Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.7% → 89.7%

Time: 7.2s

Alternatives: 11

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: 73.7% 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: 89.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{-76}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{\ell} \cdot \sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-310}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{-\ell} \cdot \frac{A}{V}} \cdot c0\\ \end{array} \end{array} \]

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 (<= (* V l) -5e-76)
   (/ (* (sqrt (- A)) c0) (* (sqrt l) (sqrt (- V))))
   (if (<= (* V l) 1e-310)
     (/ c0 (* (sqrt (/ V A)) (sqrt l)))
     (if (<= (* V l) 4e+307)
       (/ c0 (/ (sqrt (* V l)) (sqrt A)))
       (* (sqrt (* (/ -1.0 (- l)) (/ A V))) c0)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e-76) {
		tmp = (sqrt(-A) * c0) / (sqrt(l) * sqrt(-V));
	} else if ((V * l) <= 1e-310) {
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	} else if ((V * l) <= 4e+307) {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = sqrt(((-1.0 / -l) * (A / 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 ((v * l) <= (-5d-76)) then
        tmp = (sqrt(-a) * c0) / (sqrt(l) * sqrt(-v))
    else if ((v * l) <= 1d-310) then
        tmp = c0 / (sqrt((v / a)) * sqrt(l))
    else if ((v * l) <= 4d+307) then
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    else
        tmp = sqrt((((-1.0d0) / -l) * (a / 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 ((V * l) <= -5e-76) {
		tmp = (Math.sqrt(-A) * c0) / (Math.sqrt(l) * Math.sqrt(-V));
	} else if ((V * l) <= 1e-310) {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	} else if ((V * l) <= 4e+307) {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	} else {
		tmp = Math.sqrt(((-1.0 / -l) * (A / V))) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -5e-76:
		tmp = (math.sqrt(-A) * c0) / (math.sqrt(l) * math.sqrt(-V))
	elif (V * l) <= 1e-310:
		tmp = c0 / (math.sqrt((V / A)) * math.sqrt(l))
	elif (V * l) <= 4e+307:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	else:
		tmp = math.sqrt(((-1.0 / -l) * (A / V))) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -5e-76)
		tmp = Float64(Float64(sqrt(Float64(-A)) * c0) / Float64(sqrt(l) * sqrt(Float64(-V))));
	elseif (Float64(V * l) <= 1e-310)
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * sqrt(l)));
	elseif (Float64(V * l) <= 4e+307)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	else
		tmp = Float64(sqrt(Float64(Float64(-1.0 / Float64(-l)) * Float64(A / 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 ((V * l) <= -5e-76)
		tmp = (sqrt(-A) * c0) / (sqrt(l) * sqrt(-V));
	elseif ((V * l) <= 1e-310)
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	elseif ((V * l) <= 4e+307)
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	else
		tmp = sqrt(((-1.0 / -l) * (A / 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[N[(V * l), $MachinePrecision], -5e-76], N[(N[(N[Sqrt[(-A)], $MachinePrecision] * c0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-310], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e+307], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(-1.0 / (-l)), $MachinePrecision] * N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -4.9999999999999998e-76

   1. Initial program 80.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 77.0

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

   4. Applied rewrites 77.0%

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

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

      6. lift-sqrt.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. frac-2neg N/A

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

      13. sqrt-div N/A

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

      14. frac-2neg N/A

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

      15. frac-times N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-neg.f64 N/A

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

      21. lower-*.f64 N/A

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

   6. Applied rewrites 51.9%

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

   if -4.9999999999999998e-76 < (*.f64 V l) < 9.999999999999969e-311

   1. Initial program 64.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 78.5

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

   4. Applied rewrites 78.5%

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

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

      6. lift-sqrt.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. lift-/.f64 N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lift-sqrt.f64 N/A

