Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.4% → 90.1%

Time: 7.2s

Alternatives: 15

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \end{array} \]

FPCore

(FPCore (c0 A V l) :precision binary64 (* c0 (sqrt (/ A (* V l)))))

C

double code(double c0, double A, double V, double l) {
	return c0 * sqrt((A / (V * l)));
}

Fortran

real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    code = c0 * sqrt((a / (v * l)))
end function

Java

public static double code(double c0, double A, double V, double l) {
	return c0 * Math.sqrt((A / (V * l)));
}

Python

def code(c0, A, V, l):
	return c0 * math.sqrt((A / (V * l)))

Julia

function code(c0, A, V, l)
	return Float64(c0 * sqrt(Float64(A / Float64(V * l))))
end

MATLAB

function tmp = code(c0, A, V, l)
	tmp = c0 * sqrt((A / (V * l)));
end

Wolfram

code[c0_, A_, V_, l_] := N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \end{array} \]

FPCore

(FPCore (c0 A V l) :precision binary64 (* c0 (sqrt (/ A (* V l)))))

C

double code(double c0, double A, double V, double l) {
	return c0 * sqrt((A / (V * l)));
}

Fortran

real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    code = c0 * sqrt((a / (v * l)))
end function

Java

public static double code(double c0, double A, double V, double l) {
	return c0 * Math.sqrt((A / (V * l)));
}

Python

def code(c0, A, V, l):
	return c0 * math.sqrt((A / (V * l)))

Julia

function code(c0, A, V, l)
	return Float64(c0 * sqrt(Float64(A / Float64(V * l))))
end

MATLAB

function tmp = code(c0, A, V, l)
	tmp = c0 * sqrt((A / (V * l)));
end

Wolfram

code[c0_, A_, V_, l_] := N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 90.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -2 \cdot 10^{-228}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{-A}}{\sqrt{\ell} \cdot \sqrt{-V}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \left(\sqrt{\frac{1}{\ell}} \cdot c0\right)\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+274}:\\ \;\;\;\;\left(\sqrt{A} \cdot {\left(\ell \cdot V\right)}^{-0.5}\right) \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) -2e-228)
   (/ (* c0 (sqrt (- A))) (* (sqrt l) (sqrt (- V))))
   (if (<= (* l V) 0.0)
     (* (sqrt (/ A V)) (* (sqrt (/ 1.0 l)) c0))
     (if (<= (* l V) 1e+274)
       (* (* (sqrt A) (pow (* l V) -0.5)) c0)
       (* (sqrt (/ (/ A V) l)) c0)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -2e-228) {
		tmp = (c0 * sqrt(-A)) / (sqrt(l) * sqrt(-V));
	} else if ((l * V) <= 0.0) {
		tmp = sqrt((A / V)) * (sqrt((1.0 / l)) * c0);
	} else if ((l * V) <= 1e+274) {
		tmp = (sqrt(A) * pow((l * V), -0.5)) * c0;
	} else {
		tmp = sqrt(((A / V) / l)) * c0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((l * v) <= (-2d-228)) then
        tmp = (c0 * sqrt(-a)) / (sqrt(l) * sqrt(-v))
    else if ((l * v) <= 0.0d0) then
        tmp = sqrt((a / v)) * (sqrt((1.0d0 / l)) * c0)
    else if ((l * v) <= 1d+274) then
        tmp = (sqrt(a) * ((l * v) ** (-0.5d0))) * c0
    else
        tmp = sqrt(((a / v) / l)) * c0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -2e-228) {
		tmp = (c0 * Math.sqrt(-A)) / (Math.sqrt(l) * Math.sqrt(-V));
	} else if ((l * V) <= 0.0) {
		tmp = Math.sqrt((A / V)) * (Math.sqrt((1.0 / l)) * c0);
	} else if ((l * V) <= 1e+274) {
		tmp = (Math.sqrt(A) * Math.pow((l * V), -0.5)) * c0;
	} else {
		tmp = Math.sqrt(((A / V) / l)) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -2e-228:
		tmp = (c0 * math.sqrt(-A)) / (math.sqrt(l) * math.sqrt(-V))
	elif (l * V) <= 0.0:
		tmp = math.sqrt((A / V)) * (math.sqrt((1.0 / l)) * c0)
	elif (l * V) <= 1e+274:
		tmp = (math.sqrt(A) * math.pow((l * V), -0.5)) * c0
	else:
		tmp = math.sqrt(((A / V) / l)) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= -2e-228)
		tmp = Float64(Float64(c0 * sqrt(Float64(-A))) / Float64(sqrt(l) * sqrt(Float64(-V))));
	elseif (Float64(l * V) <= 0.0)
		tmp = Float64(sqrt(Float64(A / V)) * Float64(sqrt(Float64(1.0 / l)) * c0));
	elseif (Float64(l * V) <= 1e+274)
		tmp = Float64(Float64(sqrt(A) * (Float64(l * V) ^ -0.5)) * c0);
	else
		tmp = Float64(sqrt(Float64(Float64(A / V) / l)) * c0);
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -2e-228)
		tmp = (c0 * sqrt(-A)) / (sqrt(l) * sqrt(-V));
	elseif ((l * V) <= 0.0)
		tmp = sqrt((A / V)) * (sqrt((1.0 / l)) * c0);
	elseif ((l * V) <= 1e+274)
		tmp = (sqrt(A) * ((l * V) ^ -0.5)) * c0;
	else
		tmp = sqrt(((A / V) / l)) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], -2e-228], N[(N[(c0 * N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 1e+274], N[(N[(N[Sqrt[A], $MachinePrecision] * N[Power[N[(l * V), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -2.00000000000000007e-228

   1. Initial program 74.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(\frac{A}{V}\right)}{\mathsf{neg}\left(\ell\right)}}} \]

      5. div-inv N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(\frac{A}{V}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}} \]

      6. *-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot \left(\mathsf{neg}\left(\frac{A}{V}\right)\right)}} \]

      7. neg-mul-1 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot \color{blue}{\left(-1 \cdot \frac{A}{V}\right)}} \]

      8. associate-*r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}}} \]

      9. lower-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}}} \]

      10. lower-*.f64 N/A

          \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right)} \cdot \frac{A}{V}} \]

      11. metadata-eval N/A

          \[\leadsto c0 \cdot \sqrt{\left(\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}} \]

      12. frac-2neg N/A

          \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      13. lower-/.f64 N/A

          \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      14. lower-/.f64 72.5

          \[\leadsto c0 \cdot \sqrt{\left(\frac{-1}{\ell} \cdot -1\right) \cdot \color{blue}{\frac{A}{V}}} \]

   4. Applied rewrites 72.5%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{-1}{\ell} \cdot -1\right) \cdot \frac{A}{V}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{-1}{\ell} \cdot -1\right)} \cdot \frac{A}{V}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      3. associate-*l/ N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{-1 \cdot -1}{\ell}} \cdot \frac{A}{V}} \]

      4. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1}}{\ell} \cdot \frac{A}{V}} \]

      5. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\ell} \cdot \frac{A}{V}} \]

      6. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\ell}} \cdot \frac{A}{V}} \]

      7. metadata-eval 72.5

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1}}{\ell} \cdot \frac{A}{V}} \]

   6. Applied rewrites 72.5%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell}} \cdot \frac{A}{V}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{1}{\ell} \cdot \frac{A}{V}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{1}{\ell} \cdot \frac{A}{V}}} \]

      3. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell} \cdot \frac{A}{V}}} \]

      4. sqrt-prod N/A

         \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\frac{1}{\ell}} \cdot \sqrt{\frac{A}{V}}\right)} \]

      5. pow1/2 N/A

         \[\leadsto c0 \cdot \left(\color{blue}{{\left(\frac{1}{\ell}\right)}^{\frac{1}{2}}} \cdot \sqrt{\frac{A}{V}}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{\frac{A}{V}}}\right) \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(c0 \cdot {\left(\frac{1}{\ell}\right)}^{\frac{1}{2}}\right) \cdot \sqrt{\frac{A}{V}}} \]

      8. pow1/2 N/A

         \[\leadsto \left(c0 \cdot \color{blue}{\sqrt{\frac{1}{\ell}}}\right) \cdot \sqrt{\frac{A}{V}} \]

      9. lift-/.f64 N/A

         \[\leadsto \left(c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell}}}\right) \cdot \sqrt{\frac{A}{V}} \]

      10. sqrt-div N/A

          \[\leadsto \left(c0 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell}}}\right) \cdot \sqrt{\frac{A}{V}} \]

      11. metadata-eval N/A

          \[\leadsto \left(c0 \cdot \frac{\color{blue}{1}}{\sqrt{\ell}}\right) \cdot \sqrt{\frac{A}{V}} \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \left(c0 \cdot \frac{1}{\color{blue}{\sqrt{\ell}}}\right) \cdot \sqrt{\frac{A}{V}} \]

      13. div-inv N/A

          \[\leadsto \color{blue}{\frac{c0}{\sqrt{\ell}}} \cdot \sqrt{\frac{A}{V}} \]

      14. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{c0}{\sqrt{\ell}}} \cdot \sqrt{\frac{A}{V}} \]

      15. *-commutative N/A

          \[\leadsto \color{blue}{\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}} \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{A}{V}}} \cdot \frac{c0}{\sqrt{\ell}} \]

      17. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{A}{V}}} \cdot \frac{c0}{\sqrt{\ell}} \]

      18. frac-2neg N/A

          \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{neg}\left(A\right)}{\mathsf{neg}\left(V\right)}}} \cdot \frac{c0}{\sqrt{\ell}} \]

      19. lift-neg.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{-A}}{\mathsf{neg}\left(V\right)}} \cdot \frac{c0}{\sqrt{\ell}} \]

      20. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{-A}}{\sqrt{\mathsf{neg}\left(V\right)}}} \cdot \frac{c0}{\sqrt{\ell}} \]

      21. pow1/2 N/A

          \[\leadsto \frac{\color{blue}{{\left(-A\right)}^{\frac{1}{2}}}}{\sqrt{\mathsf{neg}\left(V\right)}} \cdot \frac{c0}{\sqrt{\ell}} \]

      22. lift-/.f64 N/A

          \[\leadsto \frac{{\left(-A\right)}^{\frac{1}{2}}}{\sqrt{\mathsf{neg}\left(V\right)}} \cdot \color{blue}{\frac{c0}{\sqrt{\ell}}} \]

   8. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{\sqrt{-A} \cdot c0}{\sqrt{\left(-\ell\right) \cdot V}}} \]
   9. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{-A} \cdot c0}{\color{blue}{\sqrt{\left(-\ell\right) \cdot V}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{-A} \cdot c0}{\sqrt{\color{blue}{\left(-\ell\right) \cdot V}}} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{-A} \cdot c0}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(\ell\right)\right)} \cdot V}} \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \frac{\sqrt{-A} \cdot c0}{\sqrt{\color{blue}{\mathsf{neg}\left(\ell \cdot V\right)}}} \]

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

         \[\leadsto \frac{\sqrt{-A} \cdot c0}{\sqrt{\color{blue}{\ell \cdot \left(\mathsf{neg}\left(V\right)\right)}}} \]

      6. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{-A} \cdot c0}{\sqrt{\ell \cdot \color{blue}{\left(-V\right)}}} \]

      7. sqrt-prod N/A

         \[\leadsto \frac{\sqrt{-A} \cdot c0}{\color{blue}{\sqrt{\ell} \cdot \sqrt{-V}}} \]

      8. pow1/2 N/A

         \[\leadsto \frac{\sqrt{-A} \cdot c0}{\color{blue}{{\ell}^{\frac{1}{2}}} \cdot \sqrt{-V}} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{-A} \cdot c0}{{\ell}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{-V}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{-A} \cdot c0}{\color{blue}{\sqrt{-V} \cdot {\ell}^{\frac{1}{2}}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{-A} \cdot c0}{\color{blue}{\sqrt{-V} \cdot {\ell}^{\frac{1}{2}}}} \]

      12. pow1/2 N/A

          \[\leadsto \frac{\sqrt{-A} \cdot c0}{\sqrt{-V} \cdot \color{blue}{\sqrt{\ell}}} \]

      13. lower-sqrt.f64 54.8

          \[\leadsto \frac{\sqrt{-A} \cdot c0}{\sqrt{-V} \cdot \color{blue}{\sqrt{\ell}}} \]

   10. Applied rewrites 54.8%

       \[\leadsto \frac{\sqrt{-A} \cdot c0}{\color{blue}{\sqrt{-V} \cdot \sqrt{\ell}}} \]

   if -2.00000000000000007e-228 < (*.f64 V l) < -0.0

   1. Initial program 45.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \]

      3. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      4. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      5. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      6. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{c0}{\sqrt{\ell}}} \cdot \sqrt{\frac{A}{V}} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{c0}{\color{blue}{\sqrt{\ell}}} \cdot \sqrt{\frac{A}{V}} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\ell}} \cdot \color{blue}{\sqrt{\frac{A}{V}}} \]

      13. lower-/.f64 39.8

          \[\leadsto \frac{c0}{\sqrt{\ell}} \cdot \sqrt{\color{blue}{\frac{A}{V}}} \]

   4. Applied rewrites 39.8%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}} \]

   5. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\left(c0 \cdot \sqrt{\frac{1}{\ell}}\right)} \cdot \sqrt{\frac{A}{V}} \]
   6. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(c0 \cdot \sqrt{\frac{1}{\ell}}\right)\right)\right)\right)} \cdot \sqrt{\frac{A}{V}} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\sqrt{\frac{1}{\ell}} \cdot c0}\right)\right)\right)\right) \cdot \sqrt{\frac{A}{V}} \]

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

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot c0}\right)\right) \cdot \sqrt{\frac{A}{V}} \]

      4. distribute-rgt-neg-out N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(\mathsf{neg}\left(c0\right)\right)\right)} \cdot \sqrt{\frac{A}{V}} \]

