Toniolo and Linder, Equation (13)

Percentage Accurate: 50.3% → 64.1%

Time: 17.5s

Alternatives: 20

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(n, u, t, l, om, u_42)
    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: 64.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := U \cdot \left(n \cdot 2\right)\\ t_3 := \frac{\ell \cdot \ell}{Om}\\ t_4 := \mathsf{fma}\left(-2, t\_3, t\right)\\ t_5 := \sqrt{\left(\left(U* - U\right) \cdot \left(t\_1 \cdot n\right) - \left(t\_3 \cdot 2 - t\right)\right) \cdot t\_2}\\ \mathbf{if}\;t\_5 \leq 10^{-160}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\left(t\_4 - \left(\left(U - U*\right) \cdot n\right) \cdot t\_1\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;t\_5 \leq 2 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \frac{\ell}{Om}, t\_4\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(\left(-2 \cdot U\right) \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)}{Om}\right) \cdot n, \ell, \left(\left(t \cdot n\right) \cdot 2\right) \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (pow (/ l Om) 2.0))
        (t_2 (* U (* n 2.0)))
        (t_3 (/ (* l l) Om))
        (t_4 (fma -2.0 t_3 t))
        (t_5 (sqrt (* (- (* (- U* U) (* t_1 n)) (- (* t_3 2.0) t)) t_2))))
   (if (<= t_5 1e-160)
     (* (sqrt U) (sqrt (* (- t_4 (* (* (- U U*) n) t_1)) (* n 2.0))))
     (if (<= t_5 2e+137)
       (sqrt (* (fma (* (* (/ l Om) n) (- U* U)) (/ l Om) t_4) t_2))
       (sqrt
        (fma
         (* (* (* (* -2.0 U) l) (/ (fma (/ n Om) (- U U*) 2.0) Om)) n)
         l
         (* (* (* t n) 2.0) U)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = pow((l / Om), 2.0);
	double t_2 = U * (n * 2.0);
	double t_3 = (l * l) / Om;
	double t_4 = fma(-2.0, t_3, t);
	double t_5 = sqrt(((((U_42_ - U) * (t_1 * n)) - ((t_3 * 2.0) - t)) * t_2));
	double tmp;
	if (t_5 <= 1e-160) {
		tmp = sqrt(U) * sqrt(((t_4 - (((U - U_42_) * n) * t_1)) * (n * 2.0)));
	} else if (t_5 <= 2e+137) {
		tmp = sqrt((fma((((l / Om) * n) * (U_42_ - U)), (l / Om), t_4) * t_2));
	} else {
		tmp = sqrt(fma(((((-2.0 * U) * l) * (fma((n / Om), (U - U_42_), 2.0) / Om)) * n), l, (((t * n) * 2.0) * U)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(l / Om) ^ 2.0
	t_2 = Float64(U * Float64(n * 2.0))
	t_3 = Float64(Float64(l * l) / Om)
	t_4 = fma(-2.0, t_3, t)
	t_5 = sqrt(Float64(Float64(Float64(Float64(U_42_ - U) * Float64(t_1 * n)) - Float64(Float64(t_3 * 2.0) - t)) * t_2))
	tmp = 0.0
	if (t_5 <= 1e-160)
		tmp = Float64(sqrt(U) * sqrt(Float64(Float64(t_4 - Float64(Float64(Float64(U - U_42_) * n) * t_1)) * Float64(n * 2.0))));
	elseif (t_5 <= 2e+137)
		tmp = sqrt(Float64(fma(Float64(Float64(Float64(l / Om) * n) * Float64(U_42_ - U)), Float64(l / Om), t_4) * t_2));
	else
		tmp = sqrt(fma(Float64(Float64(Float64(Float64(-2.0 * U) * l) * Float64(fma(Float64(n / Om), Float64(U - U_42_), 2.0) / Om)) * n), l, Float64(Float64(Float64(t * n) * 2.0) * U)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$4 = N[(-2.0 * t$95$3 + t), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(t$95$1 * n), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$3 * 2.0), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$5, 1e-160], N[(N[Sqrt[U], $MachinePrecision] * N[Sqrt[N[(N[(t$95$4 - N[(N[(N[(U - U$42$), $MachinePrecision] * n), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2e+137], N[Sqrt[N[(N[(N[(N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision] + t$95$4), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(-2.0 * U), $MachinePrecision] * l), $MachinePrecision] * N[(N[(N[(n / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + 2.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * l + N[(N[(N[(t * n), $MachinePrecision] * 2.0), $MachinePrecision] * U), $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*))))) < 9.9999999999999999e-161

   1. Initial program 9.2%

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

      1. lift-sqrt.f64 N/A

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

      2. 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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 45.4%

