 * N[(N[Power[N[Abs[l], $MachinePrecision], 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1e+302], N[Sqrt[N[(t$95$2 * N[(N[(N[(U$42$ - U), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$3 * n), $MachinePrecision] + N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(N[(t - N[(N[Abs[l], $MachinePrecision] * N[(N[(N[(N[Abs[l], $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + N[(N[(N[Abs[l], $MachinePrecision] + N[Abs[l], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(-2.0 * N[(U * N[(n * N[(2.0 * N[(1.0 / Om), $MachinePrecision] + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 49.9%

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

   3. Applied rewrites 23.9%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 25.4%

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

   6. Applied rewrites 25.4%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 1.0000000000000001e302

   1. Initial program 49.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   3. Applied rewrites 51.4%

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

   if 1.0000000000000001e302 < (*.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 49.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   3. Applied rewrites 51.4%

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

   5. Applied rewrites 51.8%

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

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

   1. Initial program 49.9%

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

   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. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 15.0%

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

   4. Applied rewrites 15.0%

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

Alternative 2: 64.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := -\left|\ell\right|\\ t_2 := \left(n \cdot {\left(\frac{\left|\ell\right|}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\\ t_3 := \left(2 \cdot n\right) \cdot U\\ t_4 := t\_3 \cdot \left(\left(t - 2 \cdot \frac{\left|\ell\right| \cdot \left|\ell\right|}{Om}\right) - t\_2\right)\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \left(U \cdot \left(t + -2 \cdot \frac{{\left(\left|\ell\right|\right)}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{t\_3 \cdot \left(\left(t - t\_1 \cdot \left(t\_1 \cdot \frac{2}{Om}\right)\right) - t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \sqrt{-2 \cdot \left(U \cdot \left(n \cdot \mathsf{fma}\left(2, \frac{1}{Om}, \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (- (fabs l)))
        (t_2 (* (* n (pow (/ (fabs l) Om) 2.0)) (- U U*)))
        (t_3 (* (* 2.0 n) U))
        (t_4 (* t_3 (- (- t (* 2.0 (/ (* (fabs l) (fabs l)) Om))) t_2))))
   (if (<= t_4 0.0)
     (* (sqrt n) (sqrt (* 2.0 (* U (+ t (* -2.0 (/ (pow (fabs l) 2.0) Om)))))))
     (if (<= t_4 INFINITY)
       (sqrt (* t_3 (- (- t (* t_1 (* t_1 (/ 2.0 Om)))) t_2)))
       (*
        (fabs l)
        (sqrt
         (*
          -2.0
          (*
           U
           (* n (fma 2.0 (/ 1.0 Om) (/ (* n (- U U*)) (pow Om 2.0))))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = -fabs(l);
	double t_2 = (n * pow((fabs(l) / Om), 2.0)) * (U - U_42_);
	double t_3 = (2.0 * n) * U;
	double t_4 = t_3 * ((t - (2.0 * ((fabs(l) * fabs(l)) / Om))) - t_2);
	double tmp;
	if (t_4 <= 0.0) {
		tmp = sqrt(n) * sqrt((2.0 * (U * (t + (-2.0 * (pow(fabs(l), 2.0) / Om))))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_3 * ((t - (t_1 * (t_1 * (2.0 / Om)))) - t_2)));
	} else {
		tmp = fabs(l) * sqrt((-2.0 * (U * (n * fma(2.0, (1.0 / Om), ((n * (U - U_42_)) / pow(Om, 2.0)))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(-abs(l))
	t_2 = Float64(Float64(n * (Float64(abs(l) / Om) ^ 2.0)) * Float64(U - U_42_))
	t_3 = Float64(Float64(2.0 * n) * U)
	t_4 = Float64(t_3 * Float64(Float64(t - Float64(2.0 * Float64(Float64(abs(l) * abs(l)) / Om))) - t_2))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = Float64(sqrt(n) * sqrt(Float64(2.0 * Float64(U * Float64(t + Float64(-2.0 * Float64((abs(l) ^ 2.0) / Om)))))));
	elseif (t_4 <= Inf)
		tmp = sqrt(Float64(t_3 * Float64(Float64(t - Float64(t_1 * Float64(t_1 * Float64(2.0 / Om)))) - t_2)));
	else
		tmp = Float64(abs(l) * sqrt(Float64(-2.0 * Float64(U * Float64(n * fma(2.0, Float64(1.0 / Om), Float64(Float64(n * Float64(U - U_42_)) / (Om ^ 2.0))))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = (-N[Abs[l], $MachinePrecision])}, Block[{t$95$2 = N[(N[(n * N[Power[N[(N[Abs[l], $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[(N[(t - N[(2.0 * N[(N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(U * N[(t + N[(-2.0 * N[(N[Power[N[Abs[l], $MachinePrecision], 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Sqrt[N[(t$95$3 * N[(N[(t - N[(t$95$1 * N[(t$95$1 * N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(-2.0 * N[(U * N[(n * N[(2.0 * N[(1.0 / Om), $MachinePrecision] + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[Power[Om, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 49.9%