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

      15. div-inv N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

      19. lift-sqrt.f64 N/A

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

      20. sqrt-unprod N/A

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

      21. lift-/.f64 N/A

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

      22. associate-/r/ N/A

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

      23. lift-/.f64 N/A

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

   6. Applied rewrites 79.5%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. pow-prod-down N/A

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

      9. pow1/2 N/A

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

      10. pow1/2 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-/.f64 53.8

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

   8. Applied rewrites 53.8%

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

   if 9.999999999999969e-311 < (*.f64 V l) < 3.99999999999999994e307

   1. Initial program 85.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 73.7

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

   4. Applied rewrites 73.7%

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

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

      6. lift-sqrt.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. lift-/.f64 N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lift-sqrt.f64 N/A

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

      15. div-inv N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

      19. lift-sqrt.f64 N/A

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

      20. sqrt-unprod N/A

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

      21. lift-/.f64 N/A

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

      22. associate-/r/ N/A

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

      23. lift-/.f64 N/A

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

   6. Applied rewrites 78.4%

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

   8. Applied rewrites 99.5%

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

   if 3.99999999999999994e307 < (*.f64 V l)

   1. Initial program 32.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 60.1

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

   4. Applied rewrites 60.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. distribute-frac-neg2 N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 60.0

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

   6. Applied rewrites 60.0%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{-A}{V} \cdot \frac{-1}{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{-76}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{\ell} \cdot \sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-310}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{-\ell} \cdot \frac{A}{V}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 10^{-317}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+291}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \end{array} \end{array} \]

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 (/ A (* V l))))
   (if (<= t_0 1e-317)
     (* (sqrt (/ (/ A V) l)) c0)
     (if (<= t_0 2e+291)
       (/ c0 (sqrt (/ (* V l) A)))
       (/ 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 = A / (V * l);
	double tmp;
	if (t_0 <= 1e-317) {
		tmp = sqrt(((A / V) / l)) * c0;
	} else if (t_0 <= 2e+291) {
		tmp = c0 / sqrt(((V * l) / A));
	} 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 = a / (v * l)
    if (t_0 <= 1d-317) then
        tmp = sqrt(((a / v) / l)) * c0
    else if (t_0 <= 2d+291) then
        tmp = c0 / sqrt(((v * l) / a))
    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 = A / (V * l);
	double tmp;
	if (t_0 <= 1e-317) {
		tmp = Math.sqrt(((A / V) / l)) * c0;
	} else if (t_0 <= 2e+291) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} 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 = A / (V * l)
	tmp = 0
	if t_0 <= 1e-317:
		tmp = math.sqrt(((A / V) / l)) * c0
	elif t_0 <= 2e+291:
		tmp = c0 / math.sqrt(((V * l) / A))
	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(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 1e-317)
		tmp = Float64(sqrt(Float64(Float64(A / V) / l)) * c0);
	elseif (t_0 <= 2e+291)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	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 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 1e-317)
		tmp = sqrt(((A / V) / l)) * c0;
	elseif (t_0 <= 2e+291)
		tmp = c0 / sqrt(((V * l) / A));
	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[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-317], N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[t$95$0, 2e+291], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 1.00000023e-317

   1. Initial program 44.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 56.8

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

   4. Applied rewrites 56.8%

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

   if 1.00000023e-317 < (/.f64 A (*.f64 V l)) < 1.9999999999999999e291

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 99.6

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if 1.9999999999999999e291 < (/.f64 A (*.f64 V l))

   1. Initial program 41.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 56.3

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

   4. Applied rewrites 56.3%

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

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

      6. lift-sqrt.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. lift-/.f64 N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lift-sqrt.f64 N/A

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

      15. div-inv N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

      19. lift-sqrt.f64 N/A

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

      20. sqrt-unprod N/A

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

      21. lift-/.f64 N/A

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

      22. associate-/r/ N/A

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

      23. lift-/.f64 N/A

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

   6. Applied rewrites 58.4%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 10^{-317}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+291}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+291}:\\ \;\;\;\;\sqrt{t\_0} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \end{array} \end{array} \]