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

   if 9.9999999999999999e-161 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2.0000000000000001e137

   1. Initial program 95.9%

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

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 97.8

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 97.8%

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

   if 2.0000000000000001e137 < (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 20.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 36.9%

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

   7. Applied rewrites 38.7%

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

   9. Applied rewrites 49.1%

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

   11. Applied rewrites 51.2%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq 10^{-160}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) - \left(\left(U - U*\right) \cdot n\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(n \cdot 2\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq 2 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(\left(-2 \cdot U\right) \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)}{Om}\right) \cdot n, \ell, \left(\left(t \cdot n\right) \cdot 2\right) \cdot U\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 55.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{Om}\\ t_2 := U \cdot \left(n \cdot 2\right)\\ t_3 := \left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(t\_1 \cdot 2 - t\right)\right) \cdot t\_2\\ \mathbf{if}\;t\_3 \leq 10^{-320}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-2, t\_1, t\right) \cdot n} \cdot \sqrt{U}\right) \cdot \sqrt{2}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+274}:\\ \;\;\;\;\sqrt{\left(t - \frac{\mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right) \cdot \left(\ell \cdot \ell\right)}{Om}\right) \cdot t\_2}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{2}{Om} \cdot \ell\right) \cdot \left(\left(-2 \cdot U\right) \cdot \ell\right), n, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\ell \cdot n}{Om} \cdot \frac{\left(\left(U* \cdot U\right) \cdot \ell\right) \cdot n}{Om}\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l l) Om))
        (t_2 (* U (* n 2.0)))
        (t_3
         (* (- (* (- U* U) (* (pow (/ l Om) 2.0) n)) (- (* t_1 2.0) t)) t_2)))
   (if (<= t_3 1e-320)
     (* (* (sqrt (* (fma -2.0 t_1 t) n)) (sqrt U)) (sqrt 2.0))
     (if (<= t_3 5e+274)
       (sqrt (* (- t (/ (* (fma (- U U*) (/ n Om) 2.0) (* l l)) Om)) t_2))
       (if (<= t_3 INFINITY)
         (sqrt
          (fma (* (* (/ 2.0 Om) l) (* (* -2.0 U) l)) n (* (* (* t n) U) 2.0)))
         (sqrt (* (* (/ (* l n) Om) (/ (* (* (* U* U) l) n) Om)) 2.0)))))))

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 = U * (n * 2.0);
	double t_3 = (((U_42_ - U) * (pow((l / Om), 2.0) * n)) - ((t_1 * 2.0) - t)) * t_2;
	double tmp;
	if (t_3 <= 1e-320) {
		tmp = (sqrt((fma(-2.0, t_1, t) * n)) * sqrt(U)) * sqrt(2.0);
	} else if (t_3 <= 5e+274) {
		tmp = sqrt(((t - ((fma((U - U_42_), (n / Om), 2.0) * (l * l)) / Om)) * t_2));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(fma((((2.0 / Om) * l) * ((-2.0 * U) * l)), n, (((t * n) * U) * 2.0)));
	} else {
		tmp = sqrt(((((l * n) / Om) * ((((U_42_ * U) * l) * n) / Om)) * 2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(l * l) / Om)
	t_2 = Float64(U * Float64(n * 2.0))
	t_3 = Float64(Float64(Float64(Float64(U_42_ - U) * Float64((Float64(l / Om) ^ 2.0) * n)) - Float64(Float64(t_1 * 2.0) - t)) * t_2)
	tmp = 0.0
	if (t_3 <= 1e-320)
		tmp = Float64(Float64(sqrt(Float64(fma(-2.0, t_1, t) * n)) * sqrt(U)) * sqrt(2.0));
	elseif (t_3 <= 5e+274)
		tmp = sqrt(Float64(Float64(t - Float64(Float64(fma(Float64(U - U_42_), Float64(n / Om), 2.0) * Float64(l * l)) / Om)) * t_2));
	elseif (t_3 <= Inf)
		tmp = sqrt(fma(Float64(Float64(Float64(2.0 / Om) * l) * Float64(Float64(-2.0 * U) * l)), n, Float64(Float64(Float64(t * n) * U) * 2.0)));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(l * n) / Om) * Float64(Float64(Float64(Float64(U_42_ * U) * l) * n) / Om)) * 2.0));
	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[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$1 * 2.0), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, 1e-320], N[(N[(N[Sqrt[N[(N[(-2.0 * t$95$1 + t), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+274], N[Sqrt[N[(N[(t - N[(N[(N[(N[(U - U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision] + 2.0), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(N[(N[(N[(2.0 / Om), $MachinePrecision] * l), $MachinePrecision] * N[(N[(-2.0 * U), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * n + N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(l * n), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(N[(N[(U$42$ * U), $MachinePrecision] * l), $MachinePrecision] * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 9.99989e-321

