Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.2% → 91.1%

Time: 6.9s

Alternatives: 14

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: 73.2% 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: 91.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -2 \cdot 10^{+286}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-312}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\frac{V}{A}}}{\ell}} \cdot c0\\ \end{array} \end{array} \]

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+286)
   (/ c0 (* (sqrt (/ V A)) (sqrt l)))
   (if (<= (* l V) -2e-312)
     (* (/ (sqrt (- A)) (sqrt (* l (- V)))) c0)
     (if (<= (* l V) 0.0)
       (/ c0 (sqrt (* (/ V A) l)))
       (if (<= (* l V) 2e+293)
         (* (/ (sqrt A) (sqrt (* l V))) c0)
         (* (sqrt (/ (/ 1.0 (/ V A)) 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+286) {
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	} else if ((l * V) <= -2e-312) {
		tmp = (sqrt(-A) / sqrt((l * -V))) * c0;
	} else if ((l * V) <= 0.0) {
		tmp = c0 / sqrt(((V / A) * l));
	} else if ((l * V) <= 2e+293) {
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	} else {
		tmp = sqrt(((1.0 / (V / A)) / 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+286)) then
        tmp = c0 / (sqrt((v / a)) * sqrt(l))
    else if ((l * v) <= (-2d-312)) then
        tmp = (sqrt(-a) / sqrt((l * -v))) * c0
    else if ((l * v) <= 0.0d0) then
        tmp = c0 / sqrt(((v / a) * l))
    else if ((l * v) <= 2d+293) then
        tmp = (sqrt(a) / sqrt((l * v))) * c0
    else
        tmp = sqrt(((1.0d0 / (v / a)) / 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+286) {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	} else if ((l * V) <= -2e-312) {
		tmp = (Math.sqrt(-A) / Math.sqrt((l * -V))) * c0;
	} else if ((l * V) <= 0.0) {
		tmp = c0 / Math.sqrt(((V / A) * l));
	} else if ((l * V) <= 2e+293) {
		tmp = (Math.sqrt(A) / Math.sqrt((l * V))) * c0;
	} else {
		tmp = Math.sqrt(((1.0 / (V / A)) / 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+286:
		tmp = c0 / (math.sqrt((V / A)) * math.sqrt(l))
	elif (l * V) <= -2e-312:
		tmp = (math.sqrt(-A) / math.sqrt((l * -V))) * c0
	elif (l * V) <= 0.0:
		tmp = c0 / math.sqrt(((V / A) * l))
	elif (l * V) <= 2e+293:
		tmp = (math.sqrt(A) / math.sqrt((l * V))) * c0
	else:
		tmp = math.sqrt(((1.0 / (V / A)) / 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+286)
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * sqrt(l)));
	elseif (Float64(l * V) <= -2e-312)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(l * Float64(-V)))) * c0);
	elseif (Float64(l * V) <= 0.0)
		tmp = Float64(c0 / sqrt(Float64(Float64(V / A) * l)));
	elseif (Float64(l * V) <= 2e+293)
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(l * V))) * c0);
	else
		tmp = Float64(sqrt(Float64(Float64(1.0 / Float64(V / A)) / 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+286)
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	elseif ((l * V) <= -2e-312)
		tmp = (sqrt(-A) / sqrt((l * -V))) * c0;
	elseif ((l * V) <= 0.0)
		tmp = c0 / sqrt(((V / A) * l));
	elseif ((l * V) <= 2e+293)
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	else
		tmp = sqrt(((1.0 / (V / A)) / 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+286], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -2e-312], 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[(c0 / N[Sqrt[N[(N[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 2e+293], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], N[(N[Sqrt[N[(N[(1.0 / N[(V / A), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -2.00000000000000007e286

   1. Initial program 38.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 79.5

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

   4. Applied rewrites 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. metadata-eval N/A

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

      11. sqrt-div N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-times N/A

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

      16. lift-sqrt.f64 N/A

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

      17. sqrt-prod N/A

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

      18. *-commutative N/A

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

      19. associate-/r/ N/A

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

      20. lift-/.f64 N/A

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

      21. metadata-eval N/A

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

      22. *-rgt-identity N/A

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

   6. Applied rewrites 79.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-unprod N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 36.1

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

   8. Applied rewrites 36.1%

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

   if -2.00000000000000007e286 < (*.f64 V l) < -2.0000000000019e-312

   1. Initial program 87.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. lower-/.f64 N/A

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

      5. lower-/.f64 79.6

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

   4. Applied rewrites 79.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. neg-mul-1 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-neg.f64 N/A

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

      13. sqrt-div N/A

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

      14. frac-times N/A

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

      15. lift-/.f64 N/A

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

      16. associate-/r/ N/A

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

      17. lift-/.f64 N/A

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

      18. frac-2neg N/A

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

      19. sqrt-div N/A

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

      20. pow1/2 N/A

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

   6. Applied rewrites 99.2%

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

   if -2.0000000000019e-312 < (*.f64 V l) < -0.0

   1. Initial program 53.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 72.2

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

   4. Applied rewrites 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. metadata-eval N/A

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

      11. sqrt-div N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-times N/A

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

      16. lift-sqrt.f64 N/A

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

      17. sqrt-prod N/A

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

      18. *-commutative N/A

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

      19. associate-/r/ N/A

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

      20. lift-/.f64 N/A

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

      21. metadata-eval N/A

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

      22. *-rgt-identity N/A

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

   6. Applied rewrites 74.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ 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. *-commutative N/A