      5. neg-mul-1 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot \color{blue}{\left(-1 \cdot c0\right)}\right) \cdot \sqrt{\frac{A}{V}} \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot c0\right)\right) \cdot \sqrt{\frac{A}{V}} \]

      7. associate-*l* N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot \color{blue}{\left(\sqrt{-1} \cdot \left(\sqrt{-1} \cdot c0\right)\right)}\right) \cdot \sqrt{\frac{A}{V}} \]

      8. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(\sqrt{-1} \cdot \color{blue}{\left(c0 \cdot \sqrt{-1}\right)}\right)\right) \cdot \sqrt{\frac{A}{V}} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot \sqrt{-1}\right) \cdot \left(c0 \cdot \sqrt{-1}\right)\right)} \cdot \sqrt{\frac{A}{V}} \]

      10. *-commutative N/A

          \[\leadsto \left(\left(\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot \sqrt{-1}\right) \cdot \color{blue}{\left(\sqrt{-1} \cdot c0\right)}\right) \cdot \sqrt{\frac{A}{V}} \]

      11. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\left(\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot \sqrt{-1}\right) \cdot \sqrt{-1}\right) \cdot c0\right)} \cdot \sqrt{\frac{A}{V}} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\left(\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot \sqrt{-1}\right) \cdot \sqrt{-1}\right) \cdot c0\right)} \cdot \sqrt{\frac{A}{V}} \]

   7. Applied rewrites 39.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell}} \cdot c0\right)} \cdot \sqrt{\frac{A}{V}} \]

   if -0.0 < (*.f64 V l) < 9.99999999999999921e273

   1. Initial program 82.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      3. clear-num N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\frac{V \cdot \ell}{A}}}} \]

      4. associate-/r/ N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V \cdot \ell} \cdot A}} \]

      5. sqrt-prod N/A

         \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\frac{1}{V \cdot \ell}} \cdot \sqrt{A}\right)} \]

      6. pow1/2 N/A

         \[\leadsto c0 \cdot \left(\color{blue}{{\left(\frac{1}{V \cdot \ell}\right)}^{\frac{1}{2}}} \cdot \sqrt{A}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\left({\left(\frac{1}{V \cdot \ell}\right)}^{\frac{1}{2}} \cdot \sqrt{A}\right)} \]

      8. inv-pow N/A

         \[\leadsto c0 \cdot \left({\color{blue}{\left({\left(V \cdot \ell\right)}^{-1}\right)}}^{\frac{1}{2}} \cdot \sqrt{A}\right) \]

      9. pow-pow N/A

         \[\leadsto c0 \cdot \left(\color{blue}{{\left(V \cdot \ell\right)}^{\left(-1 \cdot \frac{1}{2}\right)}} \cdot \sqrt{A}\right) \]

      10. lower-pow.f64 N/A

          \[\leadsto c0 \cdot \left(\color{blue}{{\left(V \cdot \ell\right)}^{\left(-1 \cdot \frac{1}{2}\right)}} \cdot \sqrt{A}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto c0 \cdot \left({\color{blue}{\left(V \cdot \ell\right)}}^{\left(-1 \cdot \frac{1}{2}\right)} \cdot \sqrt{A}\right) \]

      12. *-commutative N/A

          \[\leadsto c0 \cdot \left({\color{blue}{\left(\ell \cdot V\right)}}^{\left(-1 \cdot \frac{1}{2}\right)} \cdot \sqrt{A}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto c0 \cdot \left({\color{blue}{\left(\ell \cdot V\right)}}^{\left(-1 \cdot \frac{1}{2}\right)} \cdot \sqrt{A}\right) \]

      14. metadata-eval N/A

          \[\leadsto c0 \cdot \left({\left(\ell \cdot V\right)}^{\color{blue}{\frac{-1}{2}}} \cdot \sqrt{A}\right) \]

      15. lower-sqrt.f64 99.4

          \[\leadsto c0 \cdot \left({\left(\ell \cdot V\right)}^{-0.5} \cdot \color{blue}{\sqrt{A}}\right) \]

   4. Applied rewrites 99.4%

      \[\leadsto c0 \cdot \color{blue}{\left({\left(\ell \cdot V\right)}^{-0.5} \cdot \sqrt{A}\right)} \]

   if 9.99999999999999921e273 < (*.f64 V l)

   1. Initial program 52.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 84.6

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   4. Applied rewrites 84.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -2 \cdot 10^{-228}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{-A}}{\sqrt{\ell} \cdot \sqrt{-V}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \left(\sqrt{\frac{1}{\ell}} \cdot c0\right)\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+274}:\\ \;\;\;\;\left(\sqrt{A} \cdot {\left(\ell \cdot V\right)}^{-0.5}\right) \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* (sqrt (/ A (* l V))) c0)))
   (if (<= t_0 2e-276)
     (* (sqrt (/ (/ A V) l)) c0)
     (if (<= t_0 2e+305) t_0 (/ c0 (sqrt (* (/ l A) V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / (l * V))) * c0;
	double tmp;
	if (t_0 <= 2e-276) {
		tmp = sqrt(((A / V) / l)) * c0;
	} else if (t_0 <= 2e+305) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt(((l / A) * V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a / (l * v))) * c0
    if (t_0 <= 2d-276) then
        tmp = sqrt(((a / v) / l)) * c0
    else if (t_0 <= 2d+305) then
        tmp = t_0
    else
        tmp = c0 / sqrt(((l / a) * v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / (l * V))) * c0;
	double tmp;
	if (t_0 <= 2e-276) {
		tmp = Math.sqrt(((A / V) / l)) * c0;
	} else if (t_0 <= 2e+305) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt(((l / A) * V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / (l * V))) * c0
	tmp = 0
	if t_0 <= 2e-276:
		tmp = math.sqrt(((A / V) / l)) * c0
	elif t_0 <= 2e+305:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt(((l / A) * V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(sqrt(Float64(A / Float64(l * V))) * c0)
	tmp = 0.0
	if (t_0 <= 2e-276)
		tmp = Float64(sqrt(Float64(Float64(A / V) / l)) * c0);
	elseif (t_0 <= 2e+305)
		tmp = t_0;
	else
		tmp = Float64(c0 / sqrt(Float64(Float64(l / A) * V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((A / (l * V))) * c0;
	tmp = 0.0;
	if (t_0 <= 2e-276)
		tmp = sqrt(((A / V) / l)) * c0;
	elseif (t_0 <= 2e+305)
		tmp = t_0;
	else
		tmp = c0 / sqrt(((l / A) * V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(N[Sqrt[N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-276], N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[t$95$0, 2e+305], t$95$0, N[(c0 / N[Sqrt[N[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2e-276

   1. Initial program 65.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 70.9

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   4. Applied rewrites 70.9%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

   if 2e-276 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.9999999999999999e305

   1. Initial program 99.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing

   if 1.9999999999999999e305 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 41.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \]

      3. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      4. clear-num N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\frac{V \cdot \ell}{A}}}} \]

      5. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      6. metadata-eval N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{1}}{\sqrt{\frac{V \cdot \ell}{A}}} \]

      7. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      10. lower-/.f64 45.3

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V \cdot \ell}{A}}}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell}}{A}}} \]

      12. *-commutative N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\ell \cdot V}}{A}}} \]

      13. lower-*.f64 45.3

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\ell \cdot V}}{A}}} \]

   4. Applied rewrites 45.3%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{\ell \cdot V}{A}}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\ell \cdot V}{A}}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\ell \cdot V}}{A}}} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\ell}{A} \cdot V}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\ell}{A} \cdot V}}} \]

      5. lower-/.f64 58.6

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\ell}{A}} \cdot V}} \]

   6. Applied rewrites 58.6%

      \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\ell}{A} \cdot V}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0 \leq 2 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{elif}\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* (sqrt (/ A (* l V))) c0)))
   (if (<= t_0 2e-276)
     (* (sqrt (/ (/ A V) l)) c0)
     (if (<= t_0 2e+305) t_0 (/ c0 (sqrt (* (/ V A) l)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / (l * V))) * c0;
	double tmp;
	if (t_0 <= 2e-276) {
		tmp = sqrt(((A / V) / l)) * c0;
	} else if (t_0 <= 2e+305) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt(((V / A) * l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a / (l * v))) * c0
    if (t_0 <= 2d-276) then
        tmp = sqrt(((a / v) / l)) * c0
    else if (t_0 <= 2d+305) then
        tmp = t_0
    else
        tmp = c0 / sqrt(((v / a) * l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / (l * V))) * c0;
	double tmp;
	if (t_0 <= 2e-276) {
		tmp = Math.sqrt(((A / V) / l)) * c0;
	} else if (t_0 <= 2e+305) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt(((V / A) * l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / (l * V))) * c0
	tmp = 0
	if t_0 <= 2e-276:
		tmp = math.sqrt(((A / V) / l)) * c0
	elif t_0 <= 2e+305:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt(((V / A) * l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(sqrt(Float64(A / Float64(l * V))) * c0)
	tmp = 0.0
	if (t_0 <= 2e-276)
		tmp = Float64(sqrt(Float64(Float64(A / V) / l)) * c0);
	elseif (t_0 <= 2e+305)
		tmp = t_0;
	else
		tmp = Float64(c0 / sqrt(Float64(Float64(V / A) * l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((A / (l * V))) * c0;
	tmp = 0.0;
	if (t_0 <= 2e-276)
		tmp = sqrt(((A / V) / l)) * c0;
	elseif (t_0 <= 2e+305)
		tmp = t_0;
	else
		tmp = c0 / sqrt(((V / A) * l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(N[Sqrt[N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-276], N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[t$95$0, 2e+305], t$95$0, N[(c0 / N[Sqrt[N[(N[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2e-276

   1. Initial program 65.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 70.9

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   4. Applied rewrites 70.9%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

   if 2e-276 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.9999999999999999e305

   1. Initial program 99.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing

   if 1.9999999999999999e305 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 41.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \]

      3. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      4. clear-num N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\frac{V \cdot \ell}{A}}}} \]

      5. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      6. metadata-eval N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{1}}{\sqrt{\frac{V \cdot \ell}{A}}} \]

      7. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      10. lower-/.f64 45.3

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V \cdot \ell}{A}}}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell}}{A}}} \]

      12. *-commutative N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\ell \cdot V}}{A}}} \]

      13. lower-*.f64 45.3

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\ell \cdot V}}{A}}} \]

   4. Applied rewrites 45.3%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{\ell \cdot V}{A}}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\ell \cdot V}{A}}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\ell \cdot V}}{A}}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\ell \cdot \frac{V}{A}}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{c0}{\sqrt{\ell \cdot \color{blue}{\frac{V}{A}}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V}{A} \cdot \ell}}} \]

      6. lower-*.f64 58.7

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V}{A} \cdot \ell}}} \]

   6. Applied rewrites 58.7%

      \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V}{A} \cdot \ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0 \leq 2 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{elif}\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ t_1 := \sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-276}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+238}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* (sqrt (/ A (* l V))) c0)) (t_1 (* (sqrt (/ (/ A V) l)) c0)))
   (if (<= t_0 2e-276) t_1 (if (<= t_0 1e+238) t_0 t_1))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / (l * V))) * c0;
	double t_1 = sqrt(((A / V) / l)) * c0;
	double tmp;
	if (t_0 <= 2e-276) {
		tmp = t_1;
	} else if (t_0 <= 1e+238) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((a / (l * v))) * c0
    t_1 = sqrt(((a / v) / l)) * c0
    if (t_0 <= 2d-276) then
        tmp = t_1
    else if (t_0 <= 1d+238) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / (l * V))) * c0;
	double t_1 = Math.sqrt(((A / V) / l)) * c0;
	double tmp;
	if (t_0 <= 2e-276) {
		tmp = t_1;
	} else if (t_0 <= 1e+238) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / (l * V))) * c0
	t_1 = math.sqrt(((A / V) / l)) * c0
	tmp = 0
	if t_0 <= 2e-276:
		tmp = t_1
	elif t_0 <= 1e+238:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(sqrt(Float64(A / Float64(l * V))) * c0)
	t_1 = Float64(sqrt(Float64(Float64(A / V) / l)) * c0)
	tmp = 0.0
	if (t_0 <= 2e-276)
		tmp = t_1;
	elseif (t_0 <= 1e+238)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((A / (l * V))) * c0;
	t_1 = sqrt(((A / V) / l)) * c0;
	tmp = 0.0;
	if (t_0 <= 2e-276)
		tmp = t_1;
	elseif (t_0 <= 1e+238)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(N[Sqrt[N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-276], t$95$1, If[LessEqual[t$95$0, 1e+238], t$95$0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2e-276 or 1e238 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 63.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 69.9