   1. Initial program 8.7%

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

   3. Taylor expanded in U* around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate--r+ N/A

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

      10. lower--.f64 N/A

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

   5. Applied rewrites 12.6%

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

   7. Applied rewrites 42.7%

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

   8. Taylor expanded in U around 0

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

   10. Applied rewrites 42.7%

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

   if 9.99989e-321 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.9999999999999998e274

   1. Initial program 95.9%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. associate-*r/ N/A

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

      8. div-sub N/A

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

      9. lower-/.f64 N/A

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

   5. Applied rewrites 91.0%

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

   if 4.9999999999999998e274 < (*.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 33.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 32.3%

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

   7. Applied rewrites 35.3%

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

   9. Applied rewrites 43.5%

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

   10. Taylor expanded in Om around inf

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

   12. Applied rewrites 40.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 42.6%

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

   7. Applied rewrites 42.6%

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

   8. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. unswap-sqr N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 45.5

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

   10. Applied rewrites 45.5%

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

   12. Applied rewrites 52.5%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 10^{-320}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n} \cdot \sqrt{U}\right) \cdot \sqrt{2}\\ \mathbf{elif}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq 5 \cdot 10^{+274}:\\ \;\;\;\;\sqrt{\left(t - \frac{\mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right) \cdot \left(\ell \cdot \ell\right)}{Om}\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{elif}\;\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{2}{Om} \cdot \ell\right) \cdot \left(\left(-2 \cdot U\right) \cdot \ell\right), n, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\ell \cdot n}{Om} \cdot \frac{\left(\left(U* \cdot U\right) \cdot \ell\right) \cdot n}{Om}\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \frac{\ell \cdot \ell}{Om}\\ t_3 := \sqrt{\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(t\_2 \cdot 2 - t\right)\right) \cdot t\_1}\\ \mathbf{if}\;t\_3 \leq 10^{-160}:\\ \;\;\;\;\sqrt{U \cdot 2} \cdot \sqrt{\left(\frac{\mathsf{fma}\left(-n, \frac{\left(\ell \cdot \ell\right) \cdot U}{Om}, \left(-2 \cdot \ell\right) \cdot \ell\right)}{Om} + t\right) \cdot n}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2, t\_2, t\right)\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(\left(-2 \cdot U\right) \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)}{Om}\right) \cdot n, \ell, \left(\left(t \cdot n\right) \cdot 2\right) \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2 (/ (* l l) Om))
        (t_3
         (sqrt
          (*
           (- (* (- U* U) (* (pow (/ l Om) 2.0) n)) (- (* t_2 2.0) t))
           t_1))))
   (if (<= t_3 1e-160)
     (*
      (sqrt (* U 2.0))
      (sqrt
       (* (+ (/ (fma (- n) (/ (* (* l l) U) Om) (* (* -2.0 l) l)) Om) t) n)))
     (if (<= t_3 2e+137)
       (sqrt
        (* (fma (* (* (/ l Om) n) (- U* U)) (/ l Om) (fma -2.0 t_2 t)) t_1))
       (sqrt
        (fma
         (* (* (* (* -2.0 U) l) (/ (fma (/ n Om) (- U U*) 2.0) Om)) n)
         l
         (* (* (* t n) 2.0) U)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = U * (n * 2.0);
	double t_2 = (l * l) / Om;
	double t_3 = sqrt(((((U_42_ - U) * (pow((l / Om), 2.0) * n)) - ((t_2 * 2.0) - t)) * t_1));
	double tmp;
	if (t_3 <= 1e-160) {
		tmp = sqrt((U * 2.0)) * sqrt((((fma(-n, (((l * l) * U) / Om), ((-2.0 * l) * l)) / Om) + t) * n));
	} else if (t_3 <= 2e+137) {
		tmp = sqrt((fma((((l / Om) * n) * (U_42_ - U)), (l / Om), fma(-2.0, t_2, t)) * t_1));
	} else {
		tmp = sqrt(fma(((((-2.0 * U) * l) * (fma((n / Om), (U - U_42_), 2.0) / Om)) * n), l, (((t * n) * 2.0) * U)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(U * Float64(n * 2.0))
	t_2 = Float64(Float64(l * l) / Om)
	t_3 = sqrt(Float64(Float64(Float64(Float64(U_42_ - U) * Float64((Float64(l / Om) ^ 2.0) * n)) - Float64(Float64(t_2 * 2.0) - t)) * t_1))
	tmp = 0.0
	if (t_3 <= 1e-160)
		tmp = Float64(sqrt(Float64(U * 2.0)) * sqrt(Float64(Float64(Float64(fma(Float64(-n), Float64(Float64(Float64(l * l) * U) / Om), Float64(Float64(-2.0 * l) * l)) / Om) + t) * n)));
	elseif (t_3 <= 2e+137)
		tmp = sqrt(Float64(fma(Float64(Float64(Float64(l / Om) * n) * Float64(U_42_ - U)), Float64(l / Om), fma(-2.0, t_2, t)) * t_1));
	else
		tmp = sqrt(fma(Float64(Float64(Float64(Float64(-2.0 * U) * l) * Float64(fma(Float64(n / Om), Float64(U - U_42_), 2.0) / Om)) * n), l, Float64(Float64(Float64(t * n) * 2.0) * U)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$2 * 2.0), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 1e-160], N[(N[Sqrt[N[(U * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(N[(N[((-n) * N[(N[(N[(l * l), $MachinePrecision] * U), $MachinePrecision] / Om), $MachinePrecision] + N[(N[(-2.0 * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+137], N[Sqrt[N[(N[(N[(N[(N[(l / Om), $MachinePrecision] * n), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision] + N[(-2.0 * t$95$2 + t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(-2.0 * U), $MachinePrecision] * l), $MachinePrecision] * N[(N[(N[(n / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + 2.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * l + N[(N[(N[(t * n), $MachinePrecision] * 2.0), $MachinePrecision] * U), $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*))))) < 9.9999999999999999e-161