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 (/ A (* V l))))
   (if (<= t_0 0.0)
     (* (sqrt (/ (/ A V) l)) c0)
     (if (<= t_0 2e+291) (* (sqrt t_0) c0) (/ 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 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = sqrt(((A / V) / l)) * c0;
	} else if (t_0 <= 2e+291) {
		tmp = sqrt(t_0) * c0;
	} 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 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = sqrt(((a / v) / l)) * c0
    else if (t_0 <= 2d+291) then
        tmp = sqrt(t_0) * c0
    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 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.sqrt(((A / V) / l)) * c0;
	} else if (t_0 <= 2e+291) {
		tmp = Math.sqrt(t_0) * c0;
	} 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 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.sqrt(((A / V) / l)) * c0
	elif t_0 <= 2e+291:
		tmp = math.sqrt(t_0) * c0
	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(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(A / V) / l)) * c0);
	elseif (t_0 <= 2e+291)
		tmp = Float64(sqrt(t_0) * c0);
	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 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = sqrt(((A / V) / l)) * c0;
	elseif (t_0 <= 2e+291)
		tmp = sqrt(t_0) * c0;
	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[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[t$95$0, 2e+291], N[(N[Sqrt[t$95$0], $MachinePrecision] * c0), $MachinePrecision], N[(c0 / N[Sqrt[N[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0

   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-/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 56.9

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

   4. Applied rewrites 56.9%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 1.9999999999999999e291

   1. Initial program 99.1%

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

   if 1.9999999999999999e291 < (/.f64 A (*.f64 V l))

   1. Initial program 41.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 56.3

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

   4. Applied rewrites 56.3%

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

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

      6. lift-sqrt.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. lift-/.f64 N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lift-sqrt.f64 N/A

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

      15. div-inv N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

      19. lift-sqrt.f64 N/A

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

      20. sqrt-unprod N/A

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

      21. lift-/.f64 N/A

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

      22. associate-/r/ N/A

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

      23. lift-/.f64 N/A

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

   6. Applied rewrites 58.4%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+291}:\\ \;\;\;\;\sqrt{\frac{A}{V \cdot \ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ t_1 := \sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1.7 \cdot 10^{+273}:\\ \;\;\;\;\sqrt{t\_0} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* V l))) (t_1 (* (sqrt (/ (/ A V) l)) c0)))
   (if (<= t_0 0.0) t_1 (if (<= t_0 1.7e+273) (* (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 / (V * l);
	double t_1 = sqrt(((A / V) / l)) * c0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 1.7e+273) {
		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 / (v * l)
    t_1 = sqrt(((a / v) / l)) * c0
    if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 1.7d+273) 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 / (V * l);
	double t_1 = Math.sqrt(((A / V) / l)) * c0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 1.7e+273) {
		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 / (V * l)
	t_1 = math.sqrt(((A / V) / l)) * c0
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 1.7e+273:
		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(V * l))
	t_1 = Float64(sqrt(Float64(Float64(A / V) / l)) * c0)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 1.7e+273)
		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 / (V * l);
	t_1 = sqrt(((A / V) / l)) * c0;
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 1.7e+273)
		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[(V * l), $MachinePrecision]), $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, 1.7e+273], 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)) < 0.0 or 1.69999999999999999e273 < (/.f64 A (*.f64 V l))

   1. Initial program 45.0%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 57.6

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

   4. Applied rewrites 57.6%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 1.69999999999999999e273

   1. Initial program 99.1%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 1.7 \cdot 10^{+273}:\\ \;\;\;\;\sqrt{\frac{A}{V \cdot \ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{-306}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-310}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{-\ell} \cdot \frac{A}{V}} \cdot c0\\ \end{array} \end{array} \]