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   4. Applied rewrites 69.9%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

   if 2e-276 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1e238

   1. Initial program 99.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0 \leq 2 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{elif}\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0 \leq 10^{+238}:\\ \;\;\;\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-265}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \left(\sqrt{\frac{1}{\ell}} \cdot c0\right)\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+274}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) (- INFINITY))
   (* (/ (sqrt (/ (- A) l)) (sqrt (- V))) c0)
   (if (<= (* l V) -2e-265)
     (* (/ (sqrt (- A)) (sqrt (* (- l) V))) c0)
     (if (<= (* l V) 0.0)
       (* (sqrt (/ A V)) (* (sqrt (/ 1.0 l)) c0))
       (if (<= (* l V) 1e+274)
         (* (/ (sqrt A) (sqrt (* l V))) c0)
         (* (sqrt (/ (/ A V) l)) c0))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = (sqrt((-A / l)) / sqrt(-V)) * c0;
	} else if ((l * V) <= -2e-265) {
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	} else if ((l * V) <= 0.0) {
		tmp = sqrt((A / V)) * (sqrt((1.0 / l)) * c0);
	} else if ((l * V) <= 1e+274) {
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	} else {
		tmp = sqrt(((A / V) / l)) * c0;
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = (Math.sqrt((-A / l)) / Math.sqrt(-V)) * c0;
	} else if ((l * V) <= -2e-265) {
		tmp = (Math.sqrt(-A) / Math.sqrt((-l * V))) * c0;
	} else if ((l * V) <= 0.0) {
		tmp = Math.sqrt((A / V)) * (Math.sqrt((1.0 / l)) * c0);
	} else if ((l * V) <= 1e+274) {
		tmp = (Math.sqrt(A) / Math.sqrt((l * V))) * c0;
	} else {
		tmp = Math.sqrt(((A / V) / l)) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = (math.sqrt((-A / l)) / math.sqrt(-V)) * c0
	elif (l * V) <= -2e-265:
		tmp = (math.sqrt(-A) / math.sqrt((-l * V))) * c0
	elif (l * V) <= 0.0:
		tmp = math.sqrt((A / V)) * (math.sqrt((1.0 / l)) * c0)
	elif (l * V) <= 1e+274:
		tmp = (math.sqrt(A) / math.sqrt((l * V))) * c0
	else:
		tmp = math.sqrt(((A / V) / l)) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(Float64(-A) / l)) / sqrt(Float64(-V))) * c0);
	elseif (Float64(l * V) <= -2e-265)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-l) * V))) * c0);
	elseif (Float64(l * V) <= 0.0)
		tmp = Float64(sqrt(Float64(A / V)) * Float64(sqrt(Float64(1.0 / l)) * c0));
	elseif (Float64(l * V) <= 1e+274)
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(l * V))) * c0);
	else
		tmp = Float64(sqrt(Float64(Float64(A / V) / l)) * c0);
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = (sqrt((-A / l)) / sqrt(-V)) * c0;
	elseif ((l * V) <= -2e-265)
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	elseif ((l * V) <= 0.0)
		tmp = sqrt((A / V)) * (sqrt((1.0 / l)) * c0);
	elseif ((l * V) <= 1e+274)
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	else
		tmp = sqrt(((A / V) / l)) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], N[(N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -2e-265], N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 1e+274], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -inf.0

   1. Initial program 35.9%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(\frac{A}{V}\right)}{\mathsf{neg}\left(\ell\right)}}} \]

      5. div-inv N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(\frac{A}{V}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}} \]

      6. *-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot \left(\mathsf{neg}\left(\frac{A}{V}\right)\right)}} \]

      7. neg-mul-1 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot \color{blue}{\left(-1 \cdot \frac{A}{V}\right)}} \]

      8. associate-*r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}}} \]

      9. lower-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}}} \]

      10. lower-*.f64 N/A

          \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right)} \cdot \frac{A}{V}} \]

      11. metadata-eval N/A

          \[\leadsto c0 \cdot \sqrt{\left(\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}} \]

      12. frac-2neg N/A

          \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      13. lower-/.f64 N/A

          \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      14. lower-/.f64 69.3

          \[\leadsto c0 \cdot \sqrt{\left(\frac{-1}{\ell} \cdot -1\right) \cdot \color{blue}{\frac{A}{V}}} \]

   4. Applied rewrites 69.3%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{-1}{\ell} \cdot -1\right) \cdot \frac{A}{V}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{-1}{\ell} \cdot -1\right)} \cdot \frac{A}{V}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      3. associate-*l/ N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{-1 \cdot -1}{\ell}} \cdot \frac{A}{V}} \]

      4. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1}}{\ell} \cdot \frac{A}{V}} \]

      5. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\ell} \cdot \frac{A}{V}} \]

      6. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\ell}} \cdot \frac{A}{V}} \]

      7. metadata-eval 69.3

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1}}{\ell} \cdot \frac{A}{V}} \]

   6. Applied rewrites 69.3%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell}} \cdot \frac{A}{V}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{1}{\ell} \cdot \frac{A}{V}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell} \cdot \frac{A}{V}}} \]

      3. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell}} \cdot \frac{A}{V}} \]

      4. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\ell} \cdot \color{blue}{\frac{A}{V}}} \]

      5. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\ell} \cdot \color{blue}{\frac{\mathsf{neg}\left(A\right)}{\mathsf{neg}\left(V\right)}}} \]

      6. lift-neg.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\ell} \cdot \frac{\color{blue}{-A}}{\mathsf{neg}\left(V\right)}} \]

      7. frac-times N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1 \cdot \left(-A\right)}{\ell \cdot \left(\mathsf{neg}\left(V\right)\right)}}} \]

      8. *-lft-identity N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{-A}}{\ell \cdot \left(\mathsf{neg}\left(V\right)\right)}} \]

      9. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{-A}{\ell}}{\mathsf{neg}\left(V\right)}}} \]

      10. sqrt-div N/A

          \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{\mathsf{neg}\left(V\right)}}} \]

      11. pow1/2 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{{\left(\frac{-A}{\ell}\right)}^{\frac{1}{2}}}}{\sqrt{\mathsf{neg}\left(V\right)}} \]

      12. lower-/.f64 N/A

          \[\leadsto c0 \cdot \color{blue}{\frac{{\left(\frac{-A}{\ell}\right)}^{\frac{1}{2}}}{\sqrt{\mathsf{neg}\left(V\right)}}} \]

      13. pow1/2 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\frac{-A}{\ell}}}}{\sqrt{\mathsf{neg}\left(V\right)}} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\frac{-A}{\ell}}}}{\sqrt{\mathsf{neg}\left(V\right)}} \]

      15. lower-/.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{-A}{\ell}}}}{\sqrt{\mathsf{neg}\left(V\right)}} \]

      16. lower-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\color{blue}{\sqrt{\mathsf{neg}\left(V\right)}}} \]

      17. lower-neg.f64 29.5

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{\color{blue}{-V}}} \]

   8. Applied rewrites 29.5%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}} \]

   if -inf.0 < (*.f64 V l) < -1.99999999999999997e-265

   1. Initial program 85.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(\frac{A}{V}\right)}{\mathsf{neg}\left(\ell\right)}}} \]

      5. div-inv N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(\frac{A}{V}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}} \]

      6. *-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot \left(\mathsf{neg}\left(\frac{A}{V}\right)\right)}} \]

      7. neg-mul-1 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot \color{blue}{\left(-1 \cdot \frac{A}{V}\right)}} \]

      8. associate-*r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}}} \]

      9. lower-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}}} \]

      10. lower-*.f64 N/A

          \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right)} \cdot \frac{A}{V}} \]

      11. metadata-eval N/A

          \[\leadsto c0 \cdot \sqrt{\left(\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}} \]

      12. frac-2neg N/A

          \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      13. lower-/.f64 N/A

          \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      14. lower-/.f64 73.1

          \[\leadsto c0 \cdot \sqrt{\left(\frac{-1}{\ell} \cdot -1\right) \cdot \color{blue}{\frac{A}{V}}} \]

   4. Applied rewrites 73.1%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{-1}{\ell} \cdot -1\right) \cdot \frac{A}{V}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{-1}{\ell} \cdot -1\right)} \cdot \frac{A}{V}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      3. associate-*l/ N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{-1 \cdot -1}{\ell}} \cdot \frac{A}{V}} \]

      4. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1}}{\ell} \cdot \frac{A}{V}} \]

      5. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\ell} \cdot \frac{A}{V}} \]

      6. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\ell}} \cdot \frac{A}{V}} \]

      7. metadata-eval 73.1

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1}}{\ell} \cdot \frac{A}{V}} \]

   6. Applied rewrites 73.1%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell}} \cdot \frac{A}{V}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{1}{\ell} \cdot \frac{A}{V}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell} \cdot \frac{A}{V}}} \]

      3. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\ell} \cdot \color{blue}{\frac{A}{V}}} \]

      4. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell}} \cdot \frac{A}{V}} \]

      5. frac-times N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1 \cdot A}{\ell \cdot V}}} \]

      6. *-lft-identity N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{A}}{\ell \cdot V}} \]

      7. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{\ell \cdot V}}} \]

      8. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(A\right)}{\mathsf{neg}\left(\ell \cdot V\right)}}} \]

      9. lift-neg.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{-A}}{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      10. *-rgt-identity N/A

          \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\left(-A\right) \cdot 1}}{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      11. sqrt-div N/A

          \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\left(-A\right) \cdot 1}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}}} \]

      12. *-rgt-identity N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{-A}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      13. pow1/2 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{{\left(-A\right)}^{\frac{1}{2}}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      14. lower-/.f64 N/A

          \[\leadsto c0 \cdot \color{blue}{\frac{{\left(-A\right)}^{\frac{1}{2}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}}} \]

      15. pow1/2 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{-A}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      16. lower-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{-A}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      17. lower-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\color{blue}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}}} \]

      18. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\mathsf{neg}\left(\color{blue}{\ell \cdot V}\right)}} \]

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

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot V}}} \]

      20. lower-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot V}}} \]

      21. lower-neg.f64 99.5

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\left(-\ell\right)} \cdot V}} \]

   8. Applied rewrites 99.5%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}}} \]

   if -1.99999999999999997e-265 < (*.f64 V l) < -0.0

   1. Initial program 40.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \]

      3. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      4. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      5. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      6. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{c0}{\sqrt{\ell}}} \cdot \sqrt{\frac{A}{V}} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{c0}{\color{blue}{\sqrt{\ell}}} \cdot \sqrt{\frac{A}{V}} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\ell}} \cdot \color{blue}{\sqrt{\frac{A}{V}}} \]

      13. lower-/.f64 48.0

          \[\leadsto \frac{c0}{\sqrt{\ell}} \cdot \sqrt{\color{blue}{\frac{A}{V}}} \]

   4. Applied rewrites 48.0%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}} \]

   5. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\left(c0 \cdot \sqrt{\frac{1}{\ell}}\right)} \cdot \sqrt{\frac{A}{V}} \]
   6. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(c0 \cdot \sqrt{\frac{1}{\ell}}\right)\right)\right)\right)} \cdot \sqrt{\frac{A}{V}} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\sqrt{\frac{1}{\ell}} \cdot c0}\right)\right)\right)\right) \cdot \sqrt{\frac{A}{V}} \]

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

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot c0}\right)\right) \cdot \sqrt{\frac{A}{V}} \]

      4. distribute-rgt-neg-out N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(\mathsf{neg}\left(c0\right)\right)\right)} \cdot \sqrt{\frac{A}{V}} \]

      5. neg-mul-1 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot \color{blue}{\left(-1 \cdot c0\right)}\right) \cdot \sqrt{\frac{A}{V}} \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot c0\right)\right) \cdot \sqrt{\frac{A}{V}} \]

      7. associate-*l* N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot \color{blue}{\left(\sqrt{-1} \cdot \left(\sqrt{-1} \cdot c0\right)\right)}\right) \cdot \sqrt{\frac{A}{V}} \]

      8. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(\sqrt{-1} \cdot \color{blue}{\left(c0 \cdot \sqrt{-1}\right)}\right)\right) \cdot \sqrt{\frac{A}{V}} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot \sqrt{-1}\right) \cdot \left(c0 \cdot \sqrt{-1}\right)\right)} \cdot \sqrt{\frac{A}{V}} \]

      10. *-commutative N/A

          \[\leadsto \left(\left(\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot \sqrt{-1}\right) \cdot \color{blue}{\left(\sqrt{-1} \cdot c0\right)}\right) \cdot \sqrt{\frac{A}{V}} \]

      11. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\left(\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot \sqrt{-1}\right) \cdot \sqrt{-1}\right) \cdot c0\right)} \cdot \sqrt{\frac{A}{V}} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\left(\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot \sqrt{-1}\right) \cdot \sqrt{-1}\right) \cdot c0\right)} \cdot \sqrt{\frac{A}{V}} \]

   7. Applied rewrites 48.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell}} \cdot c0\right)} \cdot \sqrt{\frac{A}{V}} \]

   if -0.0 < (*.f64 V l) < 9.99999999999999921e273

   1. Initial program 82.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      3. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{A}}}{\sqrt{V \cdot \ell}} \]

      6. lower-sqrt.f64 99.4

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\color{blue}{\sqrt{V \cdot \ell}}} \]

      7. lift-*.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell}}} \]

      8. *-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

      9. lower-*.f64 99.4

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

   4. Applied rewrites 99.4%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\ell \cdot V}}} \]

   if 9.99999999999999921e273 < (*.f64 V l)

   1. Initial program 52.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 84.6

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   4. Applied rewrites 84.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-265}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \left(\sqrt{\frac{1}{\ell}} \cdot c0\right)\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+274}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}} \cdot c0\\ \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-242}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{-283}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+274}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* (/ (sqrt (/ (- A) l)) (sqrt (- V))) c0)))
   (if (<= (* l V) (- INFINITY))
     t_0
     (if (<= (* l V) -2e-242)
       (* (/ (sqrt (- A)) (sqrt (* (- l) V))) c0)
       (if (<= (* l V) 1e-283)
         t_0
         (if (<= (* l V) 1e+274)
           (* (/ (sqrt A) (sqrt (* l V))) c0)
           (* (sqrt (/ (/ A V) l)) c0)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = (sqrt((-A / l)) / sqrt(-V)) * c0;
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((l * V) <= -2e-242) {
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	} else if ((l * V) <= 1e-283) {
		tmp = t_0;
	} else if ((l * V) <= 1e+274) {
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	} else {
		tmp = sqrt(((A / V) / l)) * c0;
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = (Math.sqrt((-A / l)) / Math.sqrt(-V)) * c0;
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if ((l * V) <= -2e-242) {
		tmp = (Math.sqrt(-A) / Math.sqrt((-l * V))) * c0;
	} else if ((l * V) <= 1e-283) {
		tmp = t_0;
	} else if ((l * V) <= 1e+274) {
		tmp = (Math.sqrt(A) / Math.sqrt((l * V))) * c0;
	} else {
		tmp = Math.sqrt(((A / V) / l)) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = (math.sqrt((-A / l)) / math.sqrt(-V)) * c0
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = t_0
	elif (l * V) <= -2e-242:
		tmp = (math.sqrt(-A) / math.sqrt((-l * V))) * c0
	elif (l * V) <= 1e-283:
		tmp = t_0
	elif (l * V) <= 1e+274:
		tmp = (math.sqrt(A) / math.sqrt((l * V))) * c0
	else:
		tmp = math.sqrt(((A / V) / l)) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(Float64(sqrt(Float64(Float64(-A) / l)) / sqrt(Float64(-V))) * c0)
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(l * V) <= -2e-242)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-l) * V))) * c0);
	elseif (Float64(l * V) <= 1e-283)
		tmp = t_0;
	elseif (Float64(l * V) <= 1e+274)
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(l * V))) * c0);
	else
		tmp = Float64(sqrt(Float64(Float64(A / V) / l)) * c0);
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = (sqrt((-A / l)) / sqrt(-V)) * c0;
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = t_0;
	elseif ((l * V) <= -2e-242)
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	elseif ((l * V) <= 1e-283)
		tmp = t_0;
	elseif ((l * V) <= 1e+274)
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	else
		tmp = sqrt(((A / V) / l)) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(l * V), $MachinePrecision], -2e-242], N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 1e-283], t$95$0, If[LessEqual[N[(l * V), $MachinePrecision], 1e+274], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -inf.0 or -2e-242 < (*.f64 V l) < 9.99999999999999947e-284

