Henrywood and Agarwal, Equation (3)

Percentage Accurate: 72.8% → 89.2%

Time: 11.6s

Alternatives: 12

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: 72.8% 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: 89.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+189}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -8 \cdot 10^{-294}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-316}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e+189)
   (* (sqrt (/ A V)) (/ c0 (sqrt l)))
   (if (<= (* V l) -8e-294)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 5e-316)
       (/ c0 (* (sqrt l) (sqrt (/ V A))))
       (* c0 (/ (sqrt A) (sqrt (* V l))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+189) {
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	} else if ((V * l) <= -8e-294) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 5e-316) {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: 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+189)) then
        tmp = sqrt((a / v)) * (c0 / sqrt(l))
    else if ((v * l) <= (-8d-294)) then
        tmp = c0 * (sqrt(-a) / sqrt((v * -l)))
    else if ((v * l) <= 5d-316) then
        tmp = c0 / (sqrt(l) * sqrt((v / a)))
    else
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+189) {
		tmp = Math.sqrt((A / V)) * (c0 / Math.sqrt(l));
	} else if ((V * l) <= -8e-294) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 5e-316) {
		tmp = c0 / (Math.sqrt(l) * Math.sqrt((V / A)));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e+189:
		tmp = math.sqrt((A / V)) * (c0 / math.sqrt(l))
	elif (V * l) <= -8e-294:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 5e-316:
		tmp = c0 / (math.sqrt(l) * math.sqrt((V / A)))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e+189)
		tmp = Float64(sqrt(Float64(A / V)) * Float64(c0 / sqrt(l)));
	elseif (Float64(V * l) <= -8e-294)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 5e-316)
		tmp = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e+189)
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	elseif ((V * l) <= -8e-294)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 5e-316)
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	else
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: 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+189], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -8e-294], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-316], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -1e189

   1. Initial program 52.2%

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

      1. clear-num 52.2%

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

      2. sqrt-div 52.2%

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

      3. metadata-eval 52.2%

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

      4. *-commutative 52.2%

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

      5. associate-/l* 71.8%

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

   3. Applied egg-rr 71.8%

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

      1. associate-/l* 52.2%

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

      2. *-commutative 52.2%

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

      3. associate-/l* 72.1%

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

      4. associate-/r/ 69.6%

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

   5. Simplified 69.6%

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

      1. add-sqr-sqrt 40.6%

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

      2. sqrt-unprod 29.5%

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

      3. swap-sqr 29.2%

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

      4. unpow2 29.2%

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

      5. frac-times 29.2%

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

      6. add-sqr-sqrt 29.2%

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

      7. frac-times 29.2%

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

      8. clear-num 31.5%

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

      9. associate-*l* 34.0%

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

      10. *-commutative 34.0%

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

      11. div-inv 34.0%

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

      12. sqrt-div 24.4%

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

   7. Applied egg-rr 44.7%

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

      1. *-commutative 44.7%

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

      2. associate-*r/ 47.2%

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

   9. Simplified 47.2%

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

   if -1e189 < (*.f64 V l) < -8.00000000000000013e-294

   1. Initial program 87.5%

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

      1. frac-2neg 87.5%

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

      2. sqrt-div 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

   3. Applied egg-rr 99.4%

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

   if -8.00000000000000013e-294 < (*.f64 V l) < 5.000000017e-316

   1. Initial program 34.7%

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

      1. clear-num 34.7%

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

      2. sqrt-div 34.7%

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

      3. metadata-eval 34.7%

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

      4. *-commutative 34.7%

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

      5. associate-/l* 61.0%

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

   3. Applied egg-rr 61.0%

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

      1. associate-/l* 34.7%

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

      2. *-commutative 34.7%

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

      3. associate-/l* 61.1%

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

      4. associate-/r/ 61.0%

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

   5. Simplified 61.0%

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

      1. un-div-inv 61.2%

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

      2. sqrt-prod 43.8%

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

      3. associate-/r* 43.7%

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

   7. Applied egg-rr 43.7%

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

      1. associate-/l/ 43.8%

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

   9. Simplified 43.8%

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

   if 5.000000017e-316 < (*.f64 V l)

   1. Initial program 75.9%

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

      1. sqrt-div 94.0%

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

      2. associate-*r/ 88.4%

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

   3. Applied egg-rr 88.4%

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

      1. *-commutative 88.4%

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

      2. associate-/l* 92.2%

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

      3. associate-/r/ 94.0%

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

   5. Simplified 94.0%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+189}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -8 \cdot 10^{-294}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-316}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]