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

      7. lower-*.f64 74.2

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

   8. Applied rewrites 74.2%

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

   if -0.0 < (*.f64 V l) < 1.9999999999999998e293

   1. Initial program 86.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 99.1

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

      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 99.1

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

   4. Applied rewrites 99.1%

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

   if 1.9999999999999998e293 < (*.f64 V l)

   1. Initial program 44.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 73.6

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

   4. Applied rewrites 73.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 73.7

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

   6. Applied rewrites 73.7%

      \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{1}{\frac{V}{A}}}}{\ell}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -2 \cdot 10^{+286}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-312}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\frac{V}{A}}}{\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := {\left(-A\right)}^{0.25}\\ \mathbf{if}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{t\_0} \cdot \frac{\sqrt{-V}}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \end{array} \end{array} \]

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 (pow (- A) 0.25)))
   (if (<= A -5e-310)
     (/ c0 (* (/ (sqrt l) t_0) (/ (sqrt (- V)) t_0)))
     (* (/ (sqrt A) (sqrt (* l V))) c0))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = pow(-A, 0.25);
	double tmp;
	if (A <= -5e-310) {
		tmp = c0 / ((sqrt(l) / t_0) * (sqrt(-V) / t_0));
	} else {
		tmp = (sqrt(A) / sqrt((l * V))) * 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) :: t_0
    real(8) :: tmp
    t_0 = -a ** 0.25d0
    if (a <= (-5d-310)) then
        tmp = c0 / ((sqrt(l) / t_0) * (sqrt(-v) / t_0))
    else
        tmp = (sqrt(a) / sqrt((l * v))) * 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 t_0 = Math.pow(-A, 0.25);
	double tmp;
	if (A <= -5e-310) {
		tmp = c0 / ((Math.sqrt(l) / t_0) * (Math.sqrt(-V) / t_0));
	} else {
		tmp = (Math.sqrt(A) / Math.sqrt((l * V))) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.pow(-A, 0.25)
	tmp = 0
	if A <= -5e-310:
		tmp = c0 / ((math.sqrt(l) / t_0) * (math.sqrt(-V) / t_0))
	else:
		tmp = (math.sqrt(A) / math.sqrt((l * V))) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(-A) ^ 0.25
	tmp = 0.0
	if (A <= -5e-310)
		tmp = Float64(c0 / Float64(Float64(sqrt(l) / t_0) * Float64(sqrt(Float64(-V)) / t_0)));
	else
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(l * V))) * 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 = -A ^ 0.25;
	tmp = 0.0;
	if (A <= -5e-310)
		tmp = c0 / ((sqrt(l) / t_0) * (sqrt(-V) / t_0));
	else
		tmp = (sqrt(A) / sqrt((l * V))) * 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[Power[(-A), 0.25], $MachinePrecision]}, If[LessEqual[A, -5e-310], N[(c0 / N[(N[(N[Sqrt[l], $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[Sqrt[(-V)], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if A < -4.999999999999985e-310

   1. Initial program 72.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. lower-/.f64 N/A

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

      5. lower-/.f64 78.6

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

   4. Applied rewrites 78.6%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. metadata-eval N/A

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

      11. sqrt-div N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-times N/A

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

      16. lift-sqrt.f64 N/A

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

      17. sqrt-prod N/A

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

      18. *-commutative N/A

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

      19. associate-/r/ N/A

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

      20. lift-/.f64 N/A

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

      21. metadata-eval N/A

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

      22. *-rgt-identity N/A

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

   6. Applied rewrites 77.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lift-neg.f64 N/A

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

      6. associate-*l/ N/A

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

      7. sqrt-div N/A

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

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

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

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

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

      10. lift-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. sqrt-prod N/A

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

      13. pow1/2 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. pow1/2 N/A

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

      16. sqr-pow N/A

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

      17. times-frac N/A

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

      18. lower-*.f64 N/A

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

   8. Applied rewrites 42.9%

      \[\leadsto \frac{c0}{\color{blue}{\frac{\sqrt{-V}}{{\left(-A\right)}^{0.25}} \cdot \frac{\sqrt{\ell}}{{\left(-A\right)}^{0.25}}}} \]

   if -4.999999999999985e-310 < A

   1. Initial program 76.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 85.6

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

      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 85.6

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

   4. Applied rewrites 85.6%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{{\left(-A\right)}^{0.25}} \cdot \frac{\sqrt{-V}}{{\left(-A\right)}^{0.25}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-249}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+297}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \end{array} \end{array} \]

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 4e-249)
     (* (sqrt (/ (/ A l) V)) c0)
     (if (<= t_0 2e+297) 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 <= 4e-249) {
		tmp = sqrt(((A / l) / V)) * c0;
	} else if (t_0 <= 2e+297) {
		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 <= 4d-249) then
        tmp = sqrt(((a / l) / v)) * c0
    else if (t_0 <= 2d+297) 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 <= 4e-249) {
		tmp = Math.sqrt(((A / l) / V)) * c0;
	} else if (t_0 <= 2e+297) {
		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 <= 4e-249:
		tmp = math.sqrt(((A / l) / V)) * c0
	elif t_0 <= 2e+297:
		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 <= 4e-249)
		tmp = Float64(sqrt(Float64(Float64(A / l) / V)) * c0);
	elseif (t_0 <= 2e+297)
		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 <= 4e-249)
		tmp = sqrt(((A / l) / V)) * c0;
	elseif (t_0 <= 2e+297)
		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, 4e-249], N[(N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[t$95$0, 2e+297], 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)))) < 4.00000000000000022e-249