   1. Initial program 41.9%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(\frac{A}{V}\right)}{\mathsf{neg}\left(\ell\right)}}} \]

      5. div-inv N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(\frac{A}{V}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}} \]

      6. *-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot \left(\mathsf{neg}\left(\frac{A}{V}\right)\right)}} \]

      7. neg-mul-1 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot \color{blue}{\left(-1 \cdot \frac{A}{V}\right)}} \]

      8. associate-*r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}}} \]

      9. lower-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}}} \]

      10. lower-*.f64 N/A

          \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right)} \cdot \frac{A}{V}} \]

      11. metadata-eval N/A

          \[\leadsto c0 \cdot \sqrt{\left(\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}} \]

      12. frac-2neg N/A

          \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      13. lower-/.f64 N/A

          \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      14. lower-/.f64 68.9

          \[\leadsto c0 \cdot \sqrt{\left(\frac{-1}{\ell} \cdot -1\right) \cdot \color{blue}{\frac{A}{V}}} \]

   4. Applied rewrites 68.9%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{-1}{\ell} \cdot -1\right) \cdot \frac{A}{V}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{-1}{\ell} \cdot -1\right)} \cdot \frac{A}{V}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      3. associate-*l/ N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{-1 \cdot -1}{\ell}} \cdot \frac{A}{V}} \]

      4. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1}}{\ell} \cdot \frac{A}{V}} \]

      5. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\ell} \cdot \frac{A}{V}} \]

      6. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\ell}} \cdot \frac{A}{V}} \]

      7. metadata-eval 68.9

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1}}{\ell} \cdot \frac{A}{V}} \]

   6. Applied rewrites 68.9%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell}} \cdot \frac{A}{V}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{1}{\ell} \cdot \frac{A}{V}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell} \cdot \frac{A}{V}}} \]

      3. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell}} \cdot \frac{A}{V}} \]

      4. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\ell} \cdot \color{blue}{\frac{A}{V}}} \]

      5. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\ell} \cdot \color{blue}{\frac{\mathsf{neg}\left(A\right)}{\mathsf{neg}\left(V\right)}}} \]

      6. lift-neg.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\ell} \cdot \frac{\color{blue}{-A}}{\mathsf{neg}\left(V\right)}} \]

      7. frac-times N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1 \cdot \left(-A\right)}{\ell \cdot \left(\mathsf{neg}\left(V\right)\right)}}} \]

      8. *-lft-identity N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{-A}}{\ell \cdot \left(\mathsf{neg}\left(V\right)\right)}} \]

      9. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{-A}{\ell}}{\mathsf{neg}\left(V\right)}}} \]

      10. sqrt-div N/A

          \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{\mathsf{neg}\left(V\right)}}} \]

      11. pow1/2 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{{\left(\frac{-A}{\ell}\right)}^{\frac{1}{2}}}}{\sqrt{\mathsf{neg}\left(V\right)}} \]

      12. lower-/.f64 N/A

          \[\leadsto c0 \cdot \color{blue}{\frac{{\left(\frac{-A}{\ell}\right)}^{\frac{1}{2}}}{\sqrt{\mathsf{neg}\left(V\right)}}} \]

      13. pow1/2 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\frac{-A}{\ell}}}}{\sqrt{\mathsf{neg}\left(V\right)}} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\frac{-A}{\ell}}}}{\sqrt{\mathsf{neg}\left(V\right)}} \]

      15. lower-/.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{-A}{\ell}}}}{\sqrt{\mathsf{neg}\left(V\right)}} \]

      16. lower-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\color{blue}{\sqrt{\mathsf{neg}\left(V\right)}}} \]

      17. lower-neg.f64 39.4

          \[\leadsto c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{\color{blue}{-V}}} \]

   8. Applied rewrites 39.4%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}} \]

   if -inf.0 < (*.f64 V l) < -2e-242

   1. Initial program 85.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(\frac{A}{V}\right)}{\mathsf{neg}\left(\ell\right)}}} \]

      5. div-inv N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(\frac{A}{V}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}} \]

      6. *-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot \left(\mathsf{neg}\left(\frac{A}{V}\right)\right)}} \]

      7. neg-mul-1 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot \color{blue}{\left(-1 \cdot \frac{A}{V}\right)}} \]

      8. associate-*r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}}} \]

      9. lower-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}}} \]

      10. lower-*.f64 N/A

          \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right)} \cdot \frac{A}{V}} \]

      11. metadata-eval N/A

          \[\leadsto c0 \cdot \sqrt{\left(\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}} \]

      12. frac-2neg N/A

          \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      13. lower-/.f64 N/A

          \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      14. lower-/.f64 73.0

          \[\leadsto c0 \cdot \sqrt{\left(\frac{-1}{\ell} \cdot -1\right) \cdot \color{blue}{\frac{A}{V}}} \]

   4. Applied rewrites 73.0%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{-1}{\ell} \cdot -1\right) \cdot \frac{A}{V}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{-1}{\ell} \cdot -1\right)} \cdot \frac{A}{V}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      3. associate-*l/ N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{-1 \cdot -1}{\ell}} \cdot \frac{A}{V}} \]

      4. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1}}{\ell} \cdot \frac{A}{V}} \]

      5. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\ell} \cdot \frac{A}{V}} \]

      6. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\ell}} \cdot \frac{A}{V}} \]

      7. metadata-eval 73.0

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1}}{\ell} \cdot \frac{A}{V}} \]

   6. Applied rewrites 73.0%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell}} \cdot \frac{A}{V}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{1}{\ell} \cdot \frac{A}{V}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell} \cdot \frac{A}{V}}} \]

      3. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\ell} \cdot \color{blue}{\frac{A}{V}}} \]

      4. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell}} \cdot \frac{A}{V}} \]

      5. frac-times N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1 \cdot A}{\ell \cdot V}}} \]

      6. *-lft-identity N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{A}}{\ell \cdot V}} \]

      7. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{\ell \cdot V}}} \]

      8. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(A\right)}{\mathsf{neg}\left(\ell \cdot V\right)}}} \]

      9. lift-neg.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{-A}}{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      10. *-rgt-identity N/A

          \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\left(-A\right) \cdot 1}}{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      11. sqrt-div N/A

          \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\left(-A\right) \cdot 1}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}}} \]

      12. *-rgt-identity N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{-A}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      13. pow1/2 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{{\left(-A\right)}^{\frac{1}{2}}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      14. lower-/.f64 N/A

          \[\leadsto c0 \cdot \color{blue}{\frac{{\left(-A\right)}^{\frac{1}{2}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}}} \]

      15. pow1/2 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{-A}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      16. lower-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{-A}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      17. lower-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\color{blue}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}}} \]

      18. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\mathsf{neg}\left(\color{blue}{\ell \cdot V}\right)}} \]

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

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot V}}} \]

      20. lower-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot V}}} \]

      21. lower-neg.f64 99.5

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\left(-\ell\right)} \cdot V}} \]

   8. Applied rewrites 99.5%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}}} \]

   if 9.99999999999999947e-284 < (*.f64 V l) < 9.99999999999999921e273

   1. Initial program 82.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      3. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{A}}}{\sqrt{V \cdot \ell}} \]

      6. lower-sqrt.f64 99.4

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\color{blue}{\sqrt{V \cdot \ell}}} \]

      7. lift-*.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell}}} \]

      8. *-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

      9. lower-*.f64 99.4

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

   4. Applied rewrites 99.4%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\ell \cdot V}}} \]

   if 9.99999999999999921e273 < (*.f64 V l)

   1. Initial program 52.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 84.6

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   4. Applied rewrites 84.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-242}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{-283}:\\ \;\;\;\;\frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+274}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{t\_0 \cdot c0}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq -1 \cdot 10^{-211}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{t\_0}{\sqrt{\ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+274}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (/ A V))))
   (if (<= (* l V) (- INFINITY))
     (/ (* t_0 c0) (sqrt l))
     (if (<= (* l V) -1e-211)
       (* (/ (sqrt (- A)) (sqrt (* (- l) V))) c0)
       (if (<= (* l V) 0.0)
         (* (/ t_0 (sqrt l)) c0)
         (if (<= (* l V) 1e+274)
           (* (/ (sqrt A) (sqrt (* l V))) c0)
           (* (sqrt (/ (/ A V) l)) c0)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / V));
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = (t_0 * c0) / sqrt(l);
	} else if ((l * V) <= -1e-211) {
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	} else if ((l * V) <= 0.0) {
		tmp = (t_0 / sqrt(l)) * c0;
	} else if ((l * V) <= 1e+274) {
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	} else {
		tmp = sqrt(((A / V) / l)) * c0;
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / V));
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = (t_0 * c0) / Math.sqrt(l);
	} else if ((l * V) <= -1e-211) {
		tmp = (Math.sqrt(-A) / Math.sqrt((-l * V))) * c0;
	} else if ((l * V) <= 0.0) {
		tmp = (t_0 / Math.sqrt(l)) * c0;
	} else if ((l * V) <= 1e+274) {
		tmp = (Math.sqrt(A) / Math.sqrt((l * V))) * c0;
	} else {
		tmp = Math.sqrt(((A / V) / l)) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / V))
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = (t_0 * c0) / math.sqrt(l)
	elif (l * V) <= -1e-211:
		tmp = (math.sqrt(-A) / math.sqrt((-l * V))) * c0
	elif (l * V) <= 0.0:
		tmp = (t_0 / math.sqrt(l)) * c0
	elif (l * V) <= 1e+274:
		tmp = (math.sqrt(A) / math.sqrt((l * V))) * c0
	else:
		tmp = math.sqrt(((A / V) / l)) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(A / V))
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = Float64(Float64(t_0 * c0) / sqrt(l));
	elseif (Float64(l * V) <= -1e-211)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-l) * V))) * c0);
	elseif (Float64(l * V) <= 0.0)
		tmp = Float64(Float64(t_0 / sqrt(l)) * c0);
	elseif (Float64(l * V) <= 1e+274)
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(l * V))) * c0);
	else
		tmp = Float64(sqrt(Float64(Float64(A / V) / l)) * c0);
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((A / V));
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = (t_0 * c0) / sqrt(l);
	elseif ((l * V) <= -1e-211)
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	elseif ((l * V) <= 0.0)
		tmp = (t_0 / sqrt(l)) * c0;
	elseif ((l * V) <= 1e+274)
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	else
		tmp = sqrt(((A / V) / l)) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], N[(N[(t$95$0 * c0), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -1e-211], N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], N[(N[(t$95$0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 1e+274], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -inf.0

   1. Initial program 35.9%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \]

      3. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      4. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      5. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      6. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{A}{V}} \cdot c0}}{\sqrt{\ell}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\frac{A}{V}} \cdot c0}}{\sqrt{\ell}} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\frac{A}{V}}} \cdot c0}{\sqrt{\ell}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\frac{A}{V}}} \cdot c0}{\sqrt{\ell}} \]

      13. lower-sqrt.f64 25.7

          \[\leadsto \frac{\sqrt{\frac{A}{V}} \cdot c0}{\color{blue}{\sqrt{\ell}}} \]

   4. Applied rewrites 25.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{A}{V}} \cdot c0}{\sqrt{\ell}}} \]

   if -inf.0 < (*.f64 V l) < -1.00000000000000009e-211

   1. Initial program 86.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(\frac{A}{V}\right)}{\mathsf{neg}\left(\ell\right)}}} \]

      5. div-inv N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(\frac{A}{V}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}} \]

      6. *-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot \left(\mathsf{neg}\left(\frac{A}{V}\right)\right)}} \]

      7. neg-mul-1 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot \color{blue}{\left(-1 \cdot \frac{A}{V}\right)}} \]

      8. associate-*r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}}} \]

      9. lower-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}}} \]

      10. lower-*.f64 N/A

          \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right)} \cdot \frac{A}{V}} \]

      11. metadata-eval N/A

          \[\leadsto c0 \cdot \sqrt{\left(\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}} \]

      12. frac-2neg N/A

          \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      13. lower-/.f64 N/A

          \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      14. lower-/.f64 72.8

          \[\leadsto c0 \cdot \sqrt{\left(\frac{-1}{\ell} \cdot -1\right) \cdot \color{blue}{\frac{A}{V}}} \]

   4. Applied rewrites 72.8%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{-1}{\ell} \cdot -1\right) \cdot \frac{A}{V}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{-1}{\ell} \cdot -1\right)} \cdot \frac{A}{V}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      3. associate-*l/ N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{-1 \cdot -1}{\ell}} \cdot \frac{A}{V}} \]

      4. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1}}{\ell} \cdot \frac{A}{V}} \]

      5. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\ell} \cdot \frac{A}{V}} \]

      6. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\ell}} \cdot \frac{A}{V}} \]

