Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.4% → 86.3%

Time: 10.1s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{-267}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\left(c0 \cdot {\left(\frac{V}{A}\right)}^{-0.5}\right) \cdot {\ell}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -5e-267)
   (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
   (if (<= (* V l) 0.0)
     (* (* c0 (pow (/ V A) -0.5)) (pow l -0.5))
     (/ c0 (/ (sqrt (* V l)) (sqrt A))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e-267) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = (c0 * pow((V / A), -0.5)) * pow(l, -0.5);
	} else {
		tmp = c0 / (sqrt((V * l)) / sqrt(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) :: tmp
    if ((v * l) <= (-5d-267)) then
        tmp = c0 * (sqrt(-a) / sqrt((v * -l)))
    else if ((v * l) <= 0.0d0) then
        tmp = (c0 * ((v / a) ** (-0.5d0))) * (l ** (-0.5d0))
    else
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    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) <= -5e-267) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = (c0 * Math.pow((V / A), -0.5)) * Math.pow(l, -0.5);
	} else {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -5e-267:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = (c0 * math.pow((V / A), -0.5)) * math.pow(l, -0.5)
	else:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -5e-267)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(Float64(c0 * (Float64(V / A) ^ -0.5)) * (l ^ -0.5));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	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) <= -5e-267)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = (c0 * ((V / A) ^ -0.5)) * (l ^ -0.5);
	else
		tmp = c0 / (sqrt((V * l)) / sqrt(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_] := If[LessEqual[N[(V * l), $MachinePrecision], -5e-267], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(N[(c0 * N[Power[N[(V / A), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -4.9999999999999999e-267

   1. Initial program 88.0%

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

      1. frac-2neg 88.0%

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

      2. sqrt-div 94.3%

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

      3. *-commutative 94.3%

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

      4. distribute-rgt-neg-in 94.3%

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

   3. Applied egg-rr 94.3%

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

   if -4.9999999999999999e-267 < (*.f64 V l) < -0.0

   1. Initial program 49.5%

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

      1. sqrt-div 12.9%

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

      2. associate-*r/ 12.9%

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

   3. Applied egg-rr 12.9%

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

      1. associate-/l* 12.9%

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

   5. Simplified 12.9%

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

      1. clear-num 12.9%

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

      2. inv-pow 12.9%

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

      3. sqrt-div 49.5%

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

      4. associate-/r* 65.4%

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

   7. Applied egg-rr 65.4%

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

      1. unpow-1 65.4%

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

   9. Simplified 65.4%

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

       1. associate-/r/ 65.4%

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

       2. /-rgt-identity 65.4%

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

       3. clear-num 65.3%

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

       4. inv-pow 65.3%

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

       5. div-inv 65.3%

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

       6. clear-num 65.3%

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

       7. *-commutative 65.3%

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

       8. metadata-eval 65.3%

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

       9. pow-prod-up 65.3%

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

       10. sqrt-unprod 65.2%

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

       11. add-sqr-sqrt 65.3%

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

       12. unpow-prod-down 45.7%

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

       13. associate-*r* 45.7%

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

   11. Applied egg-rr 45.7%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 82.9%

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

      1. sqrt-div 95.7%

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

      2. associate-*r/ 90.8%

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

   3. Applied egg-rr 90.8%

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

      1. associate-/l* 95.9%

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

   5. Simplified 95.9%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{-267}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\left(c0 \cdot {\left(\frac{V}{A}\right)}^{-0.5}\right) \cdot {\ell}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \]

Alternative 2: 81.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{-241}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A \cdot \frac{1}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -5e-241)
   (* c0 (pow (/ (* V l) A) -0.5))
   (if (<= (* V l) 0.0)
     (* c0 (sqrt (/ (* A (/ 1.0 V)) l)))
     (* 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) <= -5e-241) {
		tmp = c0 * pow(((V * l) / A), -0.5);
	} else if ((V * l) <= 0.0) {
		tmp = c0 * sqrt(((A * (1.0 / V)) / 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 ((v * l) <= (-5d-241)) then
        tmp = c0 * (((v * l) / a) ** (-0.5d0))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 * sqrt(((a * (1.0d0 / v)) / 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 ((V * l) <= -5e-241) {
		tmp = c0 * Math.pow(((V * l) / A), -0.5);
	} else if ((V * l) <= 0.0) {
		tmp = c0 * Math.sqrt(((A * (1.0 / V)) / 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 (V * l) <= -5e-241:
		tmp = c0 * math.pow(((V * l) / A), -0.5)
	elif (V * l) <= 0.0:
		tmp = c0 * math.sqrt(((A * (1.0 / V)) / 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 (Float64(V * l) <= -5e-241)
		tmp = Float64(c0 * (Float64(Float64(V * l) / A) ^ -0.5));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A * Float64(1.0 / V)) / 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 ((V * l) <= -5e-241)
		tmp = c0 * (((V * l) / A) ^ -0.5);
	elseif ((V * l) <= 0.0)
		tmp = c0 * sqrt(((A * (1.0 / V)) / 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[N[(V * l), $MachinePrecision], -5e-241], N[(c0 * N[Power[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[Sqrt[N[(N[(A * N[(1.0 / V), $MachinePrecision]), $MachinePrecision] / 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 3 regimes