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 (<= (* V l) -1e-306)
   (* (/ (sqrt (- A)) (sqrt (* (- l) V))) c0)
   (if (<= (* V l) 1e-310)
     (* (sqrt (/ (/ A l) V)) c0)
     (if (<= (* V l) 4e+307)
       (* (/ (sqrt A) (sqrt (* V l))) c0)
       (* (sqrt (* (/ -1.0 (- l)) (/ A V))) c0)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e-306) {
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	} else if ((V * l) <= 1e-310) {
		tmp = sqrt(((A / l) / V)) * c0;
	} else if ((V * l) <= 4e+307) {
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	} else {
		tmp = sqrt(((-1.0 / -l) * (A / 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 ((v * l) <= (-1d-306)) then
        tmp = (sqrt(-a) / sqrt((-l * v))) * c0
    else if ((v * l) <= 1d-310) then
        tmp = sqrt(((a / l) / v)) * c0
    else if ((v * l) <= 4d+307) then
        tmp = (sqrt(a) / sqrt((v * l))) * c0
    else
        tmp = sqrt((((-1.0d0) / -l) * (a / 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 ((V * l) <= -1e-306) {
		tmp = (Math.sqrt(-A) / Math.sqrt((-l * V))) * c0;
	} else if ((V * l) <= 1e-310) {
		tmp = Math.sqrt(((A / l) / V)) * c0;
	} else if ((V * l) <= 4e+307) {
		tmp = (Math.sqrt(A) / Math.sqrt((V * l))) * c0;
	} else {
		tmp = Math.sqrt(((-1.0 / -l) * (A / V))) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e-306:
		tmp = (math.sqrt(-A) / math.sqrt((-l * V))) * c0
	elif (V * l) <= 1e-310:
		tmp = math.sqrt(((A / l) / V)) * c0
	elif (V * l) <= 4e+307:
		tmp = (math.sqrt(A) / math.sqrt((V * l))) * c0
	else:
		tmp = math.sqrt(((-1.0 / -l) * (A / V))) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e-306)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-l) * V))) * c0);
	elseif (Float64(V * l) <= 1e-310)
		tmp = Float64(sqrt(Float64(Float64(A / l) / V)) * c0);
	elseif (Float64(V * l) <= 4e+307)
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(V * l))) * c0);
	else
		tmp = Float64(sqrt(Float64(Float64(-1.0 / Float64(-l)) * Float64(A / 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 ((V * l) <= -1e-306)
		tmp = (sqrt(-A) / sqrt((-l * V))) * c0;
	elseif ((V * l) <= 1e-310)
		tmp = sqrt(((A / l) / V)) * c0;
	elseif ((V * l) <= 4e+307)
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	else
		tmp = sqrt(((-1.0 / -l) * (A / 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[N[(V * l), $MachinePrecision], -1e-306], N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-310], N[(N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e+307], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], N[(N[Sqrt[N[(N[(-1.0 / (-l)), $MachinePrecision] * N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -1.00000000000000003e-306

   1. Initial program 81.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 78.9

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

   4. Applied rewrites 78.9%

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

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. sqrt-prod N/A

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

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

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

      15. lift-*.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

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

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

      20. lower-*.f64 N/A

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

      21. lower-neg.f64 92.6

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

   6. Applied rewrites 92.6%

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

   if -1.00000000000000003e-306 < (*.f64 V l) < 9.999999999999969e-311

   1. Initial program 47.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 73.9

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

   4. Applied rewrites 73.9%

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

   if 9.999999999999969e-311 < (*.f64 V l) < 3.99999999999999994e307

   1. Initial program 85.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 99.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.4

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

   4. Applied rewrites 99.4%

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

   if 3.99999999999999994e307 < (*.f64 V l)

   1. Initial program 32.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 60.1

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

   4. Applied rewrites 60.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. distribute-frac-neg2 N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 60.0

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

   6. Applied rewrites 60.0%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{-A}{V} \cdot \frac{-1}{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{-306}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-\ell\right) \cdot V}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-310}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{-\ell} \cdot \frac{A}{V}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 10^{-310}:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{-\ell} \cdot \frac{A}{V}} \cdot c0\\ \end{array} \end{array} \]