      7. metadata-eval 72.8

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1}}{\ell} \cdot \frac{A}{V}} \]

   6. Applied rewrites 72.8%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell}} \cdot \frac{A}{V}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{1}{\ell} \cdot \frac{A}{V}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell} \cdot \frac{A}{V}}} \]

      3. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\ell} \cdot \color{blue}{\frac{A}{V}}} \]

      4. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell}} \cdot \frac{A}{V}} \]

      5. frac-times N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1 \cdot A}{\ell \cdot V}}} \]

      6. *-lft-identity N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{A}}{\ell \cdot V}} \]

      7. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{\ell \cdot V}}} \]

      8. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(A\right)}{\mathsf{neg}\left(\ell \cdot V\right)}}} \]

      9. lift-neg.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{-A}}{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      10. *-rgt-identity N/A

          \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\left(-A\right) \cdot 1}}{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      11. sqrt-div N/A

          \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\left(-A\right) \cdot 1}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}}} \]

      12. *-rgt-identity N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{-A}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      13. pow1/2 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{{\left(-A\right)}^{\frac{1}{2}}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      14. lower-/.f64 N/A

          \[\leadsto c0 \cdot \color{blue}{\frac{{\left(-A\right)}^{\frac{1}{2}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}}} \]

      15. pow1/2 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{-A}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      16. lower-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{-A}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      17. lower-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\color{blue}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}}} \]

      18. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\mathsf{neg}\left(\color{blue}{\ell \cdot V}\right)}} \]

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

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot V}}} \]

      20. lower-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot V}}} \]

      21. lower-neg.f64 99.5

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\left(-\ell\right)} \cdot V}} \]

   8. Applied rewrites 99.5%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}}} \]

   if -1.00000000000000009e-211 < (*.f64 V l) < -0.0

   1. Initial program 48.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      3. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      4. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

      6. lower-/.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\frac{A}{V}}}}{\sqrt{\ell}} \]

      8. lower-/.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A}{V}}}}{\sqrt{\ell}} \]

      9. lower-sqrt.f64 43.0

         \[\leadsto c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\color{blue}{\sqrt{\ell}}} \]

   4. Applied rewrites 43.0%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

   if -0.0 < (*.f64 V l) < 9.99999999999999921e273

   1. Initial program 82.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      3. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{A}}}{\sqrt{V \cdot \ell}} \]

      6. lower-sqrt.f64 99.4

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\color{blue}{\sqrt{V \cdot \ell}}} \]

      7. lift-*.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell}}} \]

      8. *-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

      9. lower-*.f64 99.4

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

   4. Applied rewrites 99.4%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\ell \cdot V}}} \]

   if 9.99999999999999921e273 < (*.f64 V l)

   1. Initial program 52.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 84.6

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   4. Applied rewrites 84.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}} \cdot c0}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq -1 \cdot 10^{-211}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+274}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot t\_0\\ \mathbf{elif}\;\ell \cdot V \leq -1 \cdot 10^{-211}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{t\_0}{\sqrt{\ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+274}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (/ A V))))
   (if (<= (* l V) (- INFINITY))
     (* (/ c0 (sqrt l)) t_0)
     (if (<= (* l V) -1e-211)
       (* (/ (sqrt (- A)) (sqrt (* (- l) V))) c0)
       (if (<= (* l V) 0.0)
         (* (/ t_0 (sqrt l)) c0)
         (if (<= (* l V) 1e+274)
           (* (/ (sqrt A) (sqrt (* l V))) c0)
           (* (sqrt (/ (/ A V) l)) c0)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / V));
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = (c0 / sqrt(l)) * t_0;
	} else if ((l * V) <= -1e-211) {
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	} else if ((l * V) <= 0.0) {
		tmp = (t_0 / sqrt(l)) * c0;
	} else if ((l * V) <= 1e+274) {
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	} else {
		tmp = sqrt(((A / V) / l)) * c0;
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / V));
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = (c0 / Math.sqrt(l)) * t_0;
	} else if ((l * V) <= -1e-211) {
		tmp = (Math.sqrt(-A) / Math.sqrt((-l * V))) * c0;
	} else if ((l * V) <= 0.0) {
		tmp = (t_0 / Math.sqrt(l)) * c0;
	} else if ((l * V) <= 1e+274) {
		tmp = (Math.sqrt(A) / Math.sqrt((l * V))) * c0;
	} else {
		tmp = Math.sqrt(((A / V) / l)) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / V))
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = (c0 / math.sqrt(l)) * t_0
	elif (l * V) <= -1e-211:
		tmp = (math.sqrt(-A) / math.sqrt((-l * V))) * c0
	elif (l * V) <= 0.0:
		tmp = (t_0 / math.sqrt(l)) * c0
	elif (l * V) <= 1e+274:
		tmp = (math.sqrt(A) / math.sqrt((l * V))) * c0
	else:
		tmp = math.sqrt(((A / V) / l)) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(A / V))
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = Float64(Float64(c0 / sqrt(l)) * t_0);
	elseif (Float64(l * V) <= -1e-211)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-l) * V))) * c0);
	elseif (Float64(l * V) <= 0.0)
		tmp = Float64(Float64(t_0 / sqrt(l)) * c0);
	elseif (Float64(l * V) <= 1e+274)
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(l * V))) * c0);
	else
		tmp = Float64(sqrt(Float64(Float64(A / V) / l)) * c0);
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((A / V));
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = (c0 / sqrt(l)) * t_0;
	elseif ((l * V) <= -1e-211)
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	elseif ((l * V) <= 0.0)
		tmp = (t_0 / sqrt(l)) * c0;
	elseif ((l * V) <= 1e+274)
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	else
		tmp = sqrt(((A / V) / l)) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -1e-211], N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], N[(N[(t$95$0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 1e+274], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -inf.0

   1. Initial program 35.9%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \]

      3. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      4. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      5. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      6. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{c0}{\sqrt{\ell}}} \cdot \sqrt{\frac{A}{V}} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{c0}{\color{blue}{\sqrt{\ell}}} \cdot \sqrt{\frac{A}{V}} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\ell}} \cdot \color{blue}{\sqrt{\frac{A}{V}}} \]

      13. lower-/.f64 25.7

          \[\leadsto \frac{c0}{\sqrt{\ell}} \cdot \sqrt{\color{blue}{\frac{A}{V}}} \]

   4. Applied rewrites 25.7%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}} \]

   if -inf.0 < (*.f64 V l) < -1.00000000000000009e-211

   1. Initial program 86.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(\frac{A}{V}\right)}{\mathsf{neg}\left(\ell\right)}}} \]

      5. div-inv N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(\frac{A}{V}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}} \]

      6. *-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot \left(\mathsf{neg}\left(\frac{A}{V}\right)\right)}} \]

      7. neg-mul-1 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot \color{blue}{\left(-1 \cdot \frac{A}{V}\right)}} \]

      8. associate-*r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}}} \]

      9. lower-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}}} \]

      10. lower-*.f64 N/A

          \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right)} \cdot \frac{A}{V}} \]

      11. metadata-eval N/A

          \[\leadsto c0 \cdot \sqrt{\left(\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}} \]

      12. frac-2neg N/A

          \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      13. lower-/.f64 N/A

          \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      14. lower-/.f64 72.8

          \[\leadsto c0 \cdot \sqrt{\left(\frac{-1}{\ell} \cdot -1\right) \cdot \color{blue}{\frac{A}{V}}} \]

   4. Applied rewrites 72.8%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{-1}{\ell} \cdot -1\right) \cdot \frac{A}{V}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{-1}{\ell} \cdot -1\right)} \cdot \frac{A}{V}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      3. associate-*l/ N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{-1 \cdot -1}{\ell}} \cdot \frac{A}{V}} \]

      4. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1}}{\ell} \cdot \frac{A}{V}} \]

      5. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\ell} \cdot \frac{A}{V}} \]

      6. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\ell}} \cdot \frac{A}{V}} \]

      7. metadata-eval 72.8

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1}}{\ell} \cdot \frac{A}{V}} \]

   6. Applied rewrites 72.8%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell}} \cdot \frac{A}{V}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{1}{\ell} \cdot \frac{A}{V}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell} \cdot \frac{A}{V}}} \]

      3. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\ell} \cdot \color{blue}{\frac{A}{V}}} \]

      4. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell}} \cdot \frac{A}{V}} \]

      5. frac-times N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1 \cdot A}{\ell \cdot V}}} \]

      6. *-lft-identity N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{A}}{\ell \cdot V}} \]

      7. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{\ell \cdot V}}} \]

      8. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(A\right)}{\mathsf{neg}\left(\ell \cdot V\right)}}} \]

      9. lift-neg.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{-A}}{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      10. *-rgt-identity N/A

          \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\left(-A\right) \cdot 1}}{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      11. sqrt-div N/A

          \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\left(-A\right) \cdot 1}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}}} \]

      12. *-rgt-identity N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{-A}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      13. pow1/2 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{{\left(-A\right)}^{\frac{1}{2}}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      14. lower-/.f64 N/A

          \[\leadsto c0 \cdot \color{blue}{\frac{{\left(-A\right)}^{\frac{1}{2}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}}} \]

      15. pow1/2 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{-A}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      16. lower-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{-A}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      17. lower-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\color{blue}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}}} \]

      18. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\mathsf{neg}\left(\color{blue}{\ell \cdot V}\right)}} \]

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

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot V}}} \]

      20. lower-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot V}}} \]

      21. lower-neg.f64 99.5

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\left(-\ell\right)} \cdot V}} \]

   8. Applied rewrites 99.5%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}}} \]

   if -1.00000000000000009e-211 < (*.f64 V l) < -0.0

   1. Initial program 48.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      3. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      4. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

      6. lower-/.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\frac{A}{V}}}}{\sqrt{\ell}} \]

      8. lower-/.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A}{V}}}}{\sqrt{\ell}} \]

      9. lower-sqrt.f64 43.0

         \[\leadsto c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\color{blue}{\sqrt{\ell}}} \]

   4. Applied rewrites 43.0%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

   if -0.0 < (*.f64 V l) < 9.99999999999999921e273

   1. Initial program 82.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      3. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{A}}}{\sqrt{V \cdot \ell}} \]

      6. lower-sqrt.f64 99.4

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\color{blue}{\sqrt{V \cdot \ell}}} \]

      7. lift-*.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell}}} \]

      8. *-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

      9. lower-*.f64 99.4

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

   4. Applied rewrites 99.4%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\ell \cdot V}}} \]

   if 9.99999999999999921e273 < (*.f64 V l)

   1. Initial program 52.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 84.6

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   4. Applied rewrites 84.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;\ell \cdot V \leq -1 \cdot 10^{-211}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+274}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq -1 \cdot 10^{-211}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+274}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* (/ (sqrt (/ A V)) (sqrt l)) c0)))
   (if (<= (* l V) (- INFINITY))
     t_0
     (if (<= (* l V) -1e-211)
       (* (/ (sqrt (- A)) (sqrt (* (- l) V))) c0)
       (if (<= (* l V) 0.0)
         t_0
         (if (<= (* l V) 1e+274)
           (* (/ (sqrt A) (sqrt (* l V))) c0)
           (* (sqrt (/ (/ A V) l)) c0)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = (sqrt((A / V)) / sqrt(l)) * c0;
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((l * V) <= -1e-211) {
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	} else if ((l * V) <= 0.0) {
		tmp = t_0;
	} else if ((l * V) <= 1e+274) {
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	} else {
		tmp = sqrt(((A / V) / l)) * c0;
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = (Math.sqrt((A / V)) / Math.sqrt(l)) * c0;
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if ((l * V) <= -1e-211) {
		tmp = (Math.sqrt(-A) / Math.sqrt((-l * V))) * c0;
	} else if ((l * V) <= 0.0) {
		tmp = t_0;
	} else if ((l * V) <= 1e+274) {
		tmp = (Math.sqrt(A) / Math.sqrt((l * V))) * c0;
	} else {
		tmp = Math.sqrt(((A / V) / l)) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = (math.sqrt((A / V)) / math.sqrt(l)) * c0
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = t_0
	elif (l * V) <= -1e-211:
		tmp = (math.sqrt(-A) / math.sqrt((-l * V))) * c0
	elif (l * V) <= 0.0:
		tmp = t_0
	elif (l * V) <= 1e+274:
		tmp = (math.sqrt(A) / math.sqrt((l * V))) * c0
	else:
		tmp = math.sqrt(((A / V) / l)) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(Float64(sqrt(Float64(A / V)) / sqrt(l)) * c0)
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(l * V) <= -1e-211)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-l) * V))) * c0);
	elseif (Float64(l * V) <= 0.0)
		tmp = t_0;
	elseif (Float64(l * V) <= 1e+274)
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(l * V))) * c0);
	else
		tmp = Float64(sqrt(Float64(Float64(A / V) / l)) * c0);
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = (sqrt((A / V)) / sqrt(l)) * c0;
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = t_0;
	elseif ((l * V) <= -1e-211)
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	elseif ((l * V) <= 0.0)
		tmp = t_0;
	elseif ((l * V) <= 1e+274)
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	else
		tmp = sqrt(((A / V) / l)) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(l * V), $MachinePrecision], -1e-211], N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(l * V), $MachinePrecision], 1e+274], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -inf.0 or -1.00000000000000009e-211 < (*.f64 V l) < -0.0

   1. Initial program 43.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      3. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      4. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

      6. lower-/.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\frac{A}{V}}}}{\sqrt{\ell}} \]

      8. lower-/.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{A}{V}}}}{\sqrt{\ell}} \]

      9. lower-sqrt.f64 36.0

         \[\leadsto c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\color{blue}{\sqrt{\ell}}} \]

   4. Applied rewrites 36.0%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

   if -inf.0 < (*.f64 V l) < -1.00000000000000009e-211

   1. Initial program 86.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(\frac{A}{V}\right)}{\mathsf{neg}\left(\ell\right)}}} \]

      5. div-inv N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(\frac{A}{V}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}} \]