2. if (*.f64 V l) < -4.9999999999999998e-241

   1. Initial program 87.6%

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

      1. pow1/2 87.6%

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

      2. clear-num 87.5%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5} \]

      3. inv-pow 87.5%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5} \]

      4. pow-pow 88.9%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}} \]

      5. associate-/l* 81.6%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)} \]

      6. metadata-eval 81.6%

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

   3. Applied egg-rr 81.6%

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

      1. associate-/l* 88.9%

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

   5. Simplified 88.9%

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

   if -4.9999999999999998e-241 < (*.f64 V l) < -0.0

   1. Initial program 56.0%

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

      1. clear-num 56.0%

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

      2. associate-/r/ 54.9%

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

   3. Applied egg-rr 54.9%

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

      1. associate-/r* 55.0%

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

      2. associate-*l/ 69.9%

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

   5. Applied egg-rr 69.9%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 82.9%

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

      1. pow1/2 82.9%

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

      2. clear-num 82.9%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5} \]

      3. inv-pow 82.9%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5} \]

      4. pow-pow 83.8%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}} \]

      5. associate-/l* 76.2%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)} \]

      6. metadata-eval 76.2%

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

   3. Applied egg-rr 76.2%

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

      1. associate-/l* 83.8%

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

   5. Simplified 83.8%

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

      1. add-sqr-sqrt 83.5%

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

      2. sqrt-unprod 82.8%

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

      3. pow-prod-up 82.9%

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

      4. metadata-eval 82.9%

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

      5. inv-pow 82.9%

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

      6. clear-num 82.9%

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

      7. sqrt-div 95.7%

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

      8. *-un-lft-identity 95.7%

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

      9. add-sqr-sqrt 95.4%

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

      10. times-frac 95.5%

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

      11. pow1/2 95.5%

          \[\leadsto c0 \cdot \left(\frac{1}{\sqrt{\color{blue}{{\left(V \cdot \ell\right)}^{0.5}}}} \cdot \frac{\sqrt{A}}{\sqrt{\sqrt{V \cdot \ell}}}\right) \]

      12. metadata-eval 95.5%

          \[\leadsto c0 \cdot \left(\frac{1}{\sqrt{{\left(V \cdot \ell\right)}^{\color{blue}{\left(\sqrt{0.25}\right)}}}} \cdot \frac{\sqrt{A}}{\sqrt{\sqrt{V \cdot \ell}}}\right) \]

      13. sqrt-pow1 95.5%

          \[\leadsto c0 \cdot \left(\frac{1}{\color{blue}{{\left(V \cdot \ell\right)}^{\left(\frac{\sqrt{0.25}}{2}\right)}}} \cdot \frac{\sqrt{A}}{\sqrt{\sqrt{V \cdot \ell}}}\right) \]

      14. metadata-eval 95.5%

          \[\leadsto c0 \cdot \left(\frac{1}{{\left(V \cdot \ell\right)}^{\left(\frac{\color{blue}{0.5}}{2}\right)}} \cdot \frac{\sqrt{A}}{\sqrt{\sqrt{V \cdot \ell}}}\right) \]

      15. metadata-eval 95.5%

          \[\leadsto c0 \cdot \left(\frac{1}{{\left(V \cdot \ell\right)}^{\color{blue}{0.25}}} \cdot \frac{\sqrt{A}}{\sqrt{\sqrt{V \cdot \ell}}}\right) \]

      16. pow1/2 95.5%

          \[\leadsto c0 \cdot \left(\frac{1}{{\left(V \cdot \ell\right)}^{0.25}} \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{{\left(V \cdot \ell\right)}^{0.5}}}}\right) \]

      17. metadata-eval 95.5%

          \[\leadsto c0 \cdot \left(\frac{1}{{\left(V \cdot \ell\right)}^{0.25}} \cdot \frac{\sqrt{A}}{\sqrt{{\left(V \cdot \ell\right)}^{\color{blue}{\left(\sqrt{0.25}\right)}}}}\right) \]

      18. sqrt-pow1 95.5%

          \[\leadsto c0 \cdot \left(\frac{1}{{\left(V \cdot \ell\right)}^{0.25}} \cdot \frac{\sqrt{A}}{\color{blue}{{\left(V \cdot \ell\right)}^{\left(\frac{\sqrt{0.25}}{2}\right)}}}\right) \]

      19. metadata-eval 95.5%

          \[\leadsto c0 \cdot \left(\frac{1}{{\left(V \cdot \ell\right)}^{0.25}} \cdot \frac{\sqrt{A}}{{\left(V \cdot \ell\right)}^{\left(\frac{\color{blue}{0.5}}{2}\right)}}\right) \]

      20. metadata-eval 95.5%

          \[\leadsto c0 \cdot \left(\frac{1}{{\left(V \cdot \ell\right)}^{0.25}} \cdot \frac{\sqrt{A}}{{\left(V \cdot \ell\right)}^{\color{blue}{0.25}}}\right) \]

   7. Applied egg-rr 95.5%

      \[\leadsto c0 \cdot \color{blue}{\left(\frac{1}{{\left(V \cdot \ell\right)}^{0.25}} \cdot \frac{\sqrt{A}}{{\left(V \cdot \ell\right)}^{0.25}}\right)} \]
   8. Step-by-step derivation