Alternative 2: 89.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{-V}} \cdot \left({\ell}^{-0.5} \cdot c0\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -2e-310)
   (* (/ (sqrt (- A)) (sqrt (- V))) (* (pow l -0.5) c0))
   (* c0 (/ (sqrt A) (sqrt (* V l))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -2e-310) {
		tmp = (sqrt(-A) / sqrt(-V)) * (pow(l, -0.5) * c0);
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (a <= (-2d-310)) then
        tmp = (sqrt(-a) / sqrt(-v)) * ((l ** (-0.5d0)) * c0)
    else
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -2e-310) {
		tmp = (Math.sqrt(-A) / Math.sqrt(-V)) * (Math.pow(l, -0.5) * c0);
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -2e-310:
		tmp = (math.sqrt(-A) / math.sqrt(-V)) * (math.pow(l, -0.5) * c0)
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -2e-310)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(-V))) * Float64((l ^ -0.5) * c0));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (A <= -2e-310)
		tmp = (sqrt(-A) / sqrt(-V)) * ((l ^ -0.5) * c0);
	else
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -2e-310], N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, -0.5], $MachinePrecision] * c0), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -1.999999999999994e-310

   1. Initial program 72.9%

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

      1. pow1 72.9%

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

      2. metadata-eval 72.9%

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

      3. pow-prod-up 38.0%

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

      4. pow-prod-down 29.8%

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

      5. *-commutative 29.8%

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

      6. *-commutative 29.8%

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

      7. swap-sqr 26.7%

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

      8. add-sqr-sqrt 26.7%

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

      9. associate-/r* 26.0%

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

      10. pow2 26.0%

          \[\leadsto {\left(\frac{\frac{A}{V}}{\ell} \cdot \color{blue}{{c0}^{2}}\right)}^{0.5} \]

   3. Applied egg-rr 26.0%

      \[\leadsto \color{blue}{{\left(\frac{\frac{A}{V}}{\ell} \cdot {c0}^{2}\right)}^{0.5}} \]
   4. Step-by-step derivation

      1. unpow1/2 26.0%

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

      2. associate-*l/ 26.8%

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

   5. Simplified 26.8%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{A}{V} \cdot {c0}^{2}}{\ell}}} \]
   6. Step-by-step derivation

      1. div-inv 26.8%

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

      2. sqrt-prod 14.5%

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

      3. *-commutative 14.5%

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

      4. sqrt-prod 15.2%

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

      5. unpow2 15.2%

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

      6. sqrt-prod 15.3%

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

      7. add-sqr-sqrt 38.0%

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

      8. associate-*r* 40.5%

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

      9. sqrt-prod 71.6%

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

      10. *-commutative 71.6%

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

      11. sqrt-prod 40.5%

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

      12. associate-*l* 40.5%

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

      13. inv-pow 40.5%

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

      14. sqrt-pow1 40.5%

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

      15. metadata-eval 40.5%

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

   7. Applied egg-rr 40.5%

      \[\leadsto \color{blue}{\sqrt{\frac{A}{V}} \cdot \left({\ell}^{-0.5} \cdot c0\right)} \]
   8. Step-by-step derivation

      1. frac-2neg 40.5%

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

      2. sqrt-div 46.5%

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

   9. Applied egg-rr 46.5%

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

   if -1.999999999999994e-310 < A

   1. Initial program 69.7%

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

      1. sqrt-div 85.4%

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

      2. associate-*r/ 80.6%

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

   3. Applied egg-rr 80.6%

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

      1. *-commutative 80.6%

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

      2. associate-/l* 83.4%

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

      3. associate-/r/ 85.4%

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

   5. Simplified 85.4%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{-V}} \cdot \left({\ell}^{-0.5} \cdot c0\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]

Alternative 3: 89.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{\sqrt{-A}}{\sqrt{-V}} \cdot c0}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -2e-310)
   (/ (* (/ (sqrt (- A)) (sqrt (- V))) c0) (sqrt l))
   (* c0 (/ (sqrt A) (sqrt (* V l))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -2e-310) {
		tmp = ((sqrt(-A) / sqrt(-V)) * c0) / sqrt(l);
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (a <= (-2d-310)) then
        tmp = ((sqrt(-a) / sqrt(-v)) * c0) / sqrt(l)
    else
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -2e-310) {
		tmp = ((Math.sqrt(-A) / Math.sqrt(-V)) * c0) / Math.sqrt(l);
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -2e-310:
		tmp = ((math.sqrt(-A) / math.sqrt(-V)) * c0) / math.sqrt(l)
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -2e-310)
		tmp = Float64(Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(-V))) * c0) / sqrt(l));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (A <= -2e-310)
		tmp = ((sqrt(-A) / sqrt(-V)) * c0) / sqrt(l);
	else
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -2e-310], N[(N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -1.999999999999994e-310