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

   3. Applied rewrites 23.9%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 25.4%

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

   6. Applied rewrites 25.4%

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

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

   1. Initial program 49.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. div-add-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 53.9%

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

   3. Applied rewrites 53.9%

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

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

   1. Initial program 49.9%

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

   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. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-pow.f64 15.0%

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

   4. Applied rewrites 15.0%

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

Alternative 3: 60.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := \sqrt{t\_2 \cdot \left(\left(t - 2 \cdot t\_1\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \left(U \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\\ \mathbf{elif}\;t\_3 \leq 10^{+151}:\\ \;\;\;\;\sqrt{t\_2 \cdot \mathsf{fma}\left(\left(U* - U\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2, t\_1, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - \ell \cdot \frac{\mathsf{fma}\left(2, \ell, \frac{\ell \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)}{Om}\right) \cdot \left(U \cdot \left(n + n\right)\right)}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3
         (sqrt
          (*
           t_2
           (- (- t (* 2.0 t_1)) (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_3 0.0)
     (* (sqrt n) (sqrt (* 2.0 (* U (+ t (* -2.0 (/ (pow l 2.0) Om)))))))
     (if (<= t_3 1e+151)
       (sqrt
        (* t_2 (fma (* (- U* U) (/ l Om)) (* (/ l Om) n) (fma -2.0 t_1 t))))
       (sqrt
        (*
         (- t (* l (/ (fma 2.0 l (/ (* l (* n (- U U*))) Om)) Om)))
         (* U (+ n n))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (l * l) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = sqrt((t_2 * ((t - (2.0 * t_1)) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt(n) * sqrt((2.0 * (U * (t + (-2.0 * (pow(l, 2.0) / Om))))));
	} else if (t_3 <= 1e+151) {
		tmp = sqrt((t_2 * fma(((U_42_ - U) * (l / Om)), ((l / Om) * n), fma(-2.0, t_1, t))));
	} else {
		tmp = sqrt(((t - (l * (fma(2.0, l, ((l * (n * (U - U_42_))) / Om)) / Om))) * (U * (n + n))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * l) / Om)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * t_1)) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(n) * sqrt(Float64(2.0 * Float64(U * Float64(t + Float64(-2.0 * Float64((l ^ 2.0) / Om)))))));
	elseif (t_3 <= 1e+151)
		tmp = sqrt(Float64(t_2 * fma(Float64(Float64(U_42_ - U) * Float64(l / Om)), Float64(Float64(l / Om) * n), fma(-2.0, t_1, t))));
	else
		tmp = sqrt(Float64(Float64(t - Float64(l * Float64(fma(2.0, l, Float64(Float64(l * Float64(n * Float64(U - U_42_))) / Om)) / Om))) * Float64(U * Float64(n + n))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(U * N[(t + N[(-2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+151], N[Sqrt[N[(t$95$2 * N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision] + N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(t - N[(l * N[(N[(2.0 * l + N[(N[(l * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * N[(n + n), $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*))))) < 0.0

   1. Initial program 49.9%

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

   3. Applied rewrites 23.9%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 25.4%

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

   6. Applied rewrites 25.4%

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

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1e151

   1. Initial program 49.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   3. Applied rewrites 51.4%

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

   if 1e151 < (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 49.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   3. Applied rewrites 51.4%

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in Om around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 53.3%