      1. associate-*l/ 95.5%

         \[\leadsto c0 \cdot \color{blue}{\frac{1 \cdot \frac{\sqrt{A}}{{\left(V \cdot \ell\right)}^{0.25}}}{{\left(V \cdot \ell\right)}^{0.25}}} \]

      2. *-lft-identity 95.5%

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

      3. associate-/l/ 95.4%

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

      4. pow-sqr 95.7%

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\color{blue}{{\left(V \cdot \ell\right)}^{\left(2 \cdot 0.25\right)}}} \]

      5. metadata-eval 95.7%

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

      6. unpow1/2 95.7%

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

   9. Simplified 95.7%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{-241}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A \cdot \frac{1}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]

Alternative 3: 86.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{-262}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e-262)
   (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
   (if (<= (* V l) 0.0)
     (/ c0 (* (sqrt (/ V A)) (sqrt l)))
     (/ c0 (/ (sqrt (* V l)) (sqrt A))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e-262) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	} else {
		tmp = c0 / (sqrt((V * l)) / sqrt(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) :: tmp
    if ((v * l) <= (-1d-262)) then
        tmp = c0 * (sqrt(-a) / sqrt((v * -l)))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 / (sqrt((v / a)) * sqrt(l))
    else
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    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-262) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	} else {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e-262:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = c0 / (math.sqrt((V / A)) * math.sqrt(l))
	else:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e-262)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * sqrt(l)));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	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-262)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	else
		tmp = c0 / (sqrt((V * l)) / sqrt(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_] := If[LessEqual[N[(V * l), $MachinePrecision], -1e-262], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -1.00000000000000001e-262

   1. Initial program 87.8%

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

      1. frac-2neg 87.8%

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

      2. sqrt-div 94.1%

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

      3. *-commutative 94.1%

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

      4. distribute-rgt-neg-in 94.1%

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

   3. Applied egg-rr 94.1%

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

   if -1.00000000000000001e-262 < (*.f64 V l) < -0.0

   1. Initial program 53.6%

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

      1. sqrt-div 11.8%

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

      2. associate-*r/ 11.8%

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

   3. Applied egg-rr 11.8%

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

      1. associate-/l* 11.8%

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

   5. Simplified 11.8%

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

      1. clear-num 11.8%

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

      2. inv-pow 11.8%

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

      3. sqrt-div 53.6%

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

      4. associate-/r* 68.2%

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

   7. Applied egg-rr 68.2%

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

      1. unpow-1 68.2%

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

   9. Simplified 68.2%

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

       1. sqrt-div 44.5%

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

       2. associate-/r/ 44.5%

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

       3. metadata-eval 44.5%

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

       4. sqrt-div 44.5%

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

       5. clear-num 44.6%

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

   11. Applied egg-rr 44.6%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 82.9%

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

      1. sqrt-div 95.7%

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

      2. associate-*r/ 90.8%

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

   3. Applied egg-rr 90.8%

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

      1. associate-/l* 95.9%

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

   5. Simplified 95.9%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{-262}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \]

Alternative 4: 82.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -4.6 \cdot 10^{+88}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{elif}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -4.6e+88)
   (* c0 (pow (/ (* V l) A) -0.5))
   (if (<= A -5e-310)
     (* c0 (/ (sqrt (/ A V)) (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 <= -4.6e+88) {
		tmp = c0 * pow(((V * l) / A), -0.5);
	} else if (A <= -5e-310) {
		tmp = c0 * (sqrt((A / V)) / 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 <= (-4.6d+88)) then
        tmp = c0 * (((v * l) / a) ** (-0.5d0))
    else if (a <= (-5d-310)) then
        tmp = c0 * (sqrt((a / v)) / 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 <= -4.6e+88) {
		tmp = c0 * Math.pow(((V * l) / A), -0.5);
	} else if (A <= -5e-310) {
		tmp = c0 * (Math.sqrt((A / V)) / 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 <= -4.6e+88:
		tmp = c0 * math.pow(((V * l) / A), -0.5)
	elif A <= -5e-310:
		tmp = c0 * (math.sqrt((A / V)) / 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 <= -4.6e+88)
		tmp = Float64(c0 * (Float64(Float64(V * l) / A) ^ -0.5));
	elseif (A <= -5e-310)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / 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 <= -4.6e+88)
		tmp = c0 * (((V * l) / A) ^ -0.5);
	elseif (A <= -5e-310)
		tmp = c0 * (sqrt((A / V)) / 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, -4.6e+88], N[(c0 * N[Power[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[A, -5e-310], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / 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 3 regimes

2. if A < -4.6000000000000003e88

   1. Initial program 82.1%

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

      1. pow1/2 82.1%

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

      2. clear-num 82.0%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5} \]

      3. inv-pow 82.0%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5} \]

      4. pow-pow 85.1%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}} \]

      5. associate-/l* 73.7%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)} \]

      6. metadata-eval 73.7%

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

   3. Applied egg-rr 73.7%

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

      1. associate-/l* 85.1%

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

   5. Simplified 85.1%

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

   if -4.6000000000000003e88 < A < -4.999999999999985e-310

   1. Initial program 81.7%

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

      1. associate-/r* 78.4%

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

      2. sqrt-div 40.4%

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

   3. Applied egg-rr 40.4%

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

   if -4.999999999999985e-310 < A

   1. Initial program 80.0%

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

      1. pow1/2 80.0%

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

      2. clear-num 79.9%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5} \]