   1. Initial program 72.9%

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

      1. clear-num 72.6%

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

      2. sqrt-div 73.2%

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

      3. metadata-eval 73.2%

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

      4. *-commutative 73.2%

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

      5. associate-/l* 72.1%

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

   3. Applied egg-rr 72.1%

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

      1. associate-/l* 73.2%

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

      2. *-commutative 73.2%

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

      3. associate-/l* 71.0%

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

      4. associate-/r/ 71.9%

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

   5. Simplified 71.9%

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

      1. add-sqr-sqrt 38.9%

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

      2. sqrt-unprod 28.7%

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

      3. swap-sqr 25.3%

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

      4. unpow2 25.3%

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

      5. frac-times 25.3%

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

      6. add-sqr-sqrt 25.3%

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

      7. frac-times 25.3%

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

      8. clear-num 26.0%

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

      9. associate-*l* 26.8%

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

      10. *-commutative 26.8%

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

      11. div-inv 26.8%

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

      12. sqrt-div 14.5%

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

   7. Applied egg-rr 38.0%

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

      1. frac-2neg 40.5%

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

      2. sqrt-div 46.5%

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

   9. Applied egg-rr 42.9%

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

   if -1.999999999999994e-310 < A

   1. Initial program 69.7%

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

      1. sqrt-div 85.4%

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

      2. associate-*r/ 80.6%

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

   3. Applied egg-rr 80.6%

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

      1. *-commutative 80.6%

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

      2. associate-/l* 83.4%

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

      3. associate-/r/ 85.4%

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

   5. Simplified 85.4%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{\sqrt{-A}}{\sqrt{-V}} \cdot c0}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]

Alternative 4: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+191}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-90}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-316}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (/ (sqrt (/ A V)) (sqrt l)))))
   (if (<= (* V l) -2e+191)
     t_0
     (if (<= (* V l) -2e-90)
       (* c0 (sqrt (/ A (* V l))))
       (if (<= (* V l) 5e-316) t_0 (* (sqrt A) (/ c0 (sqrt (* V l)))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (sqrt((A / V)) / sqrt(l));
	double tmp;
	if ((V * l) <= -2e+191) {
		tmp = t_0;
	} else if ((V * l) <= -2e-90) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 5e-316) {
		tmp = t_0;
	} else {
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * (sqrt((a / v)) / sqrt(l))
    if ((v * l) <= (-2d+191)) then
        tmp = t_0
    else if ((v * l) <= (-2d-90)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 5d-316) then
        tmp = t_0
    else
        tmp = sqrt(a) * (c0 / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	double tmp;
	if ((V * l) <= -2e+191) {
		tmp = t_0;
	} else if ((V * l) <= -2e-90) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 5e-316) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(A) * (c0 / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	tmp = 0
	if (V * l) <= -2e+191:
		tmp = t_0
	elif (V * l) <= -2e-90:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 5e-316:
		tmp = t_0
	else:
		tmp = math.sqrt(A) * (c0 / math.sqrt((V * l)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)))
	tmp = 0.0
	if (Float64(V * l) <= -2e+191)
		tmp = t_0;
	elseif (Float64(V * l) <= -2e-90)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 5e-316)
		tmp = t_0;
	else
		tmp = Float64(sqrt(A) * Float64(c0 / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * (sqrt((A / V)) / sqrt(l));
	tmp = 0.0;
	if ((V * l) <= -2e+191)
		tmp = t_0;
	elseif ((V * l) <= -2e-90)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 5e-316)
		tmp = t_0;
	else
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -2e+191], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -2e-90], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-316], t$95$0, N[(N[Sqrt[A], $MachinePrecision] * N[(c0 / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -2.00000000000000015e191 or -1.99999999999999999e-90 < (*.f64 V l) < 5.000000017e-316

   1. Initial program 54.8%

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

      1. associate-/r* 67.0%

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

      2. sqrt-div 46.6%

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

   3. Applied egg-rr 46.6%

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

   if -2.00000000000000015e191 < (*.f64 V l) < -1.99999999999999999e-90

   1. Initial program 97.8%

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

   if 5.000000017e-316 < (*.f64 V l)

   1. Initial program 75.9%

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

      1. clear-num 75.9%

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

      2. sqrt-div 77.4%

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

      3. metadata-eval 77.4%

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

      4. *-commutative 77.4%

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

      5. associate-/l* 75.4%

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

   3. Applied egg-rr 75.4%

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

      1. associate-/l* 77.4%

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

      2. *-commutative 77.4%

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

      3. associate-/l* 70.2%

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

      4. associate-/r/ 74.7%

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

   5. Simplified 74.7%

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

      1. un-div-inv 74.7%

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

      2. *-un-lft-identity 74.7%

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

      3. sqrt-prod 41.0%

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

      4. times-frac 40.4%

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

      5. metadata-eval 40.4%

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

      6. sqrt-div 40.4%

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

      7. clear-num 40.4%

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

      8. sqrt-div 44.1%

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

      9. times-frac 43.9%

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

      10. sqrt-prod 88.4%

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

      11. associate-/l* 92.2%

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

   7. Applied egg-rr 92.2%

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

      1. associate-/l* 88.4%

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

      2. associate-*r/ 92.0%

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

   9. Simplified 92.0%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+191}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-90}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-316}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \end{array} \]