   1. Initial program 9.2%

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

   3. Taylor expanded in U* around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate--r+ N/A

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

      10. lower--.f64 N/A

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

   5. Applied rewrites 9.1%

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

   7. Applied rewrites 43.3%

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

   9. Applied rewrites 43.4%

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

   if 9.9999999999999999e-161 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2.0000000000000001e137

   1. Initial program 95.9%

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

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 97.8

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

   4. Applied rewrites 97.8%

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

   if 2.0000000000000001e137 < (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 20.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 36.9%

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

   7. Applied rewrites 38.7%

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

   9. Applied rewrites 49.1%

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

   11. Applied rewrites 51.2%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq 10^{-160}:\\ \;\;\;\;\sqrt{U \cdot 2} \cdot \sqrt{\left(\frac{\mathsf{fma}\left(-n, \frac{\left(\ell \cdot \ell\right) \cdot U}{Om}, \left(-2 \cdot \ell\right) \cdot \ell\right)}{Om} + t\right) \cdot n}\\ \mathbf{elif}\;\sqrt{\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq 2 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U* - U\right), \frac{\ell}{Om}, \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(\left(-2 \cdot U\right) \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)}{Om}\right) \cdot n, \ell, \left(\left(t \cdot n\right) \cdot 2\right) \cdot U\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := U \cdot \left(n \cdot 2\right)\\ t_2 := \frac{\ell \cdot \ell}{Om}\\ t_3 := \sqrt{\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(t\_2 \cdot 2 - t\right)\right) \cdot t\_1}\\ \mathbf{if}\;t\_3 \leq 10^{-160}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-2, t\_2, t\right) \cdot n} \cdot \sqrt{U}\right) \cdot \sqrt{2}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{\left(t - \frac{\mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right) \cdot \left(\ell \cdot \ell\right)}{Om}\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{2 - \frac{U* \cdot n}{Om}}{Om} \cdot \ell\right) \cdot \left(\left(-2 \cdot U\right) \cdot \ell\right), n, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* U (* n 2.0)))
        (t_2 (/ (* l l) Om))
        (t_3
         (sqrt
          (*
           (- (* (- U* U) (* (pow (/ l Om) 2.0) n)) (- (* t_2 2.0) t))
           t_1))))
   (if (<= t_3 1e-160)
     (* (* (sqrt (* (fma -2.0 t_2 t) n)) (sqrt U)) (sqrt 2.0))
     (if (<= t_3 2e+137)
       (sqrt (* (- t (/ (* (fma (- U U*) (/ n Om) 2.0) (* l l)) Om)) t_1))
       (sqrt
        (fma
         (* (* (/ (- 2.0 (/ (* U* n) Om)) Om) l) (* (* -2.0 U) l))
         n
         (* (* (* t n) U) 2.0)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = U * (n * 2.0);
	double t_2 = (l * l) / Om;
	double t_3 = sqrt(((((U_42_ - U) * (pow((l / Om), 2.0) * n)) - ((t_2 * 2.0) - t)) * t_1));
	double tmp;
	if (t_3 <= 1e-160) {
		tmp = (sqrt((fma(-2.0, t_2, t) * n)) * sqrt(U)) * sqrt(2.0);
	} else if (t_3 <= 2e+137) {
		tmp = sqrt(((t - ((fma((U - U_42_), (n / Om), 2.0) * (l * l)) / Om)) * t_1));
	} else {
		tmp = sqrt(fma(((((2.0 - ((U_42_ * n) / Om)) / Om) * l) * ((-2.0 * U) * l)), n, (((t * n) * U) * 2.0)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(U * Float64(n * 2.0))
	t_2 = Float64(Float64(l * l) / Om)
	t_3 = sqrt(Float64(Float64(Float64(Float64(U_42_ - U) * Float64((Float64(l / Om) ^ 2.0) * n)) - Float64(Float64(t_2 * 2.0) - t)) * t_1))
	tmp = 0.0
	if (t_3 <= 1e-160)
		tmp = Float64(Float64(sqrt(Float64(fma(-2.0, t_2, t) * n)) * sqrt(U)) * sqrt(2.0));
	elseif (t_3 <= 2e+137)
		tmp = sqrt(Float64(Float64(t - Float64(Float64(fma(Float64(U - U_42_), Float64(n / Om), 2.0) * Float64(l * l)) / Om)) * t_1));
	else
		tmp = sqrt(fma(Float64(Float64(Float64(Float64(2.0 - Float64(Float64(U_42_ * n) / Om)) / Om) * l) * Float64(Float64(-2.0 * U) * l)), n, Float64(Float64(Float64(t * n) * U) * 2.0)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(N[(N[(U$42$ - U), $MachinePrecision] * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$2 * 2.0), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 1e-160], N[(N[(N[Sqrt[N[(N[(-2.0 * t$95$2 + t), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+137], N[Sqrt[N[(N[(t - N[(N[(N[(N[(U - U$42$), $MachinePrecision] * N[(n / Om), $MachinePrecision] + 2.0), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(2.0 - N[(N[(U$42$ * n), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * l), $MachinePrecision] * N[(N[(-2.0 * U), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * n + N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $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*))))) < 9.9999999999999999e-161