      3. inv-pow 79.9%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5} \]

      4. pow-pow 80.8%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}} \]

      5. associate-/l* 74.7%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)} \]

      6. metadata-eval 74.7%

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

   3. Applied egg-rr 74.7%

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

      1. associate-/l* 80.8%

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

   5. Simplified 80.8%

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

      1. add-sqr-sqrt 80.6%

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

      2. sqrt-unprod 79.9%

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

      3. pow-prod-up 79.9%

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

      4. metadata-eval 79.9%

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

      5. inv-pow 79.9%

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

      6. clear-num 80.0%

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

      7. sqrt-div 91.7%

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

      8. *-un-lft-identity 91.7%

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

      9. add-sqr-sqrt 91.4%

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

      10. times-frac 91.5%

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

      11. pow1/2 91.5%

          \[\leadsto c0 \cdot \left(\frac{1}{\sqrt{\color{blue}{{\left(V \cdot \ell\right)}^{0.5}}}} \cdot \frac{\sqrt{A}}{\sqrt{\sqrt{V \cdot \ell}}}\right) \]

      12. metadata-eval 91.5%

          \[\leadsto c0 \cdot \left(\frac{1}{\sqrt{{\left(V \cdot \ell\right)}^{\color{blue}{\left(\sqrt{0.25}\right)}}}} \cdot \frac{\sqrt{A}}{\sqrt{\sqrt{V \cdot \ell}}}\right) \]

      13. sqrt-pow1 91.6%

          \[\leadsto c0 \cdot \left(\frac{1}{\color{blue}{{\left(V \cdot \ell\right)}^{\left(\frac{\sqrt{0.25}}{2}\right)}}} \cdot \frac{\sqrt{A}}{\sqrt{\sqrt{V \cdot \ell}}}\right) \]

      14. metadata-eval 91.6%

          \[\leadsto c0 \cdot \left(\frac{1}{{\left(V \cdot \ell\right)}^{\left(\frac{\color{blue}{0.5}}{2}\right)}} \cdot \frac{\sqrt{A}}{\sqrt{\sqrt{V \cdot \ell}}}\right) \]

      15. metadata-eval 91.6%

          \[\leadsto c0 \cdot \left(\frac{1}{{\left(V \cdot \ell\right)}^{\color{blue}{0.25}}} \cdot \frac{\sqrt{A}}{\sqrt{\sqrt{V \cdot \ell}}}\right) \]

      16. pow1/2 91.6%

          \[\leadsto c0 \cdot \left(\frac{1}{{\left(V \cdot \ell\right)}^{0.25}} \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{{\left(V \cdot \ell\right)}^{0.5}}}}\right) \]

      17. metadata-eval 91.6%

          \[\leadsto c0 \cdot \left(\frac{1}{{\left(V \cdot \ell\right)}^{0.25}} \cdot \frac{\sqrt{A}}{\sqrt{{\left(V \cdot \ell\right)}^{\color{blue}{\left(\sqrt{0.25}\right)}}}}\right) \]

      18. sqrt-pow1 91.6%

          \[\leadsto c0 \cdot \left(\frac{1}{{\left(V \cdot \ell\right)}^{0.25}} \cdot \frac{\sqrt{A}}{\color{blue}{{\left(V \cdot \ell\right)}^{\left(\frac{\sqrt{0.25}}{2}\right)}}}\right) \]

      19. metadata-eval 91.6%

          \[\leadsto c0 \cdot \left(\frac{1}{{\left(V \cdot \ell\right)}^{0.25}} \cdot \frac{\sqrt{A}}{{\left(V \cdot \ell\right)}^{\left(\frac{\color{blue}{0.5}}{2}\right)}}\right) \]

      20. metadata-eval 91.6%

          \[\leadsto c0 \cdot \left(\frac{1}{{\left(V \cdot \ell\right)}^{0.25}} \cdot \frac{\sqrt{A}}{{\left(V \cdot \ell\right)}^{\color{blue}{0.25}}}\right) \]

   7. Applied egg-rr 91.6%

      \[\leadsto c0 \cdot \color{blue}{\left(\frac{1}{{\left(V \cdot \ell\right)}^{0.25}} \cdot \frac{\sqrt{A}}{{\left(V \cdot \ell\right)}^{0.25}}\right)} \]
   8. Step-by-step derivation

      1. associate-*l/ 91.6%

         \[\leadsto c0 \cdot \color{blue}{\frac{1 \cdot \frac{\sqrt{A}}{{\left(V \cdot \ell\right)}^{0.25}}}{{\left(V \cdot \ell\right)}^{0.25}}} \]

      2. *-lft-identity 91.6%

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

      3. associate-/l/ 91.5%

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

      4. pow-sqr 91.7%

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\color{blue}{{\left(V \cdot \ell\right)}^{\left(2 \cdot 0.25\right)}}} \]

      5. metadata-eval 91.7%

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

      6. unpow1/2 91.7%

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

   9. Simplified 91.7%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -4.6 \cdot 10^{+88}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{elif}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]