Alternative 5: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+189}:\\ \;\;\;\;t_0 \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-90}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-316}:\\ \;\;\;\;c0 \cdot \frac{t_0}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (/ A V))))
   (if (<= (* V l) -1e+189)
     (* t_0 (/ c0 (sqrt l)))
     (if (<= (* V l) -2e-90)
       (* c0 (sqrt (/ A (* V l))))
       (if (<= (* V l) 5e-316)
         (* c0 (/ t_0 (sqrt l)))
         (* (sqrt A) (/ c0 (sqrt (* V l)))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / V));
	double tmp;
	if ((V * l) <= -1e+189) {
		tmp = t_0 * (c0 / sqrt(l));
	} else if ((V * l) <= -2e-90) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 5e-316) {
		tmp = c0 * (t_0 / sqrt(l));
	} else {
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: 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))
    if ((v * l) <= (-1d+189)) then
        tmp = t_0 * (c0 / sqrt(l))
    else if ((v * l) <= (-2d-90)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 5d-316) then
        tmp = c0 * (t_0 / sqrt(l))
    else
        tmp = sqrt(a) * (c0 / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / V));
	double tmp;
	if ((V * l) <= -1e+189) {
		tmp = t_0 * (c0 / Math.sqrt(l));
	} else if ((V * l) <= -2e-90) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 5e-316) {
		tmp = c0 * (t_0 / Math.sqrt(l));
	} else {
		tmp = Math.sqrt(A) * (c0 / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / V))
	tmp = 0
	if (V * l) <= -1e+189:
		tmp = t_0 * (c0 / math.sqrt(l))
	elif (V * l) <= -2e-90:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 5e-316:
		tmp = c0 * (t_0 / math.sqrt(l))
	else:
		tmp = math.sqrt(A) * (c0 / math.sqrt((V * l)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(A / V))
	tmp = 0.0
	if (Float64(V * l) <= -1e+189)
		tmp = Float64(t_0 * Float64(c0 / sqrt(l)));
	elseif (Float64(V * l) <= -2e-90)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 5e-316)
		tmp = Float64(c0 * Float64(t_0 / sqrt(l)));
	else
		tmp = Float64(sqrt(A) * Float64(c0 / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((A / V));
	tmp = 0.0;
	if ((V * l) <= -1e+189)
		tmp = t_0 * (c0 / sqrt(l));
	elseif ((V * l) <= -2e-90)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 5e-316)
		tmp = c0 * (t_0 / sqrt(l));
	else
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -1e+189], N[(t$95$0 * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-90], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-316], N[(c0 * N[(t$95$0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[A], $MachinePrecision] * N[(c0 / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -1e189

   1. Initial program 52.2%

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

      1. clear-num 52.2%

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

      2. sqrt-div 52.2%

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

      3. metadata-eval 52.2%

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

      4. *-commutative 52.2%

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

      5. associate-/l* 71.8%

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

   3. Applied egg-rr 71.8%

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

      1. associate-/l* 52.2%

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

      2. *-commutative 52.2%

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

      3. associate-/l* 72.1%

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

      4. associate-/r/ 69.6%

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

   5. Simplified 69.6%

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

      1. add-sqr-sqrt 40.6%

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

      2. sqrt-unprod 29.5%

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

      3. swap-sqr 29.2%

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

      4. unpow2 29.2%

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

      5. frac-times 29.2%

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

      6. add-sqr-sqrt 29.2%

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

      7. frac-times 29.2%

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

      8. clear-num 31.5%

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

      9. associate-*l* 34.0%

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

      10. *-commutative 34.0%

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

      11. div-inv 34.0%

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

      12. sqrt-div 24.4%

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

   7. Applied egg-rr 44.7%

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

      1. *-commutative 44.7%

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

      2. associate-*r/ 47.2%

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

   9. Simplified 47.2%

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

   if -1e189 < (*.f64 V l) < -1.99999999999999999e-90

   1. Initial program 97.8%

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

   if -1.99999999999999999e-90 < (*.f64 V l) < 5.000000017e-316

   1. Initial program 56.9%

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

      1. associate-/r* 64.8%

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

      2. sqrt-div 45.6%

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

   3. Applied egg-rr 45.6%

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

   if 5.000000017e-316 < (*.f64 V l)