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 (<= (* V l) 1e-310)
   (* (/ (sqrt (/ A V)) (sqrt l)) c0)
   (if (<= (* V l) 4e+307)
     (/ c0 (/ (sqrt (* V l)) (sqrt A)))
     (* (sqrt (* (/ -1.0 (- l)) (/ A V))) c0))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 1e-310) {
		tmp = (sqrt((A / V)) / sqrt(l)) * c0;
	} else if ((V * l) <= 4e+307) {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = sqrt(((-1.0 / -l) * (A / 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 ((v * l) <= 1d-310) then
        tmp = (sqrt((a / v)) / sqrt(l)) * c0
    else if ((v * l) <= 4d+307) then
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    else
        tmp = sqrt((((-1.0d0) / -l) * (a / 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 ((V * l) <= 1e-310) {
		tmp = (Math.sqrt((A / V)) / Math.sqrt(l)) * c0;
	} else if ((V * l) <= 4e+307) {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	} else {
		tmp = Math.sqrt(((-1.0 / -l) * (A / V))) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= 1e-310:
		tmp = (math.sqrt((A / V)) / math.sqrt(l)) * c0
	elif (V * l) <= 4e+307:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	else:
		tmp = math.sqrt(((-1.0 / -l) * (A / V))) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= 1e-310)
		tmp = Float64(Float64(sqrt(Float64(A / V)) / sqrt(l)) * c0);
	elseif (Float64(V * l) <= 4e+307)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	else
		tmp = Float64(sqrt(Float64(Float64(-1.0 / Float64(-l)) * Float64(A / 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 ((V * l) <= 1e-310)
		tmp = (sqrt((A / V)) / sqrt(l)) * c0;
	elseif ((V * l) <= 4e+307)
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	else
		tmp = sqrt(((-1.0 / -l) * (A / 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[N[(V * l), $MachinePrecision], 1e-310], N[(N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e+307], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(-1.0 / (-l)), $MachinePrecision] * N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < 9.999999999999969e-311

   1. Initial program 73.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 49.1

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

   4. Applied rewrites 49.1%

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

   if 9.999999999999969e-311 < (*.f64 V l) < 3.99999999999999994e307

   1. Initial program 85.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 73.7

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

   4. Applied rewrites 73.7%

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

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

      6. lift-sqrt.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. lift-/.f64 N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lift-sqrt.f64 N/A

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

      15. div-inv N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lift-sqrt.f64 N/A

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

      19. lift-sqrt.f64 N/A

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

      20. sqrt-unprod N/A

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

      21. lift-/.f64 N/A

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

      22. associate-/r/ N/A

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

      23. lift-/.f64 N/A

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

   6. Applied rewrites 78.4%

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

   8. Applied rewrites 99.5%

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

   if 3.99999999999999994e307 < (*.f64 V l)

   1. Initial program 32.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 60.1

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

   4. Applied rewrites 60.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. distribute-frac-neg2 N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 60.0

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

   6. Applied rewrites 60.0%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 10^{-310}:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{-\ell} \cdot \frac{A}{V}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.1% accurate, 0.5× speedup

Math

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

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

C

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

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 74.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 73.1

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

   4. Applied rewrites 73.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. sqrt-div N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. distribute-neg-frac N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-neg.f64 80.5

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

   6. Applied rewrites 80.5%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. pow1/2 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. sqr-pow N/A

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

      9. pow-prod-down N/A

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

      10. sqr-neg N/A

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

      11. lift-neg.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lift-neg.f64 N/A

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

      14. remove-double-neg N/A

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

      15. pow-prod-down N/A

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

      16. sqr-pow N/A

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

      17. pow1/2 N/A

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

      18. associate-/l* N/A

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

      19. associate-*l/ N/A

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

   8. Applied rewrites 53.1%

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

   if -4.999999999999985e-310 < l

   1. Initial program 76.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 88.9

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

   4. Applied rewrites 88.9%

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

4. Final simplification 70.9%

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

Alternative 8: 85.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 10^{-310}:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{-\ell} \cdot \frac{A}{V}} \cdot c0\\ \end{array} \end{array} \]