Alternative 5: 84.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= l -1e-309)
   (* c0 (/ (sqrt A) (sqrt (* V l))))
   (/ c0 (* (sqrt (/ V A)) (sqrt l)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -1e-309) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 / (sqrt((V / A)) * 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 <= (-1d-309)) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 / (sqrt((v / a)) * 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 <= -1e-309) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if l <= -1e-309:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 / (math.sqrt((V / A)) * math.sqrt(l))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (l <= -1e-309)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * 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 <= -1e-309)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 / (sqrt((V / A)) * 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, -1e-309], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.000000000000002e-309

   1. Initial program 80.2%

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

      1. pow1/2 80.2%

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

      2. clear-num 80.2%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5} \]

      3. inv-pow 80.2%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5} \]

      4. pow-pow 80.7%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}} \]

      5. associate-/l* 77.9%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)} \]

      6. metadata-eval 77.9%

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

   3. Applied egg-rr 77.9%

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

      1. associate-/l* 80.7%

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

   5. Simplified 80.7%

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

      1. add-sqr-sqrt 80.5%

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

      2. sqrt-unprod 80.2%

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

      3. pow-prod-up 80.2%

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

      4. metadata-eval 80.2%

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

      5. inv-pow 80.2%

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

      6. clear-num 80.2%

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

      7. sqrt-div 35.6%

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

      8. *-un-lft-identity 35.6%

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

      9. add-sqr-sqrt 35.4%

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

      10. times-frac 35.5%

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

      11. pow1/2 35.5%

          \[\leadsto c0 \cdot \left(\frac{1}{\sqrt{\color{blue}{{\left(V \cdot \ell\right)}^{0.5}}}} \cdot \frac{\sqrt{A}}{\sqrt{\sqrt{V \cdot \ell}}}\right) \]

      12. metadata-eval 35.5%

          \[\leadsto c0 \cdot \left(\frac{1}{\sqrt{{\left(V \cdot \ell\right)}^{\color{blue}{\left(\sqrt{0.25}\right)}}}} \cdot \frac{\sqrt{A}}{\sqrt{\sqrt{V \cdot \ell}}}\right) \]

      13. sqrt-pow1 35.5%

          \[\leadsto c0 \cdot \left(\frac{1}{\color{blue}{{\left(V \cdot \ell\right)}^{\left(\frac{\sqrt{0.25}}{2}\right)}}} \cdot \frac{\sqrt{A}}{\sqrt{\sqrt{V \cdot \ell}}}\right) \]

      14. metadata-eval 35.5%

          \[\leadsto c0 \cdot \left(\frac{1}{{\left(V \cdot \ell\right)}^{\left(\frac{\color{blue}{0.5}}{2}\right)}} \cdot \frac{\sqrt{A}}{\sqrt{\sqrt{V \cdot \ell}}}\right) \]

      15. metadata-eval 35.5%

          \[\leadsto c0 \cdot \left(\frac{1}{{\left(V \cdot \ell\right)}^{\color{blue}{0.25}}} \cdot \frac{\sqrt{A}}{\sqrt{\sqrt{V \cdot \ell}}}\right) \]

      16. pow1/2 35.5%

          \[\leadsto c0 \cdot \left(\frac{1}{{\left(V \cdot \ell\right)}^{0.25}} \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{{\left(V \cdot \ell\right)}^{0.5}}}}\right) \]

      17. metadata-eval 35.5%

          \[\leadsto c0 \cdot \left(\frac{1}{{\left(V \cdot \ell\right)}^{0.25}} \cdot \frac{\sqrt{A}}{\sqrt{{\left(V \cdot \ell\right)}^{\color{blue}{\left(\sqrt{0.25}\right)}}}}\right) \]

      18. sqrt-pow1 35.5%

          \[\leadsto c0 \cdot \left(\frac{1}{{\left(V \cdot \ell\right)}^{0.25}} \cdot \frac{\sqrt{A}}{\color{blue}{{\left(V \cdot \ell\right)}^{\left(\frac{\sqrt{0.25}}{2}\right)}}}\right) \]

      19. metadata-eval 35.5%

          \[\leadsto c0 \cdot \left(\frac{1}{{\left(V \cdot \ell\right)}^{0.25}} \cdot \frac{\sqrt{A}}{{\left(V \cdot \ell\right)}^{\left(\frac{\color{blue}{0.5}}{2}\right)}}\right) \]

      20. metadata-eval 35.5%

          \[\leadsto c0 \cdot \left(\frac{1}{{\left(V \cdot \ell\right)}^{0.25}} \cdot \frac{\sqrt{A}}{{\left(V \cdot \ell\right)}^{\color{blue}{0.25}}}\right) \]

   7. Applied egg-rr 35.5%

      \[\leadsto c0 \cdot \color{blue}{\left(\frac{1}{{\left(V \cdot \ell\right)}^{0.25}} \cdot \frac{\sqrt{A}}{{\left(V \cdot \ell\right)}^{0.25}}\right)} \]
   8. Step-by-step derivation

      1. associate-*l/ 35.5%

         \[\leadsto c0 \cdot \color{blue}{\frac{1 \cdot \frac{\sqrt{A}}{{\left(V \cdot \ell\right)}^{0.25}}}{{\left(V \cdot \ell\right)}^{0.25}}} \]