   1. Initial program 75.9%

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

      1. clear-num 75.9%

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

      2. sqrt-div 77.4%

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

      3. metadata-eval 77.4%

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

      4. *-commutative 77.4%

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

      5. associate-/l* 75.4%

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

   3. Applied egg-rr 75.4%

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

      1. associate-/l* 77.4%

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

      2. *-commutative 77.4%

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

      3. associate-/l* 70.2%

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

      4. associate-/r/ 74.7%

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

   5. Simplified 74.7%

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

      1. un-div-inv 74.7%

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

      2. *-un-lft-identity 74.7%

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

      3. sqrt-prod 41.0%

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

      4. times-frac 40.4%

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

      5. metadata-eval 40.4%

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

      6. sqrt-div 40.4%

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

      7. clear-num 40.4%

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

      8. sqrt-div 44.1%

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

      9. times-frac 43.9%

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

      10. sqrt-prod 88.4%

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

      11. associate-/l* 92.2%

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

   7. Applied egg-rr 92.2%

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

      1. associate-/l* 88.4%

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

      2. associate-*r/ 92.0%

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

   9. Simplified 92.0%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+189}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-90}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-316}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \end{array} \]

Alternative 6: 86.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+189}:\\ \;\;\;\;t_0 \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-90}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-316}:\\ \;\;\;\;c0 \cdot \frac{t_0}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (/ A V))))
   (if (<= (* V l) -1e+189)
     (* t_0 (/ c0 (sqrt l)))
     (if (<= (* V l) -2e-90)
       (* c0 (sqrt (/ A (* V l))))
       (if (<= (* V l) 5e-316)
         (* c0 (/ t_0 (sqrt l)))
         (* c0 (/ (sqrt A) (sqrt (* V l)))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / V));
	double tmp;
	if ((V * l) <= -1e+189) {
		tmp = t_0 * (c0 / sqrt(l));
	} else if ((V * l) <= -2e-90) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 5e-316) {
		tmp = c0 * (t_0 / sqrt(l));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: 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))
    if ((v * l) <= (-1d+189)) then
        tmp = t_0 * (c0 / sqrt(l))
    else if ((v * l) <= (-2d-90)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 5d-316) then
        tmp = c0 * (t_0 / sqrt(l))
    else
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / V));
	double tmp;
	if ((V * l) <= -1e+189) {
		tmp = t_0 * (c0 / Math.sqrt(l));
	} else if ((V * l) <= -2e-90) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 5e-316) {
		tmp = c0 * (t_0 / Math.sqrt(l));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / V))
	tmp = 0
	if (V * l) <= -1e+189:
		tmp = t_0 * (c0 / math.sqrt(l))
	elif (V * l) <= -2e-90:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 5e-316:
		tmp = c0 * (t_0 / math.sqrt(l))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(A / V))
	tmp = 0.0
	if (Float64(V * l) <= -1e+189)
		tmp = Float64(t_0 * Float64(c0 / sqrt(l)));
	elseif (Float64(V * l) <= -2e-90)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 5e-316)
		tmp = Float64(c0 * Float64(t_0 / sqrt(l)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((A / V));
	tmp = 0.0;
	if ((V * l) <= -1e+189)
		tmp = t_0 * (c0 / sqrt(l));
	elseif ((V * l) <= -2e-90)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 5e-316)
		tmp = c0 * (t_0 / sqrt(l));
	else
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -1e+189], N[(t$95$0 * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-90], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-316], N[(c0 * N[(t$95$0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -1e189

   1. Initial program 52.2%

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

      1. clear-num 52.2%

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

      2. sqrt-div 52.2%

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

      3. metadata-eval 52.2%

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

      4. *-commutative 52.2%

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

      5. associate-/l* 71.8%

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

   3. Applied egg-rr 71.8%

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

      1. associate-/l* 52.2%

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

      2. *-commutative 52.2%

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

      3. associate-/l* 72.1%

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

      4. associate-/r/ 69.6%

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

   5. Simplified 69.6%

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

      1. add-sqr-sqrt 40.6%

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

      2. sqrt-unprod 29.5%

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

      3. swap-sqr 29.2%

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

      4. unpow2 29.2%

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

      5. frac-times 29.2%

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

      6. add-sqr-sqrt 29.2%

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

      7. frac-times 29.2%

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

      8. clear-num 31.5%

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

      9. associate-*l* 34.0%

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

      10. *-commutative 34.0%

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

      11. div-inv 34.0%

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

      12. sqrt-div 24.4%

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

   7. Applied egg-rr 44.7%

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

      1. *-commutative 44.7%

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

      2. associate-*r/ 47.2%

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

   9. Simplified 47.2%

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

   if -1e189 < (*.f64 V l) < -1.99999999999999999e-90

   1. Initial program 97.8%

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

   if -1.99999999999999999e-90 < (*.f64 V l) < 5.000000017e-316

   1. Initial program 56.9%

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

      1. associate-/r* 64.8%

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

      2. sqrt-div 45.6%

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

   3. Applied egg-rr 45.6%

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

   if 5.000000017e-316 < (*.f64 V l)