      6. *-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot \left(\mathsf{neg}\left(\frac{A}{V}\right)\right)}} \]

      7. neg-mul-1 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot \color{blue}{\left(-1 \cdot \frac{A}{V}\right)}} \]

      8. associate-*r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}}} \]

      9. lower-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}}} \]

      10. lower-*.f64 N/A

          \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right)} \cdot \frac{A}{V}} \]

      11. metadata-eval N/A

          \[\leadsto c0 \cdot \sqrt{\left(\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}} \]

      12. frac-2neg N/A

          \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      13. lower-/.f64 N/A

          \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      14. lower-/.f64 72.8

          \[\leadsto c0 \cdot \sqrt{\left(\frac{-1}{\ell} \cdot -1\right) \cdot \color{blue}{\frac{A}{V}}} \]

   4. Applied rewrites 72.8%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{-1}{\ell} \cdot -1\right) \cdot \frac{A}{V}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{-1}{\ell} \cdot -1\right)} \cdot \frac{A}{V}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      3. associate-*l/ N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{-1 \cdot -1}{\ell}} \cdot \frac{A}{V}} \]

      4. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1}}{\ell} \cdot \frac{A}{V}} \]

      5. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\ell} \cdot \frac{A}{V}} \]

      6. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\ell}} \cdot \frac{A}{V}} \]

      7. metadata-eval 72.8

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1}}{\ell} \cdot \frac{A}{V}} \]

   6. Applied rewrites 72.8%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell}} \cdot \frac{A}{V}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{1}{\ell} \cdot \frac{A}{V}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell} \cdot \frac{A}{V}}} \]

      3. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\ell} \cdot \color{blue}{\frac{A}{V}}} \]

      4. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell}} \cdot \frac{A}{V}} \]

      5. frac-times N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1 \cdot A}{\ell \cdot V}}} \]

      6. *-lft-identity N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{A}}{\ell \cdot V}} \]

      7. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{\ell \cdot V}}} \]

      8. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(A\right)}{\mathsf{neg}\left(\ell \cdot V\right)}}} \]

      9. lift-neg.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{-A}}{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      10. *-rgt-identity N/A

          \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\left(-A\right) \cdot 1}}{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      11. sqrt-div N/A

          \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\left(-A\right) \cdot 1}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}}} \]

      12. *-rgt-identity N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{-A}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      13. pow1/2 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{{\left(-A\right)}^{\frac{1}{2}}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      14. lower-/.f64 N/A

          \[\leadsto c0 \cdot \color{blue}{\frac{{\left(-A\right)}^{\frac{1}{2}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}}} \]

      15. pow1/2 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{-A}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      16. lower-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{-A}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      17. lower-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\color{blue}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}}} \]

      18. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\mathsf{neg}\left(\color{blue}{\ell \cdot V}\right)}} \]

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

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot V}}} \]

      20. lower-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot V}}} \]

      21. lower-neg.f64 99.5

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\left(-\ell\right)} \cdot V}} \]

   8. Applied rewrites 99.5%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}}} \]

   if -0.0 < (*.f64 V l) < 9.99999999999999921e273

   1. Initial program 82.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      3. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{A}}}{\sqrt{V \cdot \ell}} \]

      6. lower-sqrt.f64 99.4

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\color{blue}{\sqrt{V \cdot \ell}}} \]

      7. lift-*.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell}}} \]

      8. *-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

      9. lower-*.f64 99.4

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

   4. Applied rewrites 99.4%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\ell \cdot V}}} \]

   if 9.99999999999999921e273 < (*.f64 V l)

   1. Initial program 52.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 84.6

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   4. Applied rewrites 84.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq -1 \cdot 10^{-211}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+274}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-322}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{-283}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+274}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* (sqrt (/ (/ A V) l)) c0)))
   (if (<= (* l V) (- INFINITY))
     t_0
     (if (<= (* l V) -5e-322)
       (* (/ (sqrt (- A)) (sqrt (* (- l) V))) c0)
       (if (<= (* l V) 1e-283)
         (/ c0 (sqrt (* (/ V A) l)))
         (if (<= (* l V) 1e+274) (* (/ (sqrt A) (sqrt (* l V))) c0) t_0))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt(((A / V) / l)) * c0;
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((l * V) <= -5e-322) {
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	} else if ((l * V) <= 1e-283) {
		tmp = c0 / sqrt(((V / A) * l));
	} else if ((l * V) <= 1e+274) {
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt(((A / V) / l)) * c0;
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if ((l * V) <= -5e-322) {
		tmp = (Math.sqrt(-A) / Math.sqrt((-l * V))) * c0;
	} else if ((l * V) <= 1e-283) {
		tmp = c0 / Math.sqrt(((V / A) * l));
	} else if ((l * V) <= 1e+274) {
		tmp = (Math.sqrt(A) / Math.sqrt((l * V))) * c0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt(((A / V) / l)) * c0
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = t_0
	elif (l * V) <= -5e-322:
		tmp = (math.sqrt(-A) / math.sqrt((-l * V))) * c0
	elif (l * V) <= 1e-283:
		tmp = c0 / math.sqrt(((V / A) * l))
	elif (l * V) <= 1e+274:
		tmp = (math.sqrt(A) / math.sqrt((l * V))) * c0
	else:
		tmp = t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(sqrt(Float64(Float64(A / V) / l)) * c0)
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(l * V) <= -5e-322)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-l) * V))) * c0);
	elseif (Float64(l * V) <= 1e-283)
		tmp = Float64(c0 / sqrt(Float64(Float64(V / A) * l)));
	elseif (Float64(l * V) <= 1e+274)
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(l * V))) * c0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt(((A / V) / l)) * c0;
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = t_0;
	elseif ((l * V) <= -5e-322)
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	elseif ((l * V) <= 1e-283)
		tmp = c0 / sqrt(((V / A) * l));
	elseif ((l * V) <= 1e+274)
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(l * V), $MachinePrecision], -5e-322], N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 1e-283], N[(c0 / N[Sqrt[N[(N[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 1e+274], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -inf.0 or 9.99999999999999921e273 < (*.f64 V l)

   1. Initial program 43.0%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 75.7

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   4. Applied rewrites 75.7%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

   if -inf.0 < (*.f64 V l) < -4.99006e-322

   1. Initial program 84.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(\frac{A}{V}\right)}{\mathsf{neg}\left(\ell\right)}}} \]

      5. div-inv N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(\frac{A}{V}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}} \]

      6. *-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot \left(\mathsf{neg}\left(\frac{A}{V}\right)\right)}} \]

      7. neg-mul-1 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot \color{blue}{\left(-1 \cdot \frac{A}{V}\right)}} \]

      8. associate-*r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}}} \]

      9. lower-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}}} \]

      10. lower-*.f64 N/A

          \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right)} \cdot \frac{A}{V}} \]

      11. metadata-eval N/A

          \[\leadsto c0 \cdot \sqrt{\left(\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}} \]

      12. frac-2neg N/A

          \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      13. lower-/.f64 N/A

          \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      14. lower-/.f64 73.0

          \[\leadsto c0 \cdot \sqrt{\left(\frac{-1}{\ell} \cdot -1\right) \cdot \color{blue}{\frac{A}{V}}} \]

   4. Applied rewrites 73.0%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{-1}{\ell} \cdot -1\right) \cdot \frac{A}{V}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{-1}{\ell} \cdot -1\right)} \cdot \frac{A}{V}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      3. associate-*l/ N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{-1 \cdot -1}{\ell}} \cdot \frac{A}{V}} \]

      4. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1}}{\ell} \cdot \frac{A}{V}} \]

      5. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\ell} \cdot \frac{A}{V}} \]

      6. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\ell}} \cdot \frac{A}{V}} \]

      7. metadata-eval 73.0

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1}}{\ell} \cdot \frac{A}{V}} \]

   6. Applied rewrites 73.0%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell}} \cdot \frac{A}{V}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{1}{\ell} \cdot \frac{A}{V}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell} \cdot \frac{A}{V}}} \]

      3. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\ell} \cdot \color{blue}{\frac{A}{V}}} \]

      4. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell}} \cdot \frac{A}{V}} \]

      5. frac-times N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1 \cdot A}{\ell \cdot V}}} \]

      6. *-lft-identity N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{A}}{\ell \cdot V}} \]

      7. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{\ell \cdot V}}} \]

      8. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(A\right)}{\mathsf{neg}\left(\ell \cdot V\right)}}} \]

      9. lift-neg.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{-A}}{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      10. *-rgt-identity N/A

          \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\left(-A\right) \cdot 1}}{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      11. sqrt-div N/A

          \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\left(-A\right) \cdot 1}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}}} \]

      12. *-rgt-identity N/A

          \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{-A}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      13. pow1/2 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{{\left(-A\right)}^{\frac{1}{2}}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      14. lower-/.f64 N/A

          \[\leadsto c0 \cdot \color{blue}{\frac{{\left(-A\right)}^{\frac{1}{2}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}}} \]

      15. pow1/2 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{-A}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      16. lower-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{-A}}}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}} \]

      17. lower-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\color{blue}{\sqrt{\mathsf{neg}\left(\ell \cdot V\right)}}} \]

      18. lift-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\mathsf{neg}\left(\color{blue}{\ell \cdot V}\right)}} \]

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

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot V}}} \]

      20. lower-*.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(\ell\right)\right) \cdot V}}} \]

      21. lower-neg.f64 98.5

          \[\leadsto c0 \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{\left(-\ell\right)} \cdot V}} \]

   8. Applied rewrites 98.5%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}}} \]

   if -4.99006e-322 < (*.f64 V l) < 9.99999999999999947e-284

   1. Initial program 33.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \]

      3. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      4. clear-num N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\frac{V \cdot \ell}{A}}}} \]

      5. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      6. metadata-eval N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{1}}{\sqrt{\frac{V \cdot \ell}{A}}} \]

      7. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      10. lower-/.f64 33.4

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V \cdot \ell}{A}}}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell}}{A}}} \]

      12. *-commutative N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\ell \cdot V}}{A}}} \]

      13. lower-*.f64 33.4

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\ell \cdot V}}{A}}} \]

   4. Applied rewrites 33.4%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{\ell \cdot V}{A}}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\ell \cdot V}{A}}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\ell \cdot V}}{A}}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\ell \cdot \frac{V}{A}}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{c0}{\sqrt{\ell \cdot \color{blue}{\frac{V}{A}}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V}{A} \cdot \ell}}} \]

      6. lower-*.f64 66.6

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V}{A} \cdot \ell}}} \]

   6. Applied rewrites 66.6%

      \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V}{A} \cdot \ell}}} \]

   if 9.99999999999999947e-284 < (*.f64 V l) < 9.99999999999999921e273

   1. Initial program 82.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      3. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{A}}}{\sqrt{V \cdot \ell}} \]

      6. lower-sqrt.f64 99.4

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\color{blue}{\sqrt{V \cdot \ell}}} \]

      7. lift-*.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell}}} \]

      8. *-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

      9. lower-*.f64 99.4

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

   4. Applied rewrites 99.4%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\ell \cdot V}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-322}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{-283}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+274}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -2 \cdot 10^{-228}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{-A}}{\sqrt{\ell} \cdot \sqrt{-V}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \left(\sqrt{\frac{1}{\ell}} \cdot c0\right)\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+274}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) -2e-228)
   (/ (* c0 (sqrt (- A))) (* (sqrt l) (sqrt (- V))))
   (if (<= (* l V) 0.0)
     (* (sqrt (/ A V)) (* (sqrt (/ 1.0 l)) c0))
     (if (<= (* l V) 1e+274)
       (* (/ (sqrt A) (sqrt (* l V))) c0)
       (* (sqrt (/ (/ A V) l)) c0)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -2e-228) {
		tmp = (c0 * sqrt(-A)) / (sqrt(l) * sqrt(-V));
	} else if ((l * V) <= 0.0) {
		tmp = sqrt((A / V)) * (sqrt((1.0 / l)) * c0);
	} else if ((l * V) <= 1e+274) {
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	} else {
		tmp = sqrt(((A / V) / l)) * c0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((l * v) <= (-2d-228)) then
        tmp = (c0 * sqrt(-a)) / (sqrt(l) * sqrt(-v))
    else if ((l * v) <= 0.0d0) then
        tmp = sqrt((a / v)) * (sqrt((1.0d0 / l)) * c0)
    else if ((l * v) <= 1d+274) then
        tmp = (sqrt(a) / sqrt((l * v))) * c0
    else
        tmp = sqrt(((a / v) / l)) * c0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -2e-228) {
		tmp = (c0 * Math.sqrt(-A)) / (Math.sqrt(l) * Math.sqrt(-V));
	} else if ((l * V) <= 0.0) {
		tmp = Math.sqrt((A / V)) * (Math.sqrt((1.0 / l)) * c0);
	} else if ((l * V) <= 1e+274) {
		tmp = (Math.sqrt(A) / Math.sqrt((l * V))) * c0;
	} else {
		tmp = Math.sqrt(((A / V) / l)) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -2e-228:
		tmp = (c0 * math.sqrt(-A)) / (math.sqrt(l) * math.sqrt(-V))
	elif (l * V) <= 0.0:
		tmp = math.sqrt((A / V)) * (math.sqrt((1.0 / l)) * c0)
	elif (l * V) <= 1e+274:
		tmp = (math.sqrt(A) / math.sqrt((l * V))) * c0
	else:
		tmp = math.sqrt(((A / V) / l)) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= -2e-228)
		tmp = Float64(Float64(c0 * sqrt(Float64(-A))) / Float64(sqrt(l) * sqrt(Float64(-V))));
	elseif (Float64(l * V) <= 0.0)
		tmp = Float64(sqrt(Float64(A / V)) * Float64(sqrt(Float64(1.0 / l)) * c0));
	elseif (Float64(l * V) <= 1e+274)
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(l * V))) * c0);
	else
		tmp = Float64(sqrt(Float64(Float64(A / V) / l)) * c0);
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -2e-228)
		tmp = (c0 * sqrt(-A)) / (sqrt(l) * sqrt(-V));
	elseif ((l * V) <= 0.0)
		tmp = sqrt((A / V)) * (sqrt((1.0 / l)) * c0);
	elseif ((l * V) <= 1e+274)
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	else
		tmp = sqrt(((A / V) / l)) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], -2e-228], N[(N[(c0 * N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 1e+274], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -2.00000000000000007e-228