   1. Initial program 9.2%

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

   3. Taylor expanded in U* around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate--r+ N/A

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

      10. lower--.f64 N/A

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

   5. Applied rewrites 9.1%

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

   7. Applied rewrites 43.3%

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

   8. Taylor expanded in U around 0

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

   10. Applied rewrites 43.3%

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

   if 9.9999999999999999e-161 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2.0000000000000001e137

   1. Initial program 95.9%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. associate-*r/ N/A

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

      8. div-sub N/A

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

      9. lower-/.f64 N/A

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

   5. Applied rewrites 91.0%

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

   if 2.0000000000000001e137 < (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 20.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 36.9%

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

   7. Applied rewrites 38.7%

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

   9. Applied rewrites 49.1%

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

   10. Taylor expanded in U around 0

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

   12. Applied rewrites 50.4%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq 10^{-160}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n} \cdot \sqrt{U}\right) \cdot \sqrt{2}\\ \mathbf{elif}\;\sqrt{\left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\frac{\ell \cdot \ell}{Om} \cdot 2 - t\right)\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \leq 2 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{\left(t - \frac{\mathsf{fma}\left(U - U*, \frac{n}{Om}, 2\right) \cdot \left(\ell \cdot \ell\right)}{Om}\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{2 - \frac{U* \cdot n}{Om}}{Om} \cdot \ell\right) \cdot \left(\left(-2 \cdot U\right) \cdot \ell\right), n, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 52.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 54.9%

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

   7. Applied rewrites 56.4%

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

   9. Applied rewrites 57.5%

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

   10. Taylor expanded in Om around inf

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

   12. Applied rewrites 56.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 42.6%

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

   7. Applied rewrites 42.6%

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

   8. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. unswap-sqr N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 45.5

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

   10. Applied rewrites 45.5%

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

   12. Applied rewrites 52.5%

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

4. Final simplification 55.7%

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

Alternative 6: 50.0% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<=
      (*
       (- (* (- U* U) (* (pow (/ l Om) 2.0) n)) (- (* (/ (* l l) Om) 2.0) t))
       (* U (* n 2.0)))
      INFINITY)
   (sqrt (fma (* (* (/ U Om) l) (* l n)) -4.0 (* (* (* t n) U) 2.0)))
   (sqrt (* (* (/ (* l n) Om) (/ (* (* (* U* U) l) n) Om)) 2.0))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 52.1%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 52.3

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

   5. Applied rewrites 52.3%

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

   7. Applied rewrites 54.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 42.6%

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

   7. Applied rewrites 42.6%

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

   8. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. unswap-sqr N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 45.5

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

   10. Applied rewrites 45.5%

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

   12. Applied rewrites 52.5%

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

4. Final simplification 54.5%

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

Alternative 7: 49.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 52.1%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 52.9