   1. Initial program 68.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 80.0

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

   4. Applied rewrites 80.0%

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

   if 4.00000000000000022e-249 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2e297

   1. Initial program 99.5%

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

   if 2e297 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 56.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 65.9

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

   4. Applied rewrites 65.9%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. metadata-eval N/A

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

      11. sqrt-div N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-times N/A

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

      16. lift-sqrt.f64 N/A

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

      17. sqrt-prod N/A

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

      18. *-commutative N/A

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

      19. associate-/r/ N/A

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

      20. lift-/.f64 N/A

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

      21. metadata-eval N/A

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

      22. *-rgt-identity N/A

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

   6. Applied rewrites 67.8%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0 \leq 4 \cdot 10^{-249}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0 \leq 2 \cdot 10^{+297}:\\ \;\;\;\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ t_1 := \sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+264}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 0.0) t_1 (if (<= t_0 1e+264) 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 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 1e+264) {
		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 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 1d+264) 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 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 1e+264) {
		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 <= 0.0:
		tmp = t_1
	elif t_0 <= 1e+264:
		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 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 1e+264)
		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 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 1e+264)
		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, 0.0], t$95$1, If[LessEqual[t$95$0, 1e+264], t$95$0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 0.0 or 1.00000000000000004e264 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   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 72.8

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

   4. Applied rewrites 72.8%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.00000000000000004e264

   1. Initial program 99.4%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0 \leq 0:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{elif}\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0 \leq 10^{+264}:\\ \;\;\;\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \mathbf{if}\;\ell \cdot V \leq -2 \cdot 10^{+286}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-312}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 (/ c0 (sqrt (* (/ V A) l)))))
   (if (<= (* l V) -2e+286)
     (/ c0 (* (sqrt (/ V A)) (sqrt l)))
     (if (<= (* l V) -2e-312)
       (* (/ (sqrt (- A)) (sqrt (* l (- V)))) c0)
       (if (<= (* l V) 0.0)
         t_0
         (if (<= (* l V) 2e+293) (* (/ (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 = c0 / sqrt(((V / A) * l));
	double tmp;
	if ((l * V) <= -2e+286) {
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	} else if ((l * V) <= -2e-312) {
		tmp = (sqrt(-A) / sqrt((l * -V))) * c0;
	} else if ((l * V) <= 0.0) {
		tmp = t_0;
	} else if ((l * V) <= 2e+293) {
		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 = c0 / sqrt(((v / a) * l))
    if ((l * v) <= (-2d+286)) then
        tmp = c0 / (sqrt((v / a)) * sqrt(l))
    else if ((l * v) <= (-2d-312)) then
        tmp = (sqrt(-a) / sqrt((l * -v))) * c0
    else if ((l * v) <= 0.0d0) then
        tmp = t_0
    else if ((l * v) <= 2d+293) 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 = c0 / Math.sqrt(((V / A) * l));
	double tmp;
	if ((l * V) <= -2e+286) {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	} else if ((l * V) <= -2e-312) {
		tmp = (Math.sqrt(-A) / Math.sqrt((l * -V))) * c0;
	} else if ((l * V) <= 0.0) {
		tmp = t_0;
	} else if ((l * V) <= 2e+293) {
		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 = c0 / math.sqrt(((V / A) * l))
	tmp = 0
	if (l * V) <= -2e+286:
		tmp = c0 / (math.sqrt((V / A)) * math.sqrt(l))
	elif (l * V) <= -2e-312:
		tmp = (math.sqrt(-A) / math.sqrt((l * -V))) * c0
	elif (l * V) <= 0.0:
		tmp = t_0
	elif (l * V) <= 2e+293:
		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(c0 / sqrt(Float64(Float64(V / A) * l)))
	tmp = 0.0
	if (Float64(l * V) <= -2e+286)
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * sqrt(l)));
	elseif (Float64(l * V) <= -2e-312)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(l * Float64(-V)))) * c0);
	elseif (Float64(l * V) <= 0.0)
		tmp = t_0;
	elseif (Float64(l * V) <= 2e+293)
		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 = c0 / sqrt(((V / A) * l));
	tmp = 0.0;
	if ((l * V) <= -2e+286)
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	elseif ((l * V) <= -2e-312)
		tmp = (sqrt(-A) / sqrt((l * -V))) * c0;
	elseif ((l * V) <= 0.0)
		tmp = t_0;
	elseif ((l * V) <= 2e+293)
		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[(c0 / N[Sqrt[N[(N[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], -2e+286], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -2e-312], 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], 2e+293], 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) < -2.00000000000000007e286

   1. Initial program 38.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 79.5

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

   4. Applied rewrites 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. metadata-eval N/A

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

      11. sqrt-div N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-times N/A

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

      16. lift-sqrt.f64 N/A

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

      17. sqrt-prod N/A

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

      18. *-commutative N/A

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

      19. associate-/r/ N/A

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

      20. lift-/.f64 N/A

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

      21. metadata-eval N/A

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

      22. *-rgt-identity N/A

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

   6. Applied rewrites 79.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-unprod N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 36.1

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

   8. Applied rewrites 36.1%

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

   if -2.00000000000000007e286 < (*.f64 V l) < -2.0000000000019e-312

   1. Initial program 87.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. lower-/.f64 N/A

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

      5. lower-/.f64 79.6

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

   4. Applied rewrites 79.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. neg-mul-1 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-neg.f64 N/A