   1. Initial program 74.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(\frac{A}{V}\right)}{\mathsf{neg}\left(\ell\right)}}} \]

      5. div-inv N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(\frac{A}{V}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}} \]

      6. *-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot \left(\mathsf{neg}\left(\frac{A}{V}\right)\right)}} \]

      7. neg-mul-1 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot \color{blue}{\left(-1 \cdot \frac{A}{V}\right)}} \]

      8. associate-*r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}}} \]

      9. lower-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}}} \]

      10. lower-*.f64 N/A

          \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right)} \cdot \frac{A}{V}} \]

      11. metadata-eval N/A

          \[\leadsto c0 \cdot \sqrt{\left(\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}} \]

      12. frac-2neg N/A

          \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      13. lower-/.f64 N/A

          \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      14. lower-/.f64 72.5

          \[\leadsto c0 \cdot \sqrt{\left(\frac{-1}{\ell} \cdot -1\right) \cdot \color{blue}{\frac{A}{V}}} \]

   4. Applied rewrites 72.5%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{-1}{\ell} \cdot -1\right) \cdot \frac{A}{V}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{-1}{\ell} \cdot -1\right)} \cdot \frac{A}{V}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      3. associate-*l/ N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{-1 \cdot -1}{\ell}} \cdot \frac{A}{V}} \]

      4. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1}}{\ell} \cdot \frac{A}{V}} \]

      5. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\ell} \cdot \frac{A}{V}} \]

      6. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\ell}} \cdot \frac{A}{V}} \]

      7. metadata-eval 72.5

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1}}{\ell} \cdot \frac{A}{V}} \]

   6. Applied rewrites 72.5%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell}} \cdot \frac{A}{V}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{1}{\ell} \cdot \frac{A}{V}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{1}{\ell} \cdot \frac{A}{V}}} \]

      3. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell} \cdot \frac{A}{V}}} \]

      4. sqrt-prod N/A

         \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\frac{1}{\ell}} \cdot \sqrt{\frac{A}{V}}\right)} \]

      5. pow1/2 N/A

         \[\leadsto c0 \cdot \left(\color{blue}{{\left(\frac{1}{\ell}\right)}^{\frac{1}{2}}} \cdot \sqrt{\frac{A}{V}}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \left({\left(\frac{1}{\ell}\right)}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{\frac{A}{V}}}\right) \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(c0 \cdot {\left(\frac{1}{\ell}\right)}^{\frac{1}{2}}\right) \cdot \sqrt{\frac{A}{V}}} \]

      8. pow1/2 N/A

         \[\leadsto \left(c0 \cdot \color{blue}{\sqrt{\frac{1}{\ell}}}\right) \cdot \sqrt{\frac{A}{V}} \]

      9. lift-/.f64 N/A

         \[\leadsto \left(c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell}}}\right) \cdot \sqrt{\frac{A}{V}} \]

      10. sqrt-div N/A

          \[\leadsto \left(c0 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\ell}}}\right) \cdot \sqrt{\frac{A}{V}} \]

      11. metadata-eval N/A

          \[\leadsto \left(c0 \cdot \frac{\color{blue}{1}}{\sqrt{\ell}}\right) \cdot \sqrt{\frac{A}{V}} \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \left(c0 \cdot \frac{1}{\color{blue}{\sqrt{\ell}}}\right) \cdot \sqrt{\frac{A}{V}} \]

      13. div-inv N/A

          \[\leadsto \color{blue}{\frac{c0}{\sqrt{\ell}}} \cdot \sqrt{\frac{A}{V}} \]

      14. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{c0}{\sqrt{\ell}}} \cdot \sqrt{\frac{A}{V}} \]

      15. *-commutative N/A

          \[\leadsto \color{blue}{\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}} \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{A}{V}}} \cdot \frac{c0}{\sqrt{\ell}} \]

      17. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{A}{V}}} \cdot \frac{c0}{\sqrt{\ell}} \]

      18. frac-2neg N/A

          \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{neg}\left(A\right)}{\mathsf{neg}\left(V\right)}}} \cdot \frac{c0}{\sqrt{\ell}} \]

      19. lift-neg.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{-A}}{\mathsf{neg}\left(V\right)}} \cdot \frac{c0}{\sqrt{\ell}} \]

      20. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{-A}}{\sqrt{\mathsf{neg}\left(V\right)}}} \cdot \frac{c0}{\sqrt{\ell}} \]

      21. pow1/2 N/A

          \[\leadsto \frac{\color{blue}{{\left(-A\right)}^{\frac{1}{2}}}}{\sqrt{\mathsf{neg}\left(V\right)}} \cdot \frac{c0}{\sqrt{\ell}} \]

      22. lift-/.f64 N/A

          \[\leadsto \frac{{\left(-A\right)}^{\frac{1}{2}}}{\sqrt{\mathsf{neg}\left(V\right)}} \cdot \color{blue}{\frac{c0}{\sqrt{\ell}}} \]

   8. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{\sqrt{-A} \cdot c0}{\sqrt{\left(-\ell\right) \cdot V}}} \]
   9. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{-A} \cdot c0}{\color{blue}{\sqrt{\left(-\ell\right) \cdot V}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{-A} \cdot c0}{\sqrt{\color{blue}{\left(-\ell\right) \cdot V}}} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{-A} \cdot c0}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(\ell\right)\right)} \cdot V}} \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \frac{\sqrt{-A} \cdot c0}{\sqrt{\color{blue}{\mathsf{neg}\left(\ell \cdot V\right)}}} \]

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

         \[\leadsto \frac{\sqrt{-A} \cdot c0}{\sqrt{\color{blue}{\ell \cdot \left(\mathsf{neg}\left(V\right)\right)}}} \]

      6. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{-A} \cdot c0}{\sqrt{\ell \cdot \color{blue}{\left(-V\right)}}} \]

      7. sqrt-prod N/A

         \[\leadsto \frac{\sqrt{-A} \cdot c0}{\color{blue}{\sqrt{\ell} \cdot \sqrt{-V}}} \]

      8. pow1/2 N/A

         \[\leadsto \frac{\sqrt{-A} \cdot c0}{\color{blue}{{\ell}^{\frac{1}{2}}} \cdot \sqrt{-V}} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{-A} \cdot c0}{{\ell}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{-V}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{-A} \cdot c0}{\color{blue}{\sqrt{-V} \cdot {\ell}^{\frac{1}{2}}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{-A} \cdot c0}{\color{blue}{\sqrt{-V} \cdot {\ell}^{\frac{1}{2}}}} \]

      12. pow1/2 N/A

          \[\leadsto \frac{\sqrt{-A} \cdot c0}{\sqrt{-V} \cdot \color{blue}{\sqrt{\ell}}} \]

      13. lower-sqrt.f64 54.8

          \[\leadsto \frac{\sqrt{-A} \cdot c0}{\sqrt{-V} \cdot \color{blue}{\sqrt{\ell}}} \]

   10. Applied rewrites 54.8%

       \[\leadsto \frac{\sqrt{-A} \cdot c0}{\color{blue}{\sqrt{-V} \cdot \sqrt{\ell}}} \]

   if -2.00000000000000007e-228 < (*.f64 V l) < -0.0

   1. Initial program 45.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \]

      3. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      4. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      5. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      6. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{c0}{\sqrt{\ell}}} \cdot \sqrt{\frac{A}{V}} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{c0}{\color{blue}{\sqrt{\ell}}} \cdot \sqrt{\frac{A}{V}} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\ell}} \cdot \color{blue}{\sqrt{\frac{A}{V}}} \]

      13. lower-/.f64 39.8

          \[\leadsto \frac{c0}{\sqrt{\ell}} \cdot \sqrt{\color{blue}{\frac{A}{V}}} \]

   4. Applied rewrites 39.8%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}} \]

   5. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\left(c0 \cdot \sqrt{\frac{1}{\ell}}\right)} \cdot \sqrt{\frac{A}{V}} \]
   6. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(c0 \cdot \sqrt{\frac{1}{\ell}}\right)\right)\right)\right)} \cdot \sqrt{\frac{A}{V}} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\sqrt{\frac{1}{\ell}} \cdot c0}\right)\right)\right)\right) \cdot \sqrt{\frac{A}{V}} \]

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

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot c0}\right)\right) \cdot \sqrt{\frac{A}{V}} \]

      4. distribute-rgt-neg-out N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(\mathsf{neg}\left(c0\right)\right)\right)} \cdot \sqrt{\frac{A}{V}} \]

      5. neg-mul-1 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot \color{blue}{\left(-1 \cdot c0\right)}\right) \cdot \sqrt{\frac{A}{V}} \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot c0\right)\right) \cdot \sqrt{\frac{A}{V}} \]

      7. associate-*l* N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot \color{blue}{\left(\sqrt{-1} \cdot \left(\sqrt{-1} \cdot c0\right)\right)}\right) \cdot \sqrt{\frac{A}{V}} \]

      8. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot \left(\sqrt{-1} \cdot \color{blue}{\left(c0 \cdot \sqrt{-1}\right)}\right)\right) \cdot \sqrt{\frac{A}{V}} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot \sqrt{-1}\right) \cdot \left(c0 \cdot \sqrt{-1}\right)\right)} \cdot \sqrt{\frac{A}{V}} \]

      10. *-commutative N/A

          \[\leadsto \left(\left(\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot \sqrt{-1}\right) \cdot \color{blue}{\left(\sqrt{-1} \cdot c0\right)}\right) \cdot \sqrt{\frac{A}{V}} \]

      11. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\left(\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot \sqrt{-1}\right) \cdot \sqrt{-1}\right) \cdot c0\right)} \cdot \sqrt{\frac{A}{V}} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\left(\left(\mathsf{neg}\left(\sqrt{\frac{1}{\ell}}\right)\right) \cdot \sqrt{-1}\right) \cdot \sqrt{-1}\right) \cdot c0\right)} \cdot \sqrt{\frac{A}{V}} \]

   7. Applied rewrites 39.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\ell}} \cdot c0\right)} \cdot \sqrt{\frac{A}{V}} \]

   if -0.0 < (*.f64 V l) < 9.99999999999999921e273

   1. Initial program 82.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      3. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{A}}}{\sqrt{V \cdot \ell}} \]

      6. lower-sqrt.f64 99.4

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\color{blue}{\sqrt{V \cdot \ell}}} \]

      7. lift-*.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell}}} \]

      8. *-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

      9. lower-*.f64 99.4

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

   4. Applied rewrites 99.4%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\ell \cdot V}}} \]

   if 9.99999999999999921e273 < (*.f64 V l)

   1. Initial program 52.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 84.6

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   4. Applied rewrites 84.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -2 \cdot 10^{-228}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{-A}}{\sqrt{\ell} \cdot \sqrt{-V}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \left(\sqrt{\frac{1}{\ell}} \cdot c0\right)\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+274}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq 0:\\ \;\;\;\;\sqrt{\frac{A}{V} \cdot \frac{1}{\ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+274}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) 0.0)
   (* (sqrt (* (/ A V) (/ 1.0 l))) c0)
   (if (<= (* l V) 1e+274)
     (* (/ (sqrt A) (sqrt (* l V))) c0)
     (* (sqrt (/ (/ A V) l)) c0))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= 0.0) {
		tmp = sqrt(((A / V) * (1.0 / l))) * c0;
	} else if ((l * V) <= 1e+274) {
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	} else {
		tmp = sqrt(((A / V) / l)) * c0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((l * v) <= 0.0d0) then
        tmp = sqrt(((a / v) * (1.0d0 / l))) * c0
    else if ((l * v) <= 1d+274) then
        tmp = (sqrt(a) / sqrt((l * v))) * c0
    else
        tmp = sqrt(((a / v) / l)) * c0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= 0.0) {
		tmp = Math.sqrt(((A / V) * (1.0 / l))) * c0;
	} else if ((l * V) <= 1e+274) {
		tmp = (Math.sqrt(A) / Math.sqrt((l * V))) * c0;
	} else {
		tmp = Math.sqrt(((A / V) / l)) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= 0.0:
		tmp = math.sqrt(((A / V) * (1.0 / l))) * c0
	elif (l * V) <= 1e+274:
		tmp = (math.sqrt(A) / math.sqrt((l * V))) * c0
	else:
		tmp = math.sqrt(((A / V) / l)) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(A / V) * Float64(1.0 / l))) * c0);
	elseif (Float64(l * V) <= 1e+274)
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(l * V))) * c0);
	else
		tmp = Float64(sqrt(Float64(Float64(A / V) / l)) * c0);
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= 0.0)
		tmp = sqrt(((A / V) * (1.0 / l))) * c0;
	elseif ((l * V) <= 1e+274)
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	else
		tmp = sqrt(((A / V) / l)) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], 0.0], N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 1e+274], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -0.0

   1. Initial program 67.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(\frac{A}{V}\right)}{\mathsf{neg}\left(\ell\right)}}} \]

      5. div-inv N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(\frac{A}{V}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}} \]

      6. *-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot \left(\mathsf{neg}\left(\frac{A}{V}\right)\right)}} \]

      7. neg-mul-1 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot \color{blue}{\left(-1 \cdot \frac{A}{V}\right)}} \]

      8. associate-*r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}}} \]

      9. lower-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}}} \]

      10. lower-*.f64 N/A

          \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right)} \cdot \frac{A}{V}} \]

      11. metadata-eval N/A

          \[\leadsto c0 \cdot \sqrt{\left(\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}} \]