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

   5. Applied rewrites 52.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 42.6%

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

   7. Applied rewrites 42.6%

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

   8. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. unswap-sqr N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 45.5

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

   10. Applied rewrites 45.5%

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

4. Final simplification 51.5%

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

Alternative 8: 49.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 52.1%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 52.9

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

   5. Applied rewrites 52.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 42.6%

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

   7. Applied rewrites 42.6%

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

   8. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. unswap-sqr N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 45.5

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

   10. Applied rewrites 45.5%

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

   12. Applied rewrites 43.9%

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

4. Final simplification 51.2%

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

Alternative 9: 48.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 52.1%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 52.9

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

   5. Applied rewrites 52.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 38.7

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

   5. Applied rewrites 38.7%

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

4. Final simplification 50.3%

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

Alternative 10: 47.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 52.1%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 52.9

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

   5. Applied rewrites 52.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 24.1

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

   5. Applied rewrites 24.1%

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

4. Final simplification 47.6%

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

Alternative 11: 38.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.0%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 34.0

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

   5. Applied rewrites 34.0%

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

   7. Applied rewrites 34.0%

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

   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. Initial program 50.5%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 38.4

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

   5. Applied rewrites 38.4%

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

   7. Applied rewrites 37.7%

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

4. Final simplification 37.0%

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

Alternative 12: 55.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)\\ \mathbf{if}\;\ell \leq 1.75 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2 \cdot U, \frac{\left(t\_1 \cdot \ell\right) \cdot \ell}{Om} \cdot n, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(\left(-2 \cdot U\right) \cdot \ell\right) \cdot \frac{t\_1}{Om}\right) \cdot n, \ell, \left(\left(t \cdot n\right) \cdot 2\right) \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (fma (/ n Om) (- U U*) 2.0)))
   (if (<= l 1.75e+63)
     (sqrt (fma (* -2.0 U) (* (/ (* (* t_1 l) l) Om) n) (* (* (* t n) U) 2.0)))
     (sqrt
      (fma (* (* (* (* -2.0 U) l) (/ t_1 Om)) n) l (* (* (* t n) 2.0) U))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = fma((n / Om), (U - U_42_), 2.0);
	double tmp;
	if (l <= 1.75e+63) {
		tmp = sqrt(fma((-2.0 * U), ((((t_1 * l) * l) / Om) * n), (((t * n) * U) * 2.0)));
	} else {
		tmp = sqrt(fma(((((-2.0 * U) * l) * (t_1 / Om)) * n), l, (((t * n) * 2.0) * U)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = fma(Float64(n / Om), Float64(U - U_42_), 2.0)
	tmp = 0.0
	if (l <= 1.75e+63)
		tmp = sqrt(fma(Float64(-2.0 * U), Float64(Float64(Float64(Float64(t_1 * l) * l) / Om) * n), Float64(Float64(Float64(t * n) * U) * 2.0)));
	else
		tmp = sqrt(fma(Float64(Float64(Float64(Float64(-2.0 * U) * l) * Float64(t_1 / Om)) * n), l, Float64(Float64(Float64(t * n) * 2.0) * U)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(n / Om), $MachinePrecision] * N[(U - U$42$), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[l, 1.75e+63], N[Sqrt[N[(N[(-2.0 * U), $MachinePrecision] * N[(N[(N[(N[(t$95$1 * l), $MachinePrecision] * l), $MachinePrecision] / Om), $MachinePrecision] * n), $MachinePrecision] + N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(-2.0 * U), $MachinePrecision] * l), $MachinePrecision] * N[(t$95$1 / Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * l + N[(N[(N[(t * n), $MachinePrecision] * 2.0), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.75000000000000015e63