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

   8. Applied rewrites 53.3%

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

Alternative 4: 56.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 7.2 \cdot 10^{+53}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{U* \cdot \left|\ell\right|}{Om}, \frac{\left|\ell\right|}{Om} \cdot n, \mathsf{fma}\left(-2, \frac{\left|\ell\right| \cdot \left|\ell\right|}{Om}, t\right)\right)}\\ \mathbf{elif}\;\left|\ell\right| \leq 1.55 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-4, \frac{{\left(\left|\ell\right|\right)}^{2} \cdot n}{Om}, 2 \cdot \left(n \cdot t\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - \left|\ell\right| \cdot \frac{\mathsf{fma}\left(2, \left|\ell\right|, \frac{\left|\ell\right| \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)}{Om}\right) \cdot \left(U \cdot \left(n + n\right)\right)}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= (fabs l) 7.2e+53)
   (sqrt
    (*
     (* (* 2.0 n) U)
     (fma
      (/ (* U* (fabs l)) Om)
      (* (/ (fabs l) Om) n)
      (fma -2.0 (/ (* (fabs l) (fabs l)) Om) t))))
   (if (<= (fabs l) 1.55e+144)
     (sqrt (* (fma -4.0 (/ (* (pow (fabs l) 2.0) n) Om) (* 2.0 (* n t))) U))
     (sqrt
      (*
       (-
        t
        (*
         (fabs l)
         (/ (fma 2.0 (fabs l) (/ (* (fabs l) (* n (- U U*))) Om)) Om)))
       (* U (+ n n)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (fabs(l) <= 7.2e+53) {
		tmp = sqrt((((2.0 * n) * U) * fma(((U_42_ * fabs(l)) / Om), ((fabs(l) / Om) * n), fma(-2.0, ((fabs(l) * fabs(l)) / Om), t))));
	} else if (fabs(l) <= 1.55e+144) {
		tmp = sqrt((fma(-4.0, ((pow(fabs(l), 2.0) * n) / Om), (2.0 * (n * t))) * U));
	} else {
		tmp = sqrt(((t - (fabs(l) * (fma(2.0, fabs(l), ((fabs(l) * (n * (U - U_42_))) / Om)) / Om))) * (U * (n + n))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (abs(l) <= 7.2e+53)
		tmp = sqrt(Float64(Float64(Float64(2.0 * n) * U) * fma(Float64(Float64(U_42_ * abs(l)) / Om), Float64(Float64(abs(l) / Om) * n), fma(-2.0, Float64(Float64(abs(l) * abs(l)) / Om), t))));
	elseif (abs(l) <= 1.55e+144)
		tmp = sqrt(Float64(fma(-4.0, Float64(Float64((abs(l) ^ 2.0) * n) / Om), Float64(2.0 * Float64(n * t))) * U));
	else
		tmp = sqrt(Float64(Float64(t - Float64(abs(l) * Float64(fma(2.0, abs(l), Float64(Float64(abs(l) * Float64(n * Float64(U - U_42_))) / Om)) / Om))) * Float64(U * Float64(n + n))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[N[Abs[l], $MachinePrecision], 7.2e+53], N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(N[(U$42$ * N[Abs[l], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(N[Abs[l], $MachinePrecision] / Om), $MachinePrecision] * n), $MachinePrecision] + N[(-2.0 * N[(N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Abs[l], $MachinePrecision], 1.55e+144], N[Sqrt[N[(N[(-4.0 * N[(N[(N[Power[N[Abs[l], $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision] + N[(2.0 * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(t - N[(N[Abs[l], $MachinePrecision] * N[(N[(2.0 * N[Abs[l], $MachinePrecision] + N[(N[(N[Abs[l], $MachinePrecision] * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 7.2e53

   1. Initial program 49.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   3. Applied rewrites 51.4%

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

   4. Taylor expanded in U around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 50.5%

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

   6. Applied rewrites 50.5%

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

   if 7.2e53 < l < 1.5500000000000001e144

   1. Initial program 49.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 50.0%

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

   3. Applied rewrites 42.2%

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

   4. Taylor expanded in U around 0

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

   6. Applied rewrites 43.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2-rev N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   8. Applied rewrites 47.6%

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

   9. Taylor expanded in Om around inf

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

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 44.0%

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

   11. Applied rewrites 44.0%

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

   if 1.5500000000000001e144 < l

   1. Initial program 49.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   3. Applied rewrites 51.4%

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in Om around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 53.3%

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

   8. Applied rewrites 53.3%

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

Alternative 5: 56.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. Initial program 49.9%

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

   3. Applied rewrites 23.9%

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

   4. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 25.4%

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

   6. Applied rewrites 25.4%

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

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

   1. Initial program 49.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   3. Applied rewrites 51.4%

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in Om around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 53.3%