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

      13. sqrt-div N/A

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

      14. frac-times N/A

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

      15. lift-/.f64 N/A

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

      16. associate-/r/ N/A

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

      17. lift-/.f64 N/A

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

      18. frac-2neg N/A

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

      19. sqrt-div N/A

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

      20. pow1/2 N/A

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

   6. Applied rewrites 99.2%

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

   if -2.0000000000019e-312 < (*.f64 V l) < -0.0 or 1.9999999999999998e293 < (*.f64 V l)

   1. Initial program 50.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. lower-/.f64 N/A

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

      5. lower-/.f64 72.8

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

   4. Applied rewrites 72.8%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. metadata-eval N/A

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

      11. sqrt-div N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-times N/A

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

      16. lift-sqrt.f64 N/A

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

      17. sqrt-prod N/A

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

      18. *-commutative N/A

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

      19. associate-/r/ N/A

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

      20. lift-/.f64 N/A

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

      21. metadata-eval N/A

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

      22. *-rgt-identity N/A

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

   6. Applied rewrites 74.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ 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. *-commutative N/A

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

      7. lower-*.f64 74.0

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

   8. Applied rewrites 74.0%

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

   if -0.0 < (*.f64 V l) < 1.9999999999999998e293

   1. Initial program 86.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 99.1

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

      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 99.1

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

   4. Applied rewrites 99.1%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -2 \cdot 10^{+286}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-312}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-312}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 (/ c0 (sqrt (* (/ V A) l)))))
   (if (<= (* l V) (- INFINITY))
     (* (/ (sqrt (/ A V)) (sqrt l)) c0)
     (if (<= (* l V) -2e-312)
       (* (/ (sqrt (- A)) (sqrt (* l (- V)))) c0)
       (if (<= (* l V) 0.0)
         t_0
         (if (<= (* l V) 2e+293) (* (/ (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 = c0 / sqrt(((V / A) * l));
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = (sqrt((A / V)) / sqrt(l)) * c0;
	} else if ((l * V) <= -2e-312) {
		tmp = (sqrt(-A) / sqrt((l * -V))) * c0;
	} else if ((l * V) <= 0.0) {
		tmp = t_0;
	} else if ((l * V) <= 2e+293) {
		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 = c0 / Math.sqrt(((V / A) * l));
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = (Math.sqrt((A / V)) / Math.sqrt(l)) * c0;
	} else if ((l * V) <= -2e-312) {
		tmp = (Math.sqrt(-A) / Math.sqrt((l * -V))) * c0;
	} else if ((l * V) <= 0.0) {
		tmp = t_0;
	} else if ((l * V) <= 2e+293) {
		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 = c0 / math.sqrt(((V / A) * l))
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = (math.sqrt((A / V)) / math.sqrt(l)) * c0
	elif (l * V) <= -2e-312:
		tmp = (math.sqrt(-A) / math.sqrt((l * -V))) * c0
	elif (l * V) <= 0.0:
		tmp = t_0
	elif (l * V) <= 2e+293:
		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(c0 / sqrt(Float64(Float64(V / A) * l)))
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(A / V)) / sqrt(l)) * c0);
	elseif (Float64(l * V) <= -2e-312)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(l * Float64(-V)))) * c0);
	elseif (Float64(l * V) <= 0.0)
		tmp = t_0;
	elseif (Float64(l * V) <= 2e+293)
		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 = c0 / sqrt(((V / A) * l));
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = (sqrt((A / V)) / sqrt(l)) * c0;
	elseif ((l * V) <= -2e-312)
		tmp = (sqrt(-A) / sqrt((l * -V))) * c0;
	elseif ((l * V) <= 0.0)
		tmp = t_0;
	elseif ((l * V) <= 2e+293)
		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[(c0 / N[Sqrt[N[(N[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], N[(N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -2e-312], 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], 2e+293], 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

   1. Initial program 28.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 33.6

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

   4. Applied rewrites 33.6%

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

   if -inf.0 < (*.f64 V l) < -2.0000000000019e-312

   1. Initial program 87.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 80.5

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

   4. Applied rewrites 80.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. neg-mul-1 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-neg.f64 N/A

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

      13. sqrt-div N/A

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

      14. frac-times N/A

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

      15. lift-/.f64 N/A

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

      16. associate-/r/ N/A

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

      17. lift-/.f64 N/A

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

      18. frac-2neg N/A

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

      19. sqrt-div N/A

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

      20. pow1/2 N/A

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

   6. Applied rewrites 99.2%

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

   if -2.0000000000019e-312 < (*.f64 V l) < -0.0 or 1.9999999999999998e293 < (*.f64 V l)

   1. Initial program 50.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. lower-/.f64 N/A

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

      5. lower-/.f64 72.8

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

   4. Applied rewrites 72.8%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. metadata-eval N/A

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

      11. sqrt-div N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-times N/A

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

      16. lift-sqrt.f64 N/A

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

      17. sqrt-prod N/A

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

      18. *-commutative N/A

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

      19. associate-/r/ N/A

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

      20. lift-/.f64 N/A

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

      21. metadata-eval N/A

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

      22. *-rgt-identity N/A

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

   6. Applied rewrites 74.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ 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. *-commutative N/A