   1. Initial program 75.9%

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

      1. sqrt-div 94.0%

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

      2. associate-*r/ 88.4%

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

   3. Applied egg-rr 88.4%

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

      1. *-commutative 88.4%

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

      2. associate-/l* 92.2%

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

      3. associate-/r/ 94.0%

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

   5. Simplified 94.0%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+189}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-90}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-316}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]

Alternative 7: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+189}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-40}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-316}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e+189)
   (* (sqrt (/ A V)) (/ c0 (sqrt l)))
   (if (<= (* V l) -1e-40)
     (* c0 (sqrt (/ A (* V l))))
     (if (<= (* V l) 5e-316)
       (/ c0 (* (sqrt l) (sqrt (/ V A))))
       (* c0 (/ (sqrt A) (sqrt (* V l))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+189) {
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	} else if ((V * l) <= -1e-40) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 5e-316) {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: 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+189)) then
        tmp = sqrt((a / v)) * (c0 / sqrt(l))
    else if ((v * l) <= (-1d-40)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 5d-316) then
        tmp = c0 / (sqrt(l) * sqrt((v / a)))
    else
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+189) {
		tmp = Math.sqrt((A / V)) * (c0 / Math.sqrt(l));
	} else if ((V * l) <= -1e-40) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 5e-316) {
		tmp = c0 / (Math.sqrt(l) * Math.sqrt((V / A)));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e+189:
		tmp = math.sqrt((A / V)) * (c0 / math.sqrt(l))
	elif (V * l) <= -1e-40:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 5e-316:
		tmp = c0 / (math.sqrt(l) * math.sqrt((V / A)))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e+189)
		tmp = Float64(sqrt(Float64(A / V)) * Float64(c0 / sqrt(l)));
	elseif (Float64(V * l) <= -1e-40)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 5e-316)
		tmp = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e+189)
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	elseif ((V * l) <= -1e-40)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 5e-316)
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	else
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: 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+189], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-40], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-316], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -1e189

   1. Initial program 52.2%

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

      1. clear-num 52.2%

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

      2. sqrt-div 52.2%

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

      3. metadata-eval 52.2%

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

      4. *-commutative 52.2%

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

      5. associate-/l* 71.8%

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

   3. Applied egg-rr 71.8%

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

      1. associate-/l* 52.2%

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

      2. *-commutative 52.2%

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

      3. associate-/l* 72.1%

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

      4. associate-/r/ 69.6%

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

   5. Simplified 69.6%

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

      1. add-sqr-sqrt 40.6%

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

      2. sqrt-unprod 29.5%

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

      3. swap-sqr 29.2%

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

      4. unpow2 29.2%

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

      5. frac-times 29.2%

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

      6. add-sqr-sqrt 29.2%

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

      7. frac-times 29.2%

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

      8. clear-num 31.5%

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

      9. associate-*l* 34.0%

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

      10. *-commutative 34.0%

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

      11. div-inv 34.0%

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

      12. sqrt-div 24.4%

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

   7. Applied egg-rr 44.7%

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

      1. *-commutative 44.7%

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

      2. associate-*r/ 47.2%

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

   9. Simplified 47.2%

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

   if -1e189 < (*.f64 V l) < -9.9999999999999993e-41

   1. Initial program 99.8%

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

   if -9.9999999999999993e-41 < (*.f64 V l) < 5.000000017e-316

   1. Initial program 57.9%

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

      1. clear-num 57.9%

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

      2. sqrt-div 58.9%

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

      3. metadata-eval 58.9%

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

      4. *-commutative 58.9%

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

      5. associate-/l* 66.0%

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

   3. Applied egg-rr 66.0%

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

      1. associate-/l* 58.9%

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

      2. *-commutative 58.9%

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

      3. associate-/l* 66.6%

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

      4. associate-/r/ 66.0%

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

   5. Simplified 66.0%

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

      1. un-div-inv 66.2%

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

      2. sqrt-prod 44.5%

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

      3. associate-/r* 43.2%

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

   7. Applied egg-rr 43.2%

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

      1. associate-/l/ 44.5%

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

   9. Simplified 44.5%

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

   if 5.000000017e-316 < (*.f64 V l)