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 (<= (* V l) 1e-310)
   (* (/ (sqrt (/ A V)) (sqrt l)) c0)
   (if (<= (* V l) 4e+307)
     (* (/ (sqrt A) (sqrt (* V l))) c0)
     (* (sqrt (* (/ -1.0 (- l)) (/ A V))) c0))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 1e-310) {
		tmp = (sqrt((A / V)) / sqrt(l)) * c0;
	} else if ((V * l) <= 4e+307) {
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	} else {
		tmp = sqrt(((-1.0 / -l) * (A / 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 ((v * l) <= 1d-310) then
        tmp = (sqrt((a / v)) / sqrt(l)) * c0
    else if ((v * l) <= 4d+307) then
        tmp = (sqrt(a) / sqrt((v * l))) * c0
    else
        tmp = sqrt((((-1.0d0) / -l) * (a / 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 ((V * l) <= 1e-310) {
		tmp = (Math.sqrt((A / V)) / Math.sqrt(l)) * c0;
	} else if ((V * l) <= 4e+307) {
		tmp = (Math.sqrt(A) / Math.sqrt((V * l))) * c0;
	} else {
		tmp = Math.sqrt(((-1.0 / -l) * (A / V))) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= 1e-310:
		tmp = (math.sqrt((A / V)) / math.sqrt(l)) * c0
	elif (V * l) <= 4e+307:
		tmp = (math.sqrt(A) / math.sqrt((V * l))) * c0
	else:
		tmp = math.sqrt(((-1.0 / -l) * (A / V))) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= 1e-310)
		tmp = Float64(Float64(sqrt(Float64(A / V)) / sqrt(l)) * c0);
	elseif (Float64(V * l) <= 4e+307)
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(V * l))) * c0);
	else
		tmp = Float64(sqrt(Float64(Float64(-1.0 / Float64(-l)) * Float64(A / 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 ((V * l) <= 1e-310)
		tmp = (sqrt((A / V)) / sqrt(l)) * c0;
	elseif ((V * l) <= 4e+307)
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	else
		tmp = sqrt(((-1.0 / -l) * (A / 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[N[(V * l), $MachinePrecision], 1e-310], N[(N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e+307], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], N[(N[Sqrt[N[(N[(-1.0 / (-l)), $MachinePrecision] * N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < 9.999999999999969e-311

   1. Initial program 73.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 49.1

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

   4. Applied rewrites 49.1%

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

   if 9.999999999999969e-311 < (*.f64 V l) < 3.99999999999999994e307

   1. Initial program 85.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 99.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.4

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

   4. Applied rewrites 99.4%

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

   if 3.99999999999999994e307 < (*.f64 V l)

   1. Initial program 32.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 60.1

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

   4. Applied rewrites 60.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. distribute-frac-neg2 N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 60.0

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

   6. Applied rewrites 60.0%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 10^{-310}:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{-\ell} \cdot \frac{A}{V}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 10^{-310}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{-\ell} \cdot \frac{A}{V}} \cdot c0\\ \end{array} \end{array} \]

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 (<= (* V l) 1e-310)
   (* (sqrt (/ (/ A V) l)) c0)
   (if (<= (* V l) 4e+307)
     (* (/ (sqrt A) (sqrt (* V l))) c0)
     (* (sqrt (* (/ -1.0 (- l)) (/ A V))) c0))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= 1e-310) {
		tmp = sqrt(((A / V) / l)) * c0;
	} else if ((V * l) <= 4e+307) {
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	} else {
		tmp = sqrt(((-1.0 / -l) * (A / 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 ((v * l) <= 1d-310) then
        tmp = sqrt(((a / v) / l)) * c0
    else if ((v * l) <= 4d+307) then
        tmp = (sqrt(a) / sqrt((v * l))) * c0
    else
        tmp = sqrt((((-1.0d0) / -l) * (a / 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 ((V * l) <= 1e-310) {
		tmp = Math.sqrt(((A / V) / l)) * c0;
	} else if ((V * l) <= 4e+307) {
		tmp = (Math.sqrt(A) / Math.sqrt((V * l))) * c0;
	} else {
		tmp = Math.sqrt(((-1.0 / -l) * (A / V))) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= 1e-310:
		tmp = math.sqrt(((A / V) / l)) * c0
	elif (V * l) <= 4e+307:
		tmp = (math.sqrt(A) / math.sqrt((V * l))) * c0
	else:
		tmp = math.sqrt(((-1.0 / -l) * (A / V))) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= 1e-310)
		tmp = Float64(sqrt(Float64(Float64(A / V) / l)) * c0);
	elseif (Float64(V * l) <= 4e+307)
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(V * l))) * c0);
	else
		tmp = Float64(sqrt(Float64(Float64(-1.0 / Float64(-l)) * Float64(A / 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 ((V * l) <= 1e-310)
		tmp = sqrt(((A / V) / l)) * c0;
	elseif ((V * l) <= 4e+307)
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	else
		tmp = sqrt(((-1.0 / -l) * (A / 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[N[(V * l), $MachinePrecision], 1e-310], N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e+307], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], N[(N[Sqrt[N[(N[(-1.0 / (-l)), $MachinePrecision] * N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < 9.999999999999969e-311