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

      7. lower-*.f64 74.0

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

   8. Applied rewrites 74.0%

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

   if -0.0 < (*.f64 V l) < 1.9999999999999998e293

   1. Initial program 86.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 99.1

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

      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 99.1

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

   4. Applied rewrites 99.1%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-312}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-312}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 (/ c0 (sqrt (* (/ V A) l)))))
   (if (<= (* l V) (- INFINITY))
     (* (sqrt (/ (/ A l) V)) c0)
     (if (<= (* l V) -2e-312)
       (* (/ (sqrt (- A)) (sqrt (* l (- V)))) c0)
       (if (<= (* l V) 0.0)
         t_0
         (if (<= (* l V) 2e+293) (* (/ (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 = c0 / sqrt(((V / A) * l));
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = sqrt(((A / l) / V)) * c0;
	} else if ((l * V) <= -2e-312) {
		tmp = (sqrt(-A) / sqrt((l * -V))) * c0;
	} else if ((l * V) <= 0.0) {
		tmp = t_0;
	} else if ((l * V) <= 2e+293) {
		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 = c0 / Math.sqrt(((V / A) * l));
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((A / l) / V)) * c0;
	} else if ((l * V) <= -2e-312) {
		tmp = (Math.sqrt(-A) / Math.sqrt((l * -V))) * c0;
	} else if ((l * V) <= 0.0) {
		tmp = t_0;
	} else if ((l * V) <= 2e+293) {
		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 = c0 / math.sqrt(((V / A) * l))
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = math.sqrt(((A / l) / V)) * c0
	elif (l * V) <= -2e-312:
		tmp = (math.sqrt(-A) / math.sqrt((l * -V))) * c0
	elif (l * V) <= 0.0:
		tmp = t_0
	elif (l * V) <= 2e+293:
		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(c0 / sqrt(Float64(Float64(V / A) * l)))
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(Float64(A / l) / V)) * c0);
	elseif (Float64(l * V) <= -2e-312)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(l * Float64(-V)))) * c0);
	elseif (Float64(l * V) <= 0.0)
		tmp = t_0;
	elseif (Float64(l * V) <= 2e+293)
		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 = c0 / sqrt(((V / A) * l));
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = sqrt(((A / l) / V)) * c0;
	elseif ((l * V) <= -2e-312)
		tmp = (sqrt(-A) / sqrt((l * -V))) * c0;
	elseif ((l * V) <= 0.0)
		tmp = t_0;
	elseif ((l * V) <= 2e+293)
		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[(c0 / N[Sqrt[N[(N[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], N[(N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -2e-312], 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], 2e+293], 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

   1. Initial program 28.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 76.3

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

   4. Applied rewrites 76.3%

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

   if -inf.0 < (*.f64 V l) < -2.0000000000019e-312

   1. Initial program 87.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 80.5

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

   4. Applied rewrites 80.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. neg-mul-1 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-neg.f64 N/A

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

      13. sqrt-div N/A

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

      14. frac-times N/A

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

      15. lift-/.f64 N/A

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

      16. associate-/r/ N/A

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

      17. lift-/.f64 N/A

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

      18. frac-2neg N/A

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

      19. sqrt-div N/A

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

      20. pow1/2 N/A

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

   6. Applied rewrites 99.2%

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

   if -2.0000000000019e-312 < (*.f64 V l) < -0.0 or 1.9999999999999998e293 < (*.f64 V l)

   1. Initial program 50.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. lower-/.f64 N/A

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

      5. lower-/.f64 72.8

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

   4. Applied rewrites 72.8%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. metadata-eval N/A

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

      11. sqrt-div N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-times N/A

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

      16. lift-sqrt.f64 N/A

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

      17. sqrt-prod N/A

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

      18. *-commutative N/A

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

      19. associate-/r/ N/A

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

      20. lift-/.f64 N/A

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

      21. metadata-eval N/A

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

      22. *-rgt-identity N/A

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

   6. Applied rewrites 74.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ 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. *-commutative N/A