   1. Initial program 75.9%

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

      1. sqrt-div 94.0%

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

      2. associate-*r/ 88.4%

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

   3. Applied egg-rr 88.4%

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

      1. *-commutative 88.4%

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

      2. associate-/l* 92.2%

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

      3. associate-/r/ 94.0%

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

   5. Simplified 94.0%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+189}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-40}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-316}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]

Alternative 8: 81.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: 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 <= (-4d-310)) then
        tmp = c0 / sqrt((l / (a / v)))
    else
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 69.2%

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

      1. clear-num 69.2%

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

      2. sqrt-div 69.8%

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

      3. metadata-eval 69.8%

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

      4. *-commutative 69.8%

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

      5. associate-/l* 70.4%

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

   3. Applied egg-rr 70.4%

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

      1. associate-/l* 69.8%

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

      2. *-commutative 69.8%

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

      3. associate-/l* 69.4%

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

      4. associate-/r/ 69.2%

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

   5. Simplified 69.2%

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

      1. un-div-inv 69.3%

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

      2. clear-num 69.2%

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

      3. associate-*l/ 69.7%

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

      4. associate-/l* 69.4%

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

   7. Applied egg-rr 69.4%

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

      1. associate-/r/ 69.4%

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

      2. metadata-eval 69.4%

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

      3. associate-/r* 69.4%

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

      4. neg-mul-1 69.4%

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

      5. associate-/r/ 69.4%

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

      6. associate-/l* 69.5%

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

      7. *-commutative 69.5%

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

      8. associate-/l* 69.5%

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

      9. neg-mul-1 69.5%

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

      10. associate-/l* 69.5%

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

      11. metadata-eval 69.5%

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

      12. associate-/r* 69.5%

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

      13. neg-mul-1 69.5%

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

      14. associate-/l* 69.5%

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

      15. neg-mul-1 69.5%

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

      16. remove-double-neg 69.5%

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

      17. /-rgt-identity 69.5%

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

      18. associate-/l* 69.8%

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

      19. associate-*r/ 69.5%

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

   9. Simplified 69.5%

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

       1. *-commutative 69.5%

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

       2. associate-/r/ 70.4%

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

   11. Applied egg-rr 70.4%

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

   if -3.999999999999988e-310 < l

   1. Initial program 73.8%

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

      1. associate-/r* 74.2%

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

      2. sqrt-div 86.3%

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

   3. Applied egg-rr 86.3%

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

4. Final simplification 78.1%

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

Alternative 9: 78.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 5 \cdot 10^{+257}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: 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 (or (<= t_0 0.0) (not (<= t_0 5e+257)))
     (* c0 (sqrt (/ (/ A V) l)))
     (* c0 (sqrt t_0)))))

C

assert(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) || !(t_0 <= 5e+257)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = c0 * sqrt(t_0);
	}
	return tmp;
}

Fortran

NOTE: 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) .or. (.not. (t_0 <= 5d+257))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = c0 * sqrt(t_0)
    end if
    code = tmp
end function

Java

assert 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) || !(t_0 <= 5e+257)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 5e+257):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 5e+257))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(c0 * sqrt(t_0));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 5e+257)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = c0 * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: 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[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 5e+257]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 5.00000000000000028e257 < (/.f64 A (*.f64 V l))

   1. Initial program 41.4%

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

      1. associate-/r* 53.8%

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

      2. div-inv 53.7%

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

   3. Applied egg-rr 53.7%

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

      1. un-div-inv 53.8%

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

   5. Applied egg-rr 53.8%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 5.00000000000000028e257

   1. Initial program 99.2%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+257}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]

Alternative 10: 79.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+282}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: 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)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 5e+282) (* c0 (sqrt t_0)) (/ c0 (sqrt (* V (/ l A))))))))

C

assert(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 = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 5e+282) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: 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 = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 5d+282) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert 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 = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 5e+282) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 5e+282:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

V, l = sort([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(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 5e+282)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 5e+282)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: 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[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+282], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 39.9%

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

      1. associate-/r* 57.8%

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

      2. div-inv 57.8%

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

   3. Applied egg-rr 57.8%

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

      1. un-div-inv 57.8%

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

   5. Applied egg-rr 57.8%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.99999999999999978e282

   1. Initial program 99.2%

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

   if 4.99999999999999978e282 < (/.f64 A (*.f64 V l))