   1. Initial program 46.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 53.7%

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

   7. Applied rewrites 55.1%

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

   if 1.75000000000000015e63 < l

   1. Initial program 23.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 47.0%

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

   7. Applied rewrites 47.0%

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

   9. Applied rewrites 71.0%

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

   11. Applied rewrites 78.2%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.75 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2 \cdot U, \frac{\left(\mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right) \cdot \ell\right) \cdot \ell}{Om} \cdot n, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\left(\left(-2 \cdot U\right) \cdot \ell\right) \cdot \frac{\mathsf{fma}\left(\frac{n}{Om}, U - U*, 2\right)}{Om}\right) \cdot n, \ell, \left(\left(t \cdot n\right) \cdot 2\right) \cdot U\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 47.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{U}{Om} \cdot \ell\right) \cdot \left(\ell \cdot n\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(\frac{-2}{Om \cdot Om} \cdot n\right) \cdot \left(\left(U - U*\right) \cdot n\right)\right) \cdot \ell\right) \cdot \left(\ell \cdot U\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.15e+162)
   (sqrt (fma (* (* (/ U Om) l) (* l n)) -4.0 (* (* (* t n) U) 2.0)))
   (sqrt (* (* (* (* (/ -2.0 (* Om Om)) n) (* (- U U*) n)) l) (* l U)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.15e+162) {
		tmp = sqrt(fma((((U / Om) * l) * (l * n)), -4.0, (((t * n) * U) * 2.0)));
	} else {
		tmp = sqrt((((((-2.0 / (Om * Om)) * n) * ((U - U_42_) * n)) * l) * (l * U)));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.15e+162)
		tmp = sqrt(fma(Float64(Float64(Float64(U / Om) * l) * Float64(l * n)), -4.0, Float64(Float64(Float64(t * n) * U) * 2.0)));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(-2.0 / Float64(Om * Om)) * n) * Float64(Float64(U - U_42_) * n)) * l) * Float64(l * U)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.15e+162], N[Sqrt[N[(N[(N[(N[(U / Om), $MachinePrecision] * l), $MachinePrecision] * N[(l * n), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(-2.0 / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * N[(N[(U - U$42$), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * N[(l * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.14999999999999997e162

   1. Initial program 45.9%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 48.0

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

   5. Applied rewrites 48.0%

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

   7. Applied rewrites 51.3%

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

   if 1.14999999999999997e162 < l

   1. Initial program 12.5%

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

   3. Taylor expanded in Om around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 50.0

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

   5. Applied rewrites 50.0%

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

   7. Applied rewrites 61.9%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{U}{Om} \cdot \ell\right) \cdot \left(\ell \cdot n\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(\frac{-2}{Om \cdot Om} \cdot n\right) \cdot \left(\left(U - U*\right) \cdot n\right)\right) \cdot \ell\right) \cdot \left(\ell \cdot U\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{U}{Om} \cdot \ell\right) \cdot \left(\ell \cdot n\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot n\right)\right) \cdot \left(U* \cdot U\right)}{Om \cdot Om} \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 1.15e+162)
   (sqrt (fma (* (* (/ U Om) l) (* l n)) -4.0 (* (* (* t n) U) 2.0)))
   (sqrt (* (/ (* (* (* l n) (* l n)) (* U* U)) (* Om Om)) 2.0))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 1.15e+162) {
		tmp = sqrt(fma((((U / Om) * l) * (l * n)), -4.0, (((t * n) * U) * 2.0)));
	} else {
		tmp = sqrt((((((l * n) * (l * n)) * (U_42_ * U)) / (Om * Om)) * 2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 1.15e+162)
		tmp = sqrt(fma(Float64(Float64(Float64(U / Om) * l) * Float64(l * n)), -4.0, Float64(Float64(Float64(t * n) * U) * 2.0)));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(l * n) * Float64(l * n)) * Float64(U_42_ * U)) / Float64(Om * Om)) * 2.0));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 1.15e+162], N[Sqrt[N[(N[(N[(N[(U / Om), $MachinePrecision] * l), $MachinePrecision] * N[(l * n), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(l * n), $MachinePrecision] * N[(l * n), $MachinePrecision]), $MachinePrecision] * N[(U$42$ * U), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.14999999999999997e162

   1. Initial program 45.9%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 48.0

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

   5. Applied rewrites 48.0%

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

   7. Applied rewrites 51.3%

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

   if 1.14999999999999997e162 < l

   1. Initial program 12.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 40.7%

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

   7. Applied rewrites 40.7%

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

   8. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. unswap-sqr N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 57.6

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

   10. Applied rewrites 57.6%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{U}{Om} \cdot \ell\right) \cdot \left(\ell \cdot n\right), -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot n\right)\right) \cdot \left(U* \cdot U\right)}{Om \cdot Om} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 44.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{U}{Om} \cdot \left(\ell \cdot \ell\right)\right) \cdot n, -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot n\right)\right) \cdot \left(U* \cdot U\right)}{Om \cdot Om} \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 2e+146)
   (sqrt (fma (* (* (/ U Om) (* l l)) n) -4.0 (* (* (* t n) U) 2.0)))
   (sqrt (* (/ (* (* (* l n) (* l n)) (* U* U)) (* Om Om)) 2.0))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 2e+146) {
		tmp = sqrt(fma((((U / Om) * (l * l)) * n), -4.0, (((t * n) * U) * 2.0)));
	} else {
		tmp = sqrt((((((l * n) * (l * n)) * (U_42_ * U)) / (Om * Om)) * 2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 2e+146)
		tmp = sqrt(fma(Float64(Float64(Float64(U / Om) * Float64(l * l)) * n), -4.0, Float64(Float64(Float64(t * n) * U) * 2.0)));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(l * n) * Float64(l * n)) * Float64(U_42_ * U)) / Float64(Om * Om)) * 2.0));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 2e+146], N[Sqrt[N[(N[(N[(N[(U / Om), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * -4.0 + N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(l * n), $MachinePrecision] * N[(l * n), $MachinePrecision]), $MachinePrecision] * N[(U$42$ * U), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.99999999999999987e146