      12. frac-2neg N/A

          \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      13. lower-/.f64 N/A

          \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      14. lower-/.f64 71.1

          \[\leadsto c0 \cdot \sqrt{\left(\frac{-1}{\ell} \cdot -1\right) \cdot \color{blue}{\frac{A}{V}}} \]

   4. Applied rewrites 71.1%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{-1}{\ell} \cdot -1\right) \cdot \frac{A}{V}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{-1}{\ell} \cdot -1\right)} \cdot \frac{A}{V}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      3. associate-*l/ N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{-1 \cdot -1}{\ell}} \cdot \frac{A}{V}} \]

      4. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1}}{\ell} \cdot \frac{A}{V}} \]

      5. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\ell} \cdot \frac{A}{V}} \]

      6. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\ell}} \cdot \frac{A}{V}} \]

      7. metadata-eval 71.1

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1}}{\ell} \cdot \frac{A}{V}} \]

   6. Applied rewrites 71.1%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell}} \cdot \frac{A}{V}} \]

   if -0.0 < (*.f64 V l) < 9.99999999999999921e273

   1. Initial program 82.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      3. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{A}}}{\sqrt{V \cdot \ell}} \]

      6. lower-sqrt.f64 99.4

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\color{blue}{\sqrt{V \cdot \ell}}} \]

      7. lift-*.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell}}} \]

      8. *-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

      9. lower-*.f64 99.4

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

   4. Applied rewrites 99.4%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\ell \cdot V}}} \]

   if 9.99999999999999921e273 < (*.f64 V l)

   1. Initial program 52.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 84.6

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   4. Applied rewrites 84.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq 0:\\ \;\;\;\;\sqrt{\frac{A}{V} \cdot \frac{1}{\ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+274}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 81.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{if}\;\ell \cdot V \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+274}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* (sqrt (/ (/ A V) l)) c0)))
   (if (<= (* l V) 0.0)
     t_0
     (if (<= (* l V) 1e+274) (* (/ (sqrt A) (sqrt (* l V))) c0) t_0))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt(((A / V) / l)) * c0;
	double tmp;
	if ((l * V) <= 0.0) {
		tmp = t_0;
	} else if ((l * V) <= 1e+274) {
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((a / v) / l)) * c0
    if ((l * v) <= 0.0d0) then
        tmp = t_0
    else if ((l * v) <= 1d+274) then
        tmp = (sqrt(a) / sqrt((l * v))) * c0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt(((A / V) / l)) * c0;
	double tmp;
	if ((l * V) <= 0.0) {
		tmp = t_0;
	} else if ((l * V) <= 1e+274) {
		tmp = (Math.sqrt(A) / Math.sqrt((l * V))) * c0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt(((A / V) / l)) * c0
	tmp = 0
	if (l * V) <= 0.0:
		tmp = t_0
	elif (l * V) <= 1e+274:
		tmp = (math.sqrt(A) / math.sqrt((l * V))) * c0
	else:
		tmp = t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(sqrt(Float64(Float64(A / V) / l)) * c0)
	tmp = 0.0
	if (Float64(l * V) <= 0.0)
		tmp = t_0;
	elseif (Float64(l * V) <= 1e+274)
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(l * V))) * c0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt(((A / V) / l)) * c0;
	tmp = 0.0;
	if ((l * V) <= 0.0)
		tmp = t_0;
	elseif ((l * V) <= 1e+274)
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(l * V), $MachinePrecision], 1e+274], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 V l) < -0.0 or 9.99999999999999921e273 < (*.f64 V l)

   1. Initial program 65.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      5. lower-/.f64 72.6

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{V}}}{\ell}} \]

   4. Applied rewrites 72.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

   if -0.0 < (*.f64 V l) < 9.99999999999999921e273

   1. Initial program 82.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      3. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      4. lower-/.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{A}}}{\sqrt{V \cdot \ell}} \]

      6. lower-sqrt.f64 99.4

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\color{blue}{\sqrt{V \cdot \ell}}} \]

      7. lift-*.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell}}} \]

      8. *-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

      9. lower-*.f64 99.4

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

   4. Applied rewrites 99.4%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\ell \cdot V}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq 0:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+274}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 87.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{-\ell} \cdot \sqrt{-V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= l -5e-310)
   (* (/ (sqrt A) (* (sqrt (- l)) (sqrt (- V)))) c0)
   (/ c0 (* (sqrt (/ V A)) (sqrt l)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -5e-310) {
		tmp = (sqrt(A) / (sqrt(-l) * sqrt(-V))) * c0;
	} else {
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-5d-310)) then
        tmp = (sqrt(a) / (sqrt(-l) * sqrt(-v))) * c0
    else
        tmp = c0 / (sqrt((v / a)) * sqrt(l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -5e-310) {
		tmp = (Math.sqrt(A) / (Math.sqrt(-l) * Math.sqrt(-V))) * c0;
	} else {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if l <= -5e-310:
		tmp = (math.sqrt(A) / (math.sqrt(-l) * math.sqrt(-V))) * c0
	else:
		tmp = c0 / (math.sqrt((V / A)) * math.sqrt(l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(Float64(sqrt(A) / Float64(sqrt(Float64(-l)) * sqrt(Float64(-V)))) * c0);
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * sqrt(l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (l <= -5e-310)
		tmp = (sqrt(A) / (sqrt(-l) * sqrt(-V))) * c0;
	else
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[l, -5e-310], N[(N[(N[Sqrt[A], $MachinePrecision] / N[(N[Sqrt[(-l)], $MachinePrecision] * N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 71.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      3. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      4. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(\frac{A}{V}\right)}{\mathsf{neg}\left(\ell\right)}}} \]

      5. div-inv N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(\frac{A}{V}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\ell\right)}}} \]

      6. *-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot \left(\mathsf{neg}\left(\frac{A}{V}\right)\right)}} \]

      7. neg-mul-1 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot \color{blue}{\left(-1 \cdot \frac{A}{V}\right)}} \]

      8. associate-*r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}}} \]

      9. lower-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}}} \]

      10. lower-*.f64 N/A

          \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{\mathsf{neg}\left(\ell\right)} \cdot -1\right)} \cdot \frac{A}{V}} \]

      11. metadata-eval N/A

          \[\leadsto c0 \cdot \sqrt{\left(\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\mathsf{neg}\left(\ell\right)} \cdot -1\right) \cdot \frac{A}{V}} \]

      12. frac-2neg N/A

          \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      13. lower-/.f64 N/A

          \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      14. lower-/.f64 75.6

          \[\leadsto c0 \cdot \sqrt{\left(\frac{-1}{\ell} \cdot -1\right) \cdot \color{blue}{\frac{A}{V}}} \]

   4. Applied rewrites 75.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{-1}{\ell} \cdot -1\right) \cdot \frac{A}{V}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{-1}{\ell} \cdot -1\right)} \cdot \frac{A}{V}} \]

      2. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\left(\color{blue}{\frac{-1}{\ell}} \cdot -1\right) \cdot \frac{A}{V}} \]

      3. associate-*l/ N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{-1 \cdot -1}{\ell}} \cdot \frac{A}{V}} \]

      4. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1}}{\ell} \cdot \frac{A}{V}} \]

      5. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\ell} \cdot \frac{A}{V}} \]

      6. lower-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\ell}} \cdot \frac{A}{V}} \]

      7. metadata-eval 75.6

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1}}{\ell} \cdot \frac{A}{V}} \]

   6. Applied rewrites 75.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell}} \cdot \frac{A}{V}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{1}{\ell} \cdot \frac{A}{V}}} \]

      2. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\ell} \cdot \frac{A}{V}}} \]

      3. sqrt-prod N/A

         \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\frac{1}{\ell}} \cdot \sqrt{\frac{A}{V}}\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto c0 \cdot \left(\sqrt{\color{blue}{\frac{1}{\ell}}} \cdot \sqrt{\frac{A}{V}}\right) \]

      5. frac-2neg N/A

         \[\leadsto c0 \cdot \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\ell\right)}}} \cdot \sqrt{\frac{A}{V}}\right) \]

      6. metadata-eval N/A

         \[\leadsto c0 \cdot \left(\sqrt{\frac{\color{blue}{-1}}{\mathsf{neg}\left(\ell\right)}} \cdot \sqrt{\frac{A}{V}}\right) \]

      7. sqrt-div N/A

         \[\leadsto c0 \cdot \left(\color{blue}{\frac{\sqrt{-1}}{\sqrt{\mathsf{neg}\left(\ell\right)}}} \cdot \sqrt{\frac{A}{V}}\right) \]

      8. pow1/2 N/A

         \[\leadsto c0 \cdot \left(\frac{\color{blue}{{-1}^{\frac{1}{2}}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \sqrt{\frac{A}{V}}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto c0 \cdot \left(\frac{{-1}^{\frac{1}{2}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \sqrt{\color{blue}{\frac{A}{V}}}\right) \]

      10. frac-2neg N/A

          \[\leadsto c0 \cdot \left(\frac{{-1}^{\frac{1}{2}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(A\right)}{\mathsf{neg}\left(V\right)}}}\right) \]

      11. lift-neg.f64 N/A

          \[\leadsto c0 \cdot \left(\frac{{-1}^{\frac{1}{2}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \sqrt{\frac{\color{blue}{-A}}{\mathsf{neg}\left(V\right)}}\right) \]

      12. sqrt-div N/A

          \[\leadsto c0 \cdot \left(\frac{{-1}^{\frac{1}{2}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \color{blue}{\frac{\sqrt{-A}}{\sqrt{\mathsf{neg}\left(V\right)}}}\right) \]

      13. pow1/2 N/A

          \[\leadsto c0 \cdot \left(\frac{{-1}^{\frac{1}{2}}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot \frac{\color{blue}{{\left(-A\right)}^{\frac{1}{2}}}}{\sqrt{\mathsf{neg}\left(V\right)}}\right) \]

      14. frac-times N/A

          \[\leadsto c0 \cdot \color{blue}{\frac{{-1}^{\frac{1}{2}} \cdot {\left(-A\right)}^{\frac{1}{2}}}{\sqrt{\mathsf{neg}\left(\ell\right)} \cdot \sqrt{\mathsf{neg}\left(V\right)}}} \]

      15. unpow-prod-down N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{{\left(-1 \cdot \left(-A\right)\right)}^{\frac{1}{2}}}}{\sqrt{\mathsf{neg}\left(\ell\right)} \cdot \sqrt{\mathsf{neg}\left(V\right)}} \]

      16. mul-1-neg N/A

          \[\leadsto c0 \cdot \frac{{\color{blue}{\left(\mathsf{neg}\left(\left(-A\right)\right)\right)}}^{\frac{1}{2}}}{\sqrt{\mathsf{neg}\left(\ell\right)} \cdot \sqrt{\mathsf{neg}\left(V\right)}} \]

      17. lift-neg.f64 N/A

          \[\leadsto c0 \cdot \frac{{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(A\right)\right)}\right)\right)}^{\frac{1}{2}}}{\sqrt{\mathsf{neg}\left(\ell\right)} \cdot \sqrt{\mathsf{neg}\left(V\right)}} \]

      18. remove-double-neg N/A

          \[\leadsto c0 \cdot \frac{{\color{blue}{A}}^{\frac{1}{2}}}{\sqrt{\mathsf{neg}\left(\ell\right)} \cdot \sqrt{\mathsf{neg}\left(V\right)}} \]

      19. pow1/2 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{A}}}{\sqrt{\mathsf{neg}\left(\ell\right)} \cdot \sqrt{\mathsf{neg}\left(V\right)}} \]

      20. lower-/.f64 N/A

          \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\mathsf{neg}\left(\ell\right)} \cdot \sqrt{\mathsf{neg}\left(V\right)}}} \]

      21. lower-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{A}}}{\sqrt{\mathsf{neg}\left(\ell\right)} \cdot \sqrt{\mathsf{neg}\left(V\right)}} \]

   8. Applied rewrites 50.5%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{-\ell} \cdot \sqrt{-V}}} \]

   if -4.999999999999985e-310 < l

   1. Initial program 72.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \]

      3. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      4. clear-num N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\frac{V \cdot \ell}{A}}}} \]

      5. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      6. metadata-eval N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{1}}{\sqrt{\frac{V \cdot \ell}{A}}} \]

      7. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      10. lower-/.f64 71.9

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V \cdot \ell}{A}}}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell}}{A}}} \]

      12. *-commutative N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\ell \cdot V}}{A}}} \]

      13. lower-*.f64 71.9

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\ell \cdot V}}{A}}} \]

   4. Applied rewrites 71.9%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{\ell \cdot V}{A}}}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{\ell \cdot V}{A}}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\ell \cdot V}{A}}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\ell \cdot V}}{A}}} \]

      4. associate-/l* N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\ell \cdot \frac{V}{A}}}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{c0}{\sqrt{\ell \cdot \color{blue}{\frac{V}{A}}}} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{c0}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{c0}{\color{blue}{\sqrt{\ell}} \cdot \sqrt{\frac{V}{A}}} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{c0}{\sqrt{\ell} \cdot \color{blue}{\sqrt{\frac{V}{A}}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}} \]

      10. lower-*.f64 81.5

          \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}} \]

   6. Applied rewrites 81.5%

      \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{-\ell} \cdot \sqrt{-V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \sqrt{\frac{A}{\ell \cdot V}} \cdot c0 \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l) :precision binary64 (* (sqrt (/ A (* l V))) c0))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	return sqrt((A / (l * V))) * c0;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    code = sqrt((a / (l * v))) * c0
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	return Math.sqrt((A / (l * V))) * c0;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	return math.sqrt((A / (l * V))) * c0

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	return Float64(sqrt(Float64(A / Float64(l * V))) * c0)
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp = code(c0, A, V, l)
	tmp = sqrt((A / (l * V))) * c0;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := N[(N[Sqrt[N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]

Derivation

1. Initial program 72.0%

   \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
2. Add Preprocessing

3. Final simplification 72.0%

   \[\leadsto \sqrt{\frac{A}{\ell \cdot V}} \cdot c0 \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024248 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))