      2. *-lft-identity 35.5%

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

      3. associate-/l/ 35.5%

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

      4. pow-sqr 35.6%

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\color{blue}{{\left(V \cdot \ell\right)}^{\left(2 \cdot 0.25\right)}}} \]

      5. metadata-eval 35.6%

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

      6. unpow1/2 35.6%

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

   9. Simplified 35.6%

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

   if -1.000000000000002e-309 < l

   1. Initial program 81.9%

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

      1. sqrt-div 43.7%

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

      2. associate-*r/ 40.3%

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

   3. Applied egg-rr 40.3%

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

      1. associate-/l* 43.8%

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

   5. Simplified 43.8%

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

      1. clear-num 43.8%

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

      2. inv-pow 43.8%

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

      3. sqrt-div 82.0%

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

      4. associate-/r* 76.1%

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

   7. Applied egg-rr 76.1%

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

      1. unpow-1 76.1%

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

   9. Simplified 76.1%

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

       1. sqrt-div 84.5%

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

       2. associate-/r/ 84.5%

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

       3. metadata-eval 84.5%

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

       4. sqrt-div 84.4%

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

       5. clear-num 85.1%

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

   11. Applied egg-rr 85.1%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 6: 84.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= l -1e-309)
   (/ c0 (/ (sqrt (* V l)) (sqrt A)))
   (/ c0 (* (sqrt (/ V A)) (sqrt l)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -1e-309) {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = c0 / (sqrt((V / A)) * 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 <= (-1d-309)) then
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    else
        tmp = c0 / (sqrt((v / a)) * 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 <= -1e-309) {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	} else {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if l <= -1e-309:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	else:
		tmp = c0 / (math.sqrt((V / A)) * math.sqrt(l))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (l <= -1e-309)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * 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 <= -1e-309)
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	else
		tmp = c0 / (sqrt((V / A)) * 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, -1e-309], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.000000000000002e-309

   1. Initial program 80.2%

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

      1. sqrt-div 35.6%

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

      2. associate-*r/ 34.9%

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

   3. Applied egg-rr 34.9%

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

      1. associate-/l* 35.6%

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

   5. Simplified 35.6%

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

   if -1.000000000000002e-309 < l

   1. Initial program 81.9%

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

      1. sqrt-div 43.7%

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

      2. associate-*r/ 40.3%

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

   3. Applied egg-rr 40.3%

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

      1. associate-/l* 43.8%

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

   5. Simplified 43.8%

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

      1. clear-num 43.8%

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

      2. inv-pow 43.8%

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

      3. sqrt-div 82.0%

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

      4. associate-/r* 76.1%

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

   7. Applied egg-rr 76.1%

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

      1. unpow-1 76.1%

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

   9. Simplified 76.1%

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

       1. sqrt-div 84.5%

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

       2. associate-/r/ 84.5%

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

       3. metadata-eval 84.5%

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

       4. sqrt-div 84.4%

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

       5. clear-num 85.1%

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

   11. Applied egg-rr 85.1%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 7: 78.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\sqrt{c0 \cdot \frac{c0 \cdot \frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

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)
     (sqrt (* c0 (/ (* c0 (/ A V)) l)))
     (if (<= t_0 2e+300) (* c0 (sqrt t_0)) (/ c0 (sqrt (* l (/ V 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 = sqrt((c0 * ((c0 * (A / V)) / l)));
	} else if (t_0 <= 2e+300) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((l * (V / 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 = sqrt((c0 * ((c0 * (a / v)) / l)))
    else if (t_0 <= 2d+300) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((l * (v / 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 = Math.sqrt((c0 * ((c0 * (A / V)) / l)));
	} else if (t_0 <= 2e+300) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((l * (V / 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 = math.sqrt((c0 * ((c0 * (A / V)) / l)))
	elif t_0 <= 2e+300:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((l * (V / 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 = sqrt(Float64(c0 * Float64(Float64(c0 * Float64(A / V)) / l)));
	elseif (t_0 <= 2e+300)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / 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 = sqrt((c0 * ((c0 * (A / V)) / l)));
	elseif (t_0 <= 2e+300)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((l * (V / 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[Sqrt[N[(c0 * N[(N[(c0 * N[(A / V), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 2e+300], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 47.8%

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

      1. pow1/2 47.8%

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

      2. clear-num 47.8%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5} \]

      3. inv-pow 47.8%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5} \]

      4. pow-pow 47.8%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}} \]

      5. associate-/l* 52.0%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)} \]

      6. metadata-eval 52.0%

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

   3. Applied egg-rr 52.0%

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

      1. associate-/l* 47.8%

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

   5. Simplified 47.8%

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

      1. associate-/l* 52.0%

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

      2. associate-/r/ 52.0%

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

   7. Applied egg-rr 52.0%

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

      1. add-sqr-sqrt 52.0%

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

      2. sqrt-unprod 49.9%

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

      3. *-commutative 49.9%

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

      4. *-commutative 49.9%

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

      5. swap-sqr 49.2%

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

      6. pow-prod-up 49.2%

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

      7. *-commutative 49.2%

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

      8. clear-num 49.2%

         \[\leadsto \sqrt{{\left(\ell \cdot \color{blue}{\frac{1}{\frac{A}{V}}}\right)}^{\left(-0.5 + -0.5\right)} \cdot \left(c0 \cdot c0\right)} \]