   1. Initial program 46.1%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 48.2

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

   5. Applied rewrites 48.2%

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

   7. Applied rewrites 49.8%

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

   if 1.99999999999999987e146 < l

   1. Initial program 12.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 42.9%

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

   7. Applied rewrites 42.9%

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

   8. Taylor expanded in U* around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. unswap-sqr N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 59.2

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

   10. Applied rewrites 59.2%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(\frac{U}{Om} \cdot \left(\ell \cdot \ell\right)\right) \cdot n, -4, \left(\left(t \cdot n\right) \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(\ell \cdot n\right) \cdot \left(\ell \cdot n\right)\right) \cdot \left(U* \cdot U\right)}{Om \cdot Om} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 38.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq -7.5 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{t \cdot n} \cdot \sqrt{U}\right) \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= U -7.5e-290)
   (sqrt (* (* (* t n) U) 2.0))
   (* (* (sqrt (* t n)) (sqrt U)) (sqrt 2.0))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -7.5e-290) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = (sqrt((t * n)) * sqrt(U)) * sqrt(2.0);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (u <= (-7.5d-290)) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else
        tmp = (sqrt((t * n)) * sqrt(u)) * sqrt(2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (U <= -7.5e-290) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = (Math.sqrt((t * n)) * Math.sqrt(U)) * Math.sqrt(2.0);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if U <= -7.5e-290:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	else:
		tmp = (math.sqrt((t * n)) * math.sqrt(U)) * math.sqrt(2.0)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (U <= -7.5e-290)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	else
		tmp = Float64(Float64(sqrt(Float64(t * n)) * sqrt(U)) * sqrt(2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (U <= -7.5e-290)
		tmp = sqrt((((t * n) * U) * 2.0));
	else
		tmp = (sqrt((t * n)) * sqrt(U)) * sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[U, -7.5e-290], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[N[(t * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[U], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -7.4999999999999995e-290

   1. Initial program 40.4%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 39.5

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

   5. Applied rewrites 39.5%

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

   if -7.4999999999999995e-290 < U

   1. Initial program 44.6%

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

   3. Taylor expanded in U* around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate--r+ N/A

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

      10. lower--.f64 N/A

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

   5. Applied rewrites 34.1%

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

   7. Applied rewrites 50.4%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 45.1%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -7.5 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{t \cdot n} \cdot \sqrt{U}\right) \cdot \sqrt{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 38.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.5 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om} \cdot -4}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l 6.5e+42)
   (sqrt (* (* (* t n) U) 2.0))
   (sqrt (* (/ (* (* (* l l) n) U) Om) -4.0))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 6.5e+42) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = sqrt((((((l * l) * n) * U) / Om) * -4.0));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    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 (l <= 6.5d+42) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else
        tmp = sqrt((((((l * l) * n) * u) / om) * (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (l <= 6.5e+42) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = Math.sqrt((((((l * l) * n) * U) / Om) * -4.0));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if l <= 6.5e+42:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	else:
		tmp = math.sqrt((((((l * l) * n) * U) / Om) * -4.0))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (l <= 6.5e+42)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(l * l) * n) * U) / Om) * -4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (l <= 6.5e+42)
		tmp = sqrt((((t * n) * U) * 2.0));
	else
		tmp = sqrt((((((l * l) * n) * U) / Om) * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[l, 6.5e+42], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(l * l), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] / Om), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.50000000000000052e42

   1. Initial program 46.0%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 42.6

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

   5. Applied rewrites 42.6%

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

   if 6.50000000000000052e42 < l

   1. Initial program 25.8%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 28.6

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

   5. Applied rewrites 28.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 22.1%

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

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.5 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U}{Om} \cdot -4}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 44.8% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.5%

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

3. Taylor expanded in n around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

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

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 44.2

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

5. Applied rewrites 44.2%

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

Alternative 19: 36.1% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.5%

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

3. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 37.6

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

5. Applied rewrites 37.6%

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

6. Final simplification 37.6%

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

Alternative 20: 35.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{t \cdot \left(U \cdot \left(n \cdot 2\right)\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.5%

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

3. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 37.6

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

5. Applied rewrites 37.6%

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

7. Applied rewrites 32.2%

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

8. Final simplification 32.2%

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

Reproduce

herbie shell --seed 2024249 
(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*))))))