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

      7. lower-*.f64 74.0

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

   8. Applied rewrites 74.0%

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

   if -0.0 < (*.f64 V l) < 1.9999999999999998e293

   1. Initial program 86.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 99.1

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

      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 99.1

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

   4. Applied rewrites 99.1%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-312}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -4 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq -4 \cdot 10^{-187}:\\ \;\;\;\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) -4e+119)
   (* (sqrt (/ (/ A l) V)) c0)
   (if (<= (* l V) -4e-187)
     (* (sqrt (/ A (* l V))) c0)
     (if (<= (* l V) 0.0)
       (/ c0 (sqrt (* (/ l A) V)))
       (if (<= (* l V) 2e+293)
         (* (/ (sqrt A) (sqrt (* l V))) c0)
         (/ c0 (sqrt (* (/ V A) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -4e+119) {
		tmp = sqrt(((A / l) / V)) * c0;
	} else if ((l * V) <= -4e-187) {
		tmp = sqrt((A / (l * V))) * c0;
	} else if ((l * V) <= 0.0) {
		tmp = c0 / sqrt(((l / A) * V));
	} else if ((l * V) <= 2e+293) {
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	} 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) :: tmp
    if ((l * v) <= (-4d+119)) then
        tmp = sqrt(((a / l) / v)) * c0
    else if ((l * v) <= (-4d-187)) then
        tmp = sqrt((a / (l * v))) * c0
    else if ((l * v) <= 0.0d0) then
        tmp = c0 / sqrt(((l / a) * v))
    else if ((l * v) <= 2d+293) then
        tmp = (sqrt(a) / sqrt((l * v))) * c0
    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 tmp;
	if ((l * V) <= -4e+119) {
		tmp = Math.sqrt(((A / l) / V)) * c0;
	} else if ((l * V) <= -4e-187) {
		tmp = Math.sqrt((A / (l * V))) * c0;
	} else if ((l * V) <= 0.0) {
		tmp = c0 / Math.sqrt(((l / A) * V));
	} else if ((l * V) <= 2e+293) {
		tmp = (Math.sqrt(A) / Math.sqrt((l * V))) * c0;
	} 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):
	tmp = 0
	if (l * V) <= -4e+119:
		tmp = math.sqrt(((A / l) / V)) * c0
	elif (l * V) <= -4e-187:
		tmp = math.sqrt((A / (l * V))) * c0
	elif (l * V) <= 0.0:
		tmp = c0 / math.sqrt(((l / A) * V))
	elif (l * V) <= 2e+293:
		tmp = (math.sqrt(A) / math.sqrt((l * V))) * c0
	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)
	tmp = 0.0
	if (Float64(l * V) <= -4e+119)
		tmp = Float64(sqrt(Float64(Float64(A / l) / V)) * c0);
	elseif (Float64(l * V) <= -4e-187)
		tmp = Float64(sqrt(Float64(A / Float64(l * V))) * c0);
	elseif (Float64(l * V) <= 0.0)
		tmp = Float64(c0 / sqrt(Float64(Float64(l / A) * V)));
	elseif (Float64(l * V) <= 2e+293)
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(l * V))) * c0);
	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)
	tmp = 0.0;
	if ((l * V) <= -4e+119)
		tmp = sqrt(((A / l) / V)) * c0;
	elseif ((l * V) <= -4e-187)
		tmp = sqrt((A / (l * V))) * c0;
	elseif ((l * V) <= 0.0)
		tmp = c0 / sqrt(((l / A) * V));
	elseif ((l * V) <= 2e+293)
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	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_] := If[LessEqual[N[(l * V), $MachinePrecision], -4e+119], N[(N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -4e-187], N[(N[Sqrt[N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], N[(c0 / N[Sqrt[N[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 2e+293], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], N[(c0 / N[Sqrt[N[(N[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -3.99999999999999978e119

   1. Initial program 54.9%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 74.1

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

   4. Applied rewrites 74.1%

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

   if -3.99999999999999978e119 < (*.f64 V l) < -4.0000000000000001e-187

   1. Initial program 93.9%

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

   if -4.0000000000000001e-187 < (*.f64 V l) < -0.0

   1. Initial program 64.0%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 75.9

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

   4. Applied rewrites 75.9%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. metadata-eval N/A

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

      11. sqrt-div N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-times N/A

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

      16. lift-sqrt.f64 N/A

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

      17. sqrt-prod N/A

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

      18. *-commutative N/A

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

      19. associate-/r/ N/A

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

      20. lift-/.f64 N/A

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

      21. metadata-eval N/A

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

      22. *-rgt-identity N/A

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

   6. Applied rewrites 75.2%

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

   if -0.0 < (*.f64 V l) < 1.9999999999999998e293

   1. Initial program 86.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 99.1

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

      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 99.1

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

   4. Applied rewrites 99.1%

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

   if 1.9999999999999998e293 < (*.f64 V l)

   1. Initial program 44.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 73.6

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

   4. Applied rewrites 73.6%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. metadata-eval N/A

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

      11. sqrt-div N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-times N/A

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

      16. lift-sqrt.f64 N/A

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

      17. sqrt-prod N/A

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

      18. *-commutative N/A

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

      19. associate-/r/ N/A

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

      20. lift-/.f64 N/A

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

      21. metadata-eval N/A

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

      22. *-rgt-identity N/A

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

   6. Applied rewrites 73.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ 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. *-commutative N/A

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

      7. lower-*.f64 73.7

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

   8. Applied rewrites 73.7%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -4 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq -4 \cdot 10^{-187}:\\ \;\;\;\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-317}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+276}:\\ \;\;\;\;\sqrt{t\_0} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \end{array} \end{array} \]

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 (/ A (* l V))))
   (if (<= t_0 5e-317)
     (* (sqrt (/ (/ A l) V)) c0)
     (if (<= t_0 2e+276) (* (sqrt t_0) c0) (/ 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 = A / (l * V);
	double tmp;
	if (t_0 <= 5e-317) {
		tmp = sqrt(((A / l) / V)) * c0;
	} else if (t_0 <= 2e+276) {
		tmp = sqrt(t_0) * c0;
	} 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 = a / (l * v)
    if (t_0 <= 5d-317) then
        tmp = sqrt(((a / l) / v)) * c0
    else if (t_0 <= 2d+276) then
        tmp = sqrt(t_0) * c0
    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 = A / (l * V);
	double tmp;
	if (t_0 <= 5e-317) {
		tmp = Math.sqrt(((A / l) / V)) * c0;
	} else if (t_0 <= 2e+276) {
		tmp = Math.sqrt(t_0) * c0;
	} 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 = A / (l * V)
	tmp = 0
	if t_0 <= 5e-317:
		tmp = math.sqrt(((A / l) / V)) * c0
	elif t_0 <= 2e+276:
		tmp = math.sqrt(t_0) * c0
	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(A / Float64(l * V))
	tmp = 0.0
	if (t_0 <= 5e-317)
		tmp = Float64(sqrt(Float64(Float64(A / l) / V)) * c0);
	elseif (t_0 <= 2e+276)
		tmp = Float64(sqrt(t_0) * c0);
	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 = A / (l * V);
	tmp = 0.0;
	if (t_0 <= 5e-317)
		tmp = sqrt(((A / l) / V)) * c0;
	elseif (t_0 <= 2e+276)
		tmp = sqrt(t_0) * c0;
	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[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-317], N[(N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[t$95$0, 2e+276], N[(N[Sqrt[t$95$0], $MachinePrecision] * c0), $MachinePrecision], N[(c0 / N[Sqrt[N[(N[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 5.00000017e-317