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

   8. Applied rewrites 53.3%

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

Alternative 6: 53.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := U \cdot \left(n + n\right)\\ \mathbf{if}\;\left|\ell\right| \leq 3.4 \cdot 10^{-94}:\\ \;\;\;\;\sqrt{\left|t \cdot t\_1\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - \left|\ell\right| \cdot \frac{\mathsf{fma}\left(2, \left|\ell\right|, \frac{\left|\ell\right| \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}\right)}{Om}\right) \cdot t\_1}\\ \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* U (+ n n))))
   (if (<= (fabs l) 3.4e-94)
     (sqrt (fabs (* t t_1)))
     (sqrt
      (*
       (-
        t
        (*
         (fabs l)
         (/ (fma 2.0 (fabs l) (/ (* (fabs l) (* n (- U U*))) Om)) Om)))
       t_1)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = U * (n + n);
	double tmp;
	if (fabs(l) <= 3.4e-94) {
		tmp = sqrt(fabs((t * t_1)));
	} else {
		tmp = sqrt(((t - (fabs(l) * (fma(2.0, fabs(l), ((fabs(l) * (n * (U - U_42_))) / Om)) / Om))) * t_1));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(U * Float64(n + n))
	tmp = 0.0
	if (abs(l) <= 3.4e-94)
		tmp = sqrt(abs(Float64(t * t_1)));
	else
		tmp = sqrt(Float64(Float64(t - Float64(abs(l) * Float64(fma(2.0, abs(l), Float64(Float64(abs(l) * Float64(n * Float64(U - U_42_))) / Om)) / Om))) * t_1));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[l], $MachinePrecision], 3.4e-94], N[Sqrt[N[Abs[N[(t * t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(t - N[(N[Abs[l], $MachinePrecision] * N[(N[(2.0 * N[Abs[l], $MachinePrecision] + N[(N[(N[Abs[l], $MachinePrecision] * N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 3.3999999999999998e-94

   1. Initial program 49.9%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 36.1%

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

   4. Applied rewrites 36.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 35.7%

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. count-2-rev N/A

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

      14. lift-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. count-2-rev N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. count-2-rev N/A

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

      22. lower-+.f64 35.7%

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

   6. Applied rewrites 35.7%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

   8. Applied rewrites 38.3%

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

   if 3.3999999999999998e-94 < l

   1. Initial program 49.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   3. Applied rewrites 51.4%

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in Om around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 53.3%

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

   8. Applied rewrites 53.3%

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

Alternative 7: 49.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. Initial program 49.9%

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

   3. Applied rewrites 23.9%

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

   4. Taylor expanded in l around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 20.1%

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

   6. Applied rewrites 20.1%

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

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

   1. Initial program 49.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

   3. Applied rewrites 51.4%

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 47.2%

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

   8. Applied rewrites 47.2%

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

Alternative 8: 40.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * 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], 0.0], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[Abs[N[(t * N[(U * N[(n + n), $MachinePrecision]), $MachinePrecision]), $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*))))) < 0.0

   1. Initial program 49.9%

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

   3. Applied rewrites 23.9%

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

   4. Taylor expanded in l around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 20.1%

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

   6. Applied rewrites 20.1%

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

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

   1. Initial program 49.9%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 36.1%

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

   4. Applied rewrites 36.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 35.7%

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. count-2-rev N/A

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

      14. lift-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. count-2-rev N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. count-2-rev N/A

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

      22. lower-+.f64 35.7%

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

   6. Applied rewrites 35.7%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

   8. Applied rewrites 38.3%

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

Alternative 9: 38.3% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

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(abs((t * (u * (n + n)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.9%

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

2. Taylor expanded in t around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 36.1%

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

4. Applied rewrites 36.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-*.f64 35.7%

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. count-2-rev N/A

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

   14. lift-+.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-+.f64 N/A

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

   17. count-2-rev N/A

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

   18. associate-*r* N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 N/A

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

   21. count-2-rev N/A

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

   22. lower-+.f64 35.7%

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

6. Applied rewrites 35.7%

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

   1. rem-square-sqrt N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sqr-abs-rev N/A

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

8. Applied rewrites 38.3%

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

Alternative 10: 35.7% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

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((((u + u) * n) * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.9%

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

2. Taylor expanded in t around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 36.1%

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

4. Applied rewrites 36.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-*.f64 35.7%

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. count-2-rev N/A

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

   14. lift-+.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-+.f64 N/A

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

   17. count-2-rev N/A

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

   18. associate-*r* N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 N/A

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

   21. count-2-rev N/A

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

   22. lower-+.f64 35.7%

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

6. Applied rewrites 35.7%

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

Reproduce

herbie shell --seed 2025193 
(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*))))))