   1. Initial program 36.8%

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

      1. clear-num 36.8%

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

      2. sqrt-div 40.8%

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

      3. metadata-eval 40.8%

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

      4. *-commutative 40.8%

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

      5. associate-/l* 52.0%

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

   3. Applied egg-rr 52.0%

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

      1. associate-/l* 40.8%

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

      2. *-commutative 40.8%

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

      3. associate-/l* 51.6%

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

      4. associate-/r/ 52.0%

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

   5. Simplified 52.0%

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

      1. un-div-inv 52.1%

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

      2. clear-num 52.1%

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

      3. associate-*l/ 40.8%

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

      4. associate-/l* 51.7%

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

   7. Applied egg-rr 51.7%

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

      1. associate-/r/ 51.6%

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

      2. metadata-eval 51.6%

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

      3. associate-/r* 51.6%

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

      4. neg-mul-1 51.6%

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

      5. associate-/r/ 51.7%

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

      6. associate-/l* 51.7%

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

      7. *-commutative 51.7%

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

      8. associate-/l* 51.7%

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

      9. neg-mul-1 51.7%

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

      10. associate-/l* 51.7%

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

      11. metadata-eval 51.7%

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

      12. associate-/r* 51.7%

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

      13. neg-mul-1 51.7%

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

      14. associate-/l* 51.7%

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

      15. neg-mul-1 51.7%

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

      16. remove-double-neg 51.7%

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

      17. /-rgt-identity 51.7%

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

      18. associate-/l* 40.8%

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

      19. associate-*r/ 51.7%

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

   9. Simplified 51.7%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+282}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]

Alternative 11: 79.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+282}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: 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)
     (/ c0 (sqrt (/ V (/ A l))))
     (if (<= t_0 5e+282) (* c0 (sqrt t_0)) (/ c0 (sqrt (* V (/ l A))))))))

C

assert(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 = c0 / sqrt((V / (A / l)));
	} else if (t_0 <= 5e+282) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: 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 = c0 / sqrt((v / (a / l)))
    else if (t_0 <= 5d+282) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert 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 = c0 / Math.sqrt((V / (A / l)));
	} else if (t_0 <= 5e+282) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 / math.sqrt((V / (A / l)))
	elif t_0 <= 5e+282:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

V, l = sort([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(c0 / sqrt(Float64(V / Float64(A / l))));
	elseif (t_0 <= 5e+282)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 / sqrt((V / (A / l)));
	elseif (t_0 <= 5e+282)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: 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[(c0 / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+282], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 39.9%

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

      1. clear-num 39.9%

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

      2. sqrt-div 39.9%

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

      3. metadata-eval 39.9%

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

      4. *-commutative 39.9%

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

      5. associate-/l* 57.8%

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

   3. Applied egg-rr 57.8%

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

      1. associate-/l* 39.9%

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

      2. *-commutative 39.9%

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

      3. associate-/l* 57.8%

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

      4. associate-/r/ 57.8%

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

   5. Simplified 57.8%

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

      1. un-div-inv 57.8%

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

      2. associate-*l/ 39.9%

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

      3. associate-/l* 57.9%

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

   7. Applied egg-rr 57.9%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.99999999999999978e282

   1. Initial program 99.2%

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

   if 4.99999999999999978e282 < (/.f64 A (*.f64 V l))

   1. Initial program 36.8%

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

      1. clear-num 36.8%

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

      2. sqrt-div 40.8%

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

      3. metadata-eval 40.8%

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

      4. *-commutative 40.8%

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

      5. associate-/l* 52.0%

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

   3. Applied egg-rr 52.0%

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

      1. associate-/l* 40.8%

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

      2. *-commutative 40.8%

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

      3. associate-/l* 51.6%

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

      4. associate-/r/ 52.0%

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

   5. Simplified 52.0%

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

      1. un-div-inv 52.1%

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

      2. clear-num 52.1%

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

      3. associate-*l/ 40.8%

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

      4. associate-/l* 51.7%

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

   7. Applied egg-rr 51.7%

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

      1. associate-/r/ 51.6%

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

      2. metadata-eval 51.6%

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

      3. associate-/r* 51.6%

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

      4. neg-mul-1 51.6%

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

      5. associate-/r/ 51.7%

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

      6. associate-/l* 51.7%

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

      7. *-commutative 51.7%

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

      8. associate-/l* 51.7%

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

      9. neg-mul-1 51.7%

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

      10. associate-/l* 51.7%

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

      11. metadata-eval 51.7%

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

      12. associate-/r* 51.7%

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

      13. neg-mul-1 51.7%

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

      14. associate-/l* 51.7%

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

      15. neg-mul-1 51.7%

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

      16. remove-double-neg 51.7%

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

      17. /-rgt-identity 51.7%

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

      18. associate-/l* 40.8%

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

      19. associate-*r/ 51.7%

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

   9. Simplified 51.7%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+282}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]

Alternative 12: 72.8% accurate, 1.0× speedup

Math

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

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l) :precision binary64 (* c0 (sqrt (/ A (* V l)))))

C

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

Fortran

NOTE: 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 = c0 * sqrt((a / (v * l)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.4%

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

2. Final simplification 71.4%

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

Reproduce

herbie shell --seed 2023306 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))