   1. Initial program 36.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 63.9

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

   4. Applied rewrites 63.9%

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

   if 5.00000017e-317 < (/.f64 A (*.f64 V l)) < 2.0000000000000001e276

   1. Initial program 99.5%

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

   if 2.0000000000000001e276 < (/.f64 A (*.f64 V l))

   1. Initial program 49.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. lower-/.f64 N/A

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

      5. lower-/.f64 60.9

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

   4. Applied rewrites 60.9%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. metadata-eval N/A

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

      11. sqrt-div N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-times N/A

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

      16. lift-sqrt.f64 N/A

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

      17. sqrt-prod N/A

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

      18. *-commutative N/A

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

      19. associate-/r/ N/A

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

      20. lift-/.f64 N/A

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

      21. metadata-eval N/A

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

      22. *-rgt-identity N/A

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

   6. Applied rewrites 63.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ 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. *-commutative N/A

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

      7. lower-*.f64 62.1

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

   8. Applied rewrites 62.1%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 5 \cdot 10^{-317}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;\frac{A}{\ell \cdot V} \leq 2 \cdot 10^{+276}:\\ \;\;\;\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ t_1 := \sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-317}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+262}:\\ \;\;\;\;\sqrt{t\_0} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (/ A (* l V))) (t_1 (* (sqrt (/ (/ A l) V)) c0)))
   (if (<= t_0 5e-317) t_1 (if (<= t_0 1e+262) (* (sqrt t_0) c0) t_1))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double t_1 = sqrt(((A / l) / V)) * c0;
	double tmp;
	if (t_0 <= 5e-317) {
		tmp = t_1;
	} else if (t_0 <= 1e+262) {
		tmp = sqrt(t_0) * c0;
	} 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 = a / (l * v)
    t_1 = sqrt(((a / l) / v)) * c0
    if (t_0 <= 5d-317) then
        tmp = t_1
    else if (t_0 <= 1d+262) then
        tmp = sqrt(t_0) * c0
    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 = A / (l * V);
	double t_1 = Math.sqrt(((A / l) / V)) * c0;
	double tmp;
	if (t_0 <= 5e-317) {
		tmp = t_1;
	} else if (t_0 <= 1e+262) {
		tmp = Math.sqrt(t_0) * c0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (l * V)
	t_1 = math.sqrt(((A / l) / V)) * c0
	tmp = 0
	if t_0 <= 5e-317:
		tmp = t_1
	elif t_0 <= 1e+262:
		tmp = math.sqrt(t_0) * c0
	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(A / Float64(l * V))
	t_1 = Float64(sqrt(Float64(Float64(A / l) / V)) * c0)
	tmp = 0.0
	if (t_0 <= 5e-317)
		tmp = t_1;
	elseif (t_0 <= 1e+262)
		tmp = Float64(sqrt(t_0) * c0);
	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 = A / (l * V);
	t_1 = sqrt(((A / l) / V)) * c0;
	tmp = 0.0;
	if (t_0 <= 5e-317)
		tmp = t_1;
	elseif (t_0 <= 1e+262)
		tmp = sqrt(t_0) * c0;
	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[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-317], t$95$1, If[LessEqual[t$95$0, 1e+262], N[(N[Sqrt[t$95$0], $MachinePrecision] * c0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 5.00000017e-317 or 1e262 < (/.f64 A (*.f64 V l))

   1. Initial program 44.0%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 62.2

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

   4. Applied rewrites 62.2%

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

   if 5.00000017e-317 < (/.f64 A (*.f64 V l)) < 1e262

   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 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 5 \cdot 10^{-317}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;\frac{A}{\ell \cdot V} \leq 10^{+262}:\\ \;\;\;\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\ell} \cdot \sqrt{-V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \end{array} \end{array} \]

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 (<= A -5e-310)
   (* (/ (sqrt (- A)) (* (sqrt l) (sqrt (- V)))) c0)
   (* (/ (sqrt A) (sqrt (* l V))) c0)))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = (sqrt(-A) / (sqrt(l) * sqrt(-V))) * c0;
	} else {
		tmp = (sqrt(A) / sqrt((l * V))) * 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 (a <= (-5d-310)) then
        tmp = (sqrt(-a) / (sqrt(l) * sqrt(-v))) * c0
    else
        tmp = (sqrt(a) / sqrt((l * v))) * 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 (A <= -5e-310) {
		tmp = (Math.sqrt(-A) / (Math.sqrt(l) * Math.sqrt(-V))) * c0;
	} else {
		tmp = (Math.sqrt(A) / Math.sqrt((l * V))) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -5e-310:
		tmp = (math.sqrt(-A) / (math.sqrt(l) * math.sqrt(-V))) * c0
	else:
		tmp = (math.sqrt(A) / math.sqrt((l * V))) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -5e-310)
		tmp = Float64(Float64(sqrt(Float64(-A)) / Float64(sqrt(l) * sqrt(Float64(-V)))) * c0);
	else
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(l * V))) * 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 (A <= -5e-310)
		tmp = (sqrt(-A) / (sqrt(l) * sqrt(-V))) * c0;
	else
		tmp = (sqrt(A) / sqrt((l * V))) * 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[A, -5e-310], N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -4.999999999999985e-310