      9. div-inv 49.2%

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

      10. metadata-eval 49.2%

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

      11. inv-pow 49.2%

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

      12. clear-num 49.2%

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

      13. associate-/l/ 47.1%

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

      14. *-commutative 47.1%

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

   9. Applied egg-rr 47.1%

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

       1. associate-*r* 47.8%

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

       2. *-commutative 47.8%

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

       3. associate-/r* 49.9%

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

       4. associate-*l/ 54.1%

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

   11. Simplified 54.1%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 2.0000000000000001e300

   1. Initial program 99.2%

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

   if 2.0000000000000001e300 < (/.f64 A (*.f64 V l))

   1. Initial program 50.5%

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

      1. pow1/2 50.5%

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

      2. clear-num 50.5%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5} \]

      3. inv-pow 50.5%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5} \]

      4. pow-pow 55.5%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}} \]

      5. associate-/l* 63.6%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)} \]

      6. metadata-eval 63.6%

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

   3. Applied egg-rr 63.6%

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

      1. associate-/l* 55.5%

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

   5. Simplified 55.5%

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

      1. associate-/l* 63.6%

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

      2. associate-/r/ 63.8%

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

   7. Applied egg-rr 63.8%

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

      1. /-rgt-identity 63.8%

         \[\leadsto \color{blue}{\frac{c0}{1}} \cdot {\left(\frac{V}{A} \cdot \ell\right)}^{-0.5} \]

      2. add-sqr-sqrt 63.8%

         \[\leadsto \frac{c0}{1} \cdot \color{blue}{\left(\sqrt{{\left(\frac{V}{A} \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(\frac{V}{A} \cdot \ell\right)}^{-0.5}}\right)} \]

      3. sqrt-unprod 60.2%

         \[\leadsto \frac{c0}{1} \cdot \color{blue}{\sqrt{{\left(\frac{V}{A} \cdot \ell\right)}^{-0.5} \cdot {\left(\frac{V}{A} \cdot \ell\right)}^{-0.5}}} \]

      4. pow-prod-up 60.2%

         \[\leadsto \frac{c0}{1} \cdot \sqrt{\color{blue}{{\left(\frac{V}{A} \cdot \ell\right)}^{\left(-0.5 + -0.5\right)}}} \]

      5. *-commutative 60.2%

         \[\leadsto \frac{c0}{1} \cdot \sqrt{{\color{blue}{\left(\ell \cdot \frac{V}{A}\right)}}^{\left(-0.5 + -0.5\right)}} \]

      6. clear-num 60.2%

         \[\leadsto \frac{c0}{1} \cdot \sqrt{{\left(\ell \cdot \color{blue}{\frac{1}{\frac{A}{V}}}\right)}^{\left(-0.5 + -0.5\right)}} \]

      7. div-inv 60.2%

         \[\leadsto \frac{c0}{1} \cdot \sqrt{{\color{blue}{\left(\frac{\ell}{\frac{A}{V}}\right)}}^{\left(-0.5 + -0.5\right)}} \]

      8. metadata-eval 60.2%

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

      9. inv-pow 60.2%

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

      10. clear-num 60.3%

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

      11. associate-/r/ 60.3%

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

      12. sqrt-div 46.0%

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

      13. associate-/r/ 45.9%

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

      14. metadata-eval 45.9%

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

      15. sqrt-div 45.9%

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

      16. clear-num 46.0%

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

      17. sqrt-prod 63.9%

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

      18. *-commutative 63.9%

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

      19. associate-*r/ 55.5%

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

      20. associate-*l/ 63.6%

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

      21. sqrt-unprod 48.0%

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

      22. *-commutative 48.0%

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

      23. sqrt-unprod 63.6%

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

   9. Applied egg-rr 63.6%

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

       1. associate-*r/ 55.5%

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

       2. associate-*l/ 63.9%

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

   11. Simplified 63.9%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;\sqrt{c0 \cdot \frac{c0 \cdot \frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+300}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]

Alternative 8: 76.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

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 (/ A (* V l))))
   (if (<= t_0 2e+300) (* c0 (sqrt t_0)) (* c0 (sqrt (/ (/ A V) l))))))

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 <= 2e+300) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 * sqrt(((A / 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 = a / (v * l)
    if (t_0 <= 2d+300) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 * sqrt(((a / 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 = A / (V * l);
	double tmp;
	if (t_0 <= 2e+300) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 2e+300:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	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 <= 2e+300)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	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 <= 2e+300)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 * sqrt(((A / 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[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+300], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 2.0000000000000001e300

   1. Initial program 88.1%

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

   if 2.0000000000000001e300 < (/.f64 A (*.f64 V l))

   1. Initial program 50.5%

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

      1. sqrt-div 25.4%

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

      2. associate-*r/ 25.4%

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

   3. Applied egg-rr 25.4%

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

      1. associate-/l* 25.5%

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

   5. Simplified 25.5%

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

      1. clear-num 25.4%

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

      2. associate-/r/ 25.4%

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

      3. clear-num 25.4%

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

      4. sqrt-div 50.5%

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

      5. associate-/r* 60.3%

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

   7. Applied egg-rr 60.3%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+300}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]