   1. Initial program 73.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 77.7

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

   4. Applied rewrites 77.7%

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

   if 9.999999999999969e-311 < (*.f64 V l) < 3.99999999999999994e307

   1. Initial program 85.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 99.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.4

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

   4. Applied rewrites 99.4%

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

   if 3.99999999999999994e307 < (*.f64 V l)

   1. Initial program 32.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 60.1

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

   4. Applied rewrites 60.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. distribute-frac-neg2 N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 60.0

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

   6. Applied rewrites 60.0%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 10^{-310}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{-\ell} \cdot \frac{A}{V}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{if}\;V \cdot \ell \leq 10^{-310}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* (sqrt (/ (/ A V) l)) c0)))
   (if (<= (* V l) 1e-310)
     t_0
     (if (<= (* V l) 4e+307) (* (/ (sqrt A) (sqrt (* V l))) c0) t_0))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt(((A / V) / l)) * c0;
	double tmp;
	if ((V * l) <= 1e-310) {
		tmp = t_0;
	} else if ((V * l) <= 4e+307) {
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((a / v) / l)) * c0
    if ((v * l) <= 1d-310) then
        tmp = t_0
    else if ((v * l) <= 4d+307) then
        tmp = (sqrt(a) / sqrt((v * l))) * c0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt(((A / V) / l)) * c0;
	double tmp;
	if ((V * l) <= 1e-310) {
		tmp = t_0;
	} else if ((V * l) <= 4e+307) {
		tmp = (Math.sqrt(A) / Math.sqrt((V * l))) * c0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt(((A / V) / l)) * c0
	tmp = 0
	if (V * l) <= 1e-310:
		tmp = t_0
	elif (V * l) <= 4e+307:
		tmp = (math.sqrt(A) / math.sqrt((V * l))) * c0
	else:
		tmp = t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(sqrt(Float64(Float64(A / V) / l)) * c0)
	tmp = 0.0
	if (Float64(V * l) <= 1e-310)
		tmp = t_0;
	elseif (Float64(V * l) <= 4e+307)
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(V * l))) * c0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt(((A / V) / l)) * c0;
	tmp = 0.0;
	if ((V * l) <= 1e-310)
		tmp = t_0;
	elseif ((V * l) <= 4e+307)
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], 1e-310], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], 4e+307], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 V l) < 9.999999999999969e-311 or 3.99999999999999994e307 < (*.f64 V l)

   1. Initial program 68.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 75.7

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

   4. Applied rewrites 75.7%

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

   if 9.999999999999969e-311 < (*.f64 V l) < 3.99999999999999994e307

   1. Initial program 85.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 99.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.4

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

   4. Applied rewrites 99.4%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 10^{-310}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \sqrt{\frac{A}{V \cdot \ell}} \cdot c0 \end{array} \]

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 (* V l))) c0))

C

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

Python

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

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	return Float64(sqrt(Float64(A / Float64(V * l))) * c0)
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp = code(c0, A, V, l)
	tmp = sqrt((A / (V * l))) * 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[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]

Derivation

1. Initial program 75.2%

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

3. Final simplification 75.2%

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

Reproduce

herbie shell --seed 2024251 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))