   1. Initial program 72.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. lower-/.f64 N/A

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

      5. lower-/.f64 78.6

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

   4. Applied rewrites 78.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. neg-mul-1 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-neg.f64 N/A

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

      13. sqrt-div N/A

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

      14. frac-times N/A

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

      15. lift-/.f64 N/A

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

      16. associate-/r/ N/A

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

      17. lift-/.f64 N/A

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

      18. frac-2neg N/A

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

      19. sqrt-div N/A

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

      20. pow1/2 N/A

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

   6. Applied rewrites 80.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow1/2 N/A

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

      8. lower-sqrt.f64 43.0

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

   8. Applied rewrites 43.0%

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

   if -4.999999999999985e-310 < A

   1. Initial program 76.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 85.6

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

      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 85.6

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

   4. Applied rewrites 85.6%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\ell} \cdot \sqrt{-V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 86.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V} \cdot \sqrt{\ell}} \cdot \sqrt{A}\\ \end{array} \end{array} \]

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 (<= V -5e-310)
   (* (/ (sqrt (/ (- A) l)) (sqrt (- V))) c0)
   (* (/ c0 (* (sqrt V) (sqrt l))) (sqrt A))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (V <= -5e-310) {
		tmp = (sqrt((-A / l)) / sqrt(-V)) * c0;
	} else {
		tmp = (c0 / (sqrt(V) * sqrt(l))) * sqrt(A);
	}
	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 (v <= (-5d-310)) then
        tmp = (sqrt((-a / l)) / sqrt(-v)) * c0
    else
        tmp = (c0 / (sqrt(v) * sqrt(l))) * sqrt(a)
    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 (V <= -5e-310) {
		tmp = (Math.sqrt((-A / l)) / Math.sqrt(-V)) * c0;
	} else {
		tmp = (c0 / (Math.sqrt(V) * Math.sqrt(l))) * Math.sqrt(A);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if V <= -5e-310:
		tmp = (math.sqrt((-A / l)) / math.sqrt(-V)) * c0
	else:
		tmp = (c0 / (math.sqrt(V) * math.sqrt(l))) * math.sqrt(A)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (V <= -5e-310)
		tmp = Float64(Float64(sqrt(Float64(Float64(-A) / l)) / sqrt(Float64(-V))) * c0);
	else
		tmp = Float64(Float64(c0 / Float64(sqrt(V) * sqrt(l))) * sqrt(A));
	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 (V <= -5e-310)
		tmp = (sqrt((-A / l)) / sqrt(-V)) * c0;
	else
		tmp = (c0 / (sqrt(V) * sqrt(l))) * sqrt(A);
	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[V, -5e-310], N[(N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], N[(N[(c0 / N[(N[Sqrt[V], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if V < -4.999999999999985e-310

   1. Initial program 77.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 81.8

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

   4. Applied rewrites 81.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. sqrt-div N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-frac-neg2 N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. distribute-frac-neg2 N/A

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

      16. distribute-neg-frac N/A

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

      17. lower-/.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-neg.f64 92.9

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

   6. Applied rewrites 92.9%

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

   if -4.999999999999985e-310 < V

   1. Initial program 71.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 39.8

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

   4. Applied rewrites 39.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lower-*.f64 45.9

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

   6. Applied rewrites 45.9%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V} \cdot \sqrt{\ell}} \cdot \sqrt{A}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 83.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \leq -1.02 \cdot 10^{-237}:\\ \;\;\;\;\frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

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 (<= V -1.02e-237)
   (* (/ (sqrt (/ (- A) 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 (V <= -1.02e-237) {
		tmp = (sqrt((-A / 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 (v <= (-1.02d-237)) then
        tmp = (sqrt((-a / 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 (V <= -1.02e-237) {
		tmp = (Math.sqrt((-A / 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 V <= -1.02e-237:
		tmp = (math.sqrt((-A / 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 (V <= -1.02e-237)
		tmp = Float64(Float64(sqrt(Float64(Float64(-A) / 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 (V <= -1.02e-237)
		tmp = (sqrt((-A / 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[V, -1.02e-237], N[(N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $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 V < -1.02e-237

   1. Initial program 76.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 80.9

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

   4. Applied rewrites 80.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. sqrt-div N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-frac-neg2 N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. distribute-frac-neg2 N/A

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

      16. distribute-neg-frac N/A

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

      17. lower-/.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-neg.f64 94.0

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

   6. Applied rewrites 94.0%

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

   if -1.02e-237 < V

   1. Initial program 72.9%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. lower-/.f64 N/A

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

      5. lower-/.f64 75.2

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

   4. Applied rewrites 75.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. metadata-eval N/A

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

      11. sqrt-div N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-times N/A

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

      16. lift-sqrt.f64 N/A

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

      17. sqrt-prod N/A

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

      18. *-commutative N/A

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

      19. associate-/r/ N/A

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

      20. lift-/.f64 N/A

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

      21. metadata-eval N/A

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

      22. *-rgt-identity N/A

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

   6. Applied rewrites 75.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-unprod N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 40.7

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

   8. Applied rewrites 40.7%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \leq -1.02 \cdot 10^{-237}:\\ \;\;\;\;\frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 73.2% 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 74.4%

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

3. Final simplification 74.4%

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

Reproduce

herbie shell --seed 2024250 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))