Alternative 9: 76.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

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 2e+300) (* 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 <= 2e+300) {
		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 <= 2d+300) 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 <= 2e+300) {
		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 <= 2e+300:
		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 <= 2e+300)
		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 <= 2e+300)
		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, 2e+300], 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 2 regimes

2. if (/.f64 A (*.f64 V l)) < 2.0000000000000001e300

   1. Initial program 88.1%

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

   if 2.0000000000000001e300 < (/.f64 A (*.f64 V l))

   1. Initial program 50.5%

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

      1. pow1/2 50.5%

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

      2. clear-num 50.5%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5} \]

      3. inv-pow 50.5%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5} \]

      4. pow-pow 55.5%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}} \]

      5. associate-/l* 63.6%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)} \]

      6. metadata-eval 63.6%

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

   3. Applied egg-rr 63.6%

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

      1. associate-/l* 55.5%

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

   5. Simplified 55.5%

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

      1. add-sqr-sqrt 55.5%

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

      2. sqrt-unprod 50.5%

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

      3. pow-prod-up 50.5%

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

      4. metadata-eval 50.5%

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

      5. inv-pow 50.5%

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

      6. clear-num 50.5%

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

      7. sqrt-div 25.4%

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

      8. clear-num 25.4%

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

      9. div-inv 25.5%

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

      10. sqrt-undiv 55.5%

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

      11. associate-*r/ 63.6%

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

   7. Applied egg-rr 63.6%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+300}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]

Alternative 10: 76.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

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 2e+300) (* c0 (sqrt t_0)) (/ c0 (sqrt (* l (/ V 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 <= 2e+300) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((l * (V / 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 <= 2d+300) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((l * (v / 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 <= 2e+300) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((l * (V / 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 <= 2e+300:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((l * (V / 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 <= 2e+300)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / 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 <= 2e+300)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((l * (V / 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, 2e+300], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 2.0000000000000001e300

   1. Initial program 88.1%

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

   if 2.0000000000000001e300 < (/.f64 A (*.f64 V l))

   1. Initial program 50.5%

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

      1. pow1/2 50.5%

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

      2. clear-num 50.5%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5} \]

      3. inv-pow 50.5%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5} \]

      4. pow-pow 55.5%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}} \]

      5. associate-/l* 63.6%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)} \]

      6. metadata-eval 63.6%

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

   3. Applied egg-rr 63.6%

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

      1. associate-/l* 55.5%

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

   5. Simplified 55.5%

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

      1. associate-/l* 63.6%

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

      2. associate-/r/ 63.8%

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

   7. Applied egg-rr 63.8%

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

      1. /-rgt-identity 63.8%

         \[\leadsto \color{blue}{\frac{c0}{1}} \cdot {\left(\frac{V}{A} \cdot \ell\right)}^{-0.5} \]

      2. add-sqr-sqrt 63.8%

         \[\leadsto \frac{c0}{1} \cdot \color{blue}{\left(\sqrt{{\left(\frac{V}{A} \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(\frac{V}{A} \cdot \ell\right)}^{-0.5}}\right)} \]

      3. sqrt-unprod 60.2%

         \[\leadsto \frac{c0}{1} \cdot \color{blue}{\sqrt{{\left(\frac{V}{A} \cdot \ell\right)}^{-0.5} \cdot {\left(\frac{V}{A} \cdot \ell\right)}^{-0.5}}} \]

      4. pow-prod-up 60.2%

         \[\leadsto \frac{c0}{1} \cdot \sqrt{\color{blue}{{\left(\frac{V}{A} \cdot \ell\right)}^{\left(-0.5 + -0.5\right)}}} \]

      5. *-commutative 60.2%

         \[\leadsto \frac{c0}{1} \cdot \sqrt{{\color{blue}{\left(\ell \cdot \frac{V}{A}\right)}}^{\left(-0.5 + -0.5\right)}} \]

      6. clear-num 60.2%

         \[\leadsto \frac{c0}{1} \cdot \sqrt{{\left(\ell \cdot \color{blue}{\frac{1}{\frac{A}{V}}}\right)}^{\left(-0.5 + -0.5\right)}} \]

      7. div-inv 60.2%

         \[\leadsto \frac{c0}{1} \cdot \sqrt{{\color{blue}{\left(\frac{\ell}{\frac{A}{V}}\right)}}^{\left(-0.5 + -0.5\right)}} \]

      8. metadata-eval 60.2%

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

      9. inv-pow 60.2%

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

      10. clear-num 60.3%

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

      11. associate-/r/ 60.3%

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

      12. sqrt-div 46.0%

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

      13. associate-/r/ 45.9%

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

      14. metadata-eval 45.9%

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

      15. sqrt-div 45.9%

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

      16. clear-num 46.0%

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

      17. sqrt-prod 63.9%

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

      18. *-commutative 63.9%

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

      19. associate-*r/ 55.5%

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

      20. associate-*l/ 63.6%

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

      21. sqrt-unprod 48.0%

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

      22. *-commutative 48.0%

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

      23. sqrt-unprod 63.6%

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

   9. Applied egg-rr 63.6%

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

       1. associate-*r/ 55.5%

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

       2. associate-*l/ 63.9%

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

   11. Simplified 63.9%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+300}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]

Alternative 11: 73.4% 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 81.0%

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

2. Final simplification 81.0%

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

Reproduce

herbie shell --seed 2023255 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))