Henrywood and Agarwal, Equation (3)

Percentage Accurate: 72.7% → 92.1%

Time: 10.8s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-311}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 10^{-312}:\\ \;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{\ell}}}{\sqrt[3]{V}}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell \cdot V}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) (- INFINITY))
   (* c0 (/ (sqrt (/ (- A) l)) (sqrt (- V))))
   (if (<= (* l V) -2e-311)
     (* c0 (/ (sqrt (- A)) (sqrt (* l (- V)))))
     (if (<= (* l V) 1e-312)
       (* c0 (pow (/ (cbrt (/ A l)) (cbrt V)) 1.5))
       (/ c0 (/ (sqrt (* l V)) (sqrt A)))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = c0 * (sqrt((-A / l)) / sqrt(-V));
	} else if ((l * V) <= -2e-311) {
		tmp = c0 * (sqrt(-A) / sqrt((l * -V)));
	} else if ((l * V) <= 1e-312) {
		tmp = c0 * pow((cbrt((A / l)) / cbrt(V)), 1.5);
	} else {
		tmp = c0 / (sqrt((l * V)) / sqrt(A));
	}
	return tmp;
}

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 * (Math.sqrt((-A / l)) / Math.sqrt(-V));
	} else if ((l * V) <= -2e-311) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((l * -V)));
	} else if ((l * V) <= 1e-312) {
		tmp = c0 * Math.pow((Math.cbrt((A / l)) / Math.cbrt(V)), 1.5);
	} else {
		tmp = c0 / (Math.sqrt((l * V)) / Math.sqrt(A));
	}
	return tmp;
}

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = Float64(c0 * Float64(sqrt(Float64(Float64(-A) / l)) / sqrt(Float64(-V))));
	elseif (Float64(l * V) <= -2e-311)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(l * Float64(-V)))));
	elseif (Float64(l * V) <= 1e-312)
		tmp = Float64(c0 * (Float64(cbrt(Float64(A / l)) / cbrt(V)) ^ 1.5));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(l * V)) / sqrt(A)));
	end
	return 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[(l * V), $MachinePrecision], (-Infinity)], N[(c0 * N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -2e-311], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 1e-312], N[(c0 * N[Power[N[(N[Power[N[(A / l), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[V, 1/3], $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 63.3%

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

      1. pow1/2 63.3%

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

      2. clear-num 63.3%

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

      3. inv-pow 63.3%

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

      4. pow-pow 63.3%

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

      5. associate-/l* 79.1%

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

      6. metadata-eval 79.1%

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

   3. Applied egg-rr 79.1%

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

      1. associate-/l* 63.3%

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

      2. *-lft-identity 63.3%

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

      3. times-frac 79.0%

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

      4. /-rgt-identity 79.0%

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

   5. Simplified 79.0%

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

   6. Taylor expanded in c0 around 0 63.3%

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

      1. associate-/l/ 79.1%

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

   8. Simplified 79.1%

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

      1. frac-2neg 79.1%

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

      2. sqrt-div 39.2%

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

   10. Applied egg-rr 39.2%

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

       1. distribute-neg-frac 39.2%

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

   12. Simplified 39.2%

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

   if -inf.0 < (*.f64 V l) < -1.9999999999999e-311

   1. Initial program 83.6%

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

      1. frac-2neg 83.6%

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

      2. sqrt-div 99.3%

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

      3. *-commutative 99.3%

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

      4. distribute-rgt-neg-in 99.3%

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

   3. Applied egg-rr 99.3%

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

   if -1.9999999999999e-311 < (*.f64 V l) < 9.9999999999847e-313

   1. Initial program 33.4%

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

      1. pow1/2 33.4%

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

      2. add-cube-cbrt 33.4%

         \[\leadsto c0 \cdot {\color{blue}{\left(\left(\sqrt[3]{\frac{A}{V \cdot \ell}} \cdot \sqrt[3]{\frac{A}{V \cdot \ell}}\right) \cdot \sqrt[3]{\frac{A}{V \cdot \ell}}\right)}}^{0.5} \]

      3. pow3 33.4%

         \[\leadsto c0 \cdot {\color{blue}{\left({\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{3}\right)}}^{0.5} \]

      4. pow-pow 33.4%

         \[\leadsto c0 \cdot \color{blue}{{\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{\left(3 \cdot 0.5\right)}} \]

      5. metadata-eval 33.4%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{\color{blue}{1.5}} \]

   3. Applied egg-rr 33.4%

      \[\leadsto c0 \cdot \color{blue}{{\left(\sqrt[3]{\frac{A}{V \cdot \ell}}\right)}^{1.5}} \]
   4. Step-by-step derivation

      1. associate-/r* 63.3%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{\frac{A}{V}}{\ell}}}\right)}^{1.5} \]

      2. div-inv 63.3%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{A}{V} \cdot \frac{1}{\ell}}}\right)}^{1.5} \]

      3. frac-times 33.4%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{A \cdot 1}{V \cdot \ell}}}\right)}^{1.5} \]

      4. *-commutative 33.4%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{A \cdot 1}{\color{blue}{\ell \cdot V}}}\right)}^{1.5} \]

      5. frac-times 63.3%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\color{blue}{\frac{A}{\ell} \cdot \frac{1}{V}}}\right)}^{1.5} \]

      6. cbrt-prod 83.5%

         \[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{A}{\ell}} \cdot \sqrt[3]{\frac{1}{V}}\right)}}^{1.5} \]

      7. cbrt-div 83.5%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{A}{\ell}} \cdot \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{V}}}\right)}^{1.5} \]

      8. metadata-eval 83.5%

         \[\leadsto c0 \cdot {\left(\sqrt[3]{\frac{A}{\ell}} \cdot \frac{\color{blue}{1}}{\sqrt[3]{V}}\right)}^{1.5} \]

   5. Applied egg-rr 83.5%

      \[\leadsto c0 \cdot {\color{blue}{\left(\sqrt[3]{\frac{A}{\ell}} \cdot \frac{1}{\sqrt[3]{V}}\right)}}^{1.5} \]
   6. Step-by-step derivation

      1. associate-*r/ 83.5%

         \[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{\frac{A}{\ell}} \cdot 1}{\sqrt[3]{V}}\right)}}^{1.5} \]

      2. *-rgt-identity 83.5%

         \[\leadsto c0 \cdot {\left(\frac{\color{blue}{\sqrt[3]{\frac{A}{\ell}}}}{\sqrt[3]{V}}\right)}^{1.5} \]

   7. Simplified 83.5%

      \[\leadsto c0 \cdot {\color{blue}{\left(\frac{\sqrt[3]{\frac{A}{\ell}}}{\sqrt[3]{V}}\right)}}^{1.5} \]

   if 9.9999999999847e-313 < (*.f64 V l)

   1. Initial program 76.4%

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

      1. sqrt-div 94.5%

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

      2. associate-*r/ 92.5%

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

   3. Applied egg-rr 92.5%

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

      1. associate-/l* 94.5%

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

   5. Simplified 94.5%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-311}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 10^{-312}:\\ \;\;\;\;c0 \cdot {\left(\frac{\sqrt[3]{\frac{A}{\ell}}}{\sqrt[3]{V}}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell \cdot V}}{\sqrt{A}}}\\ \end{array} \]

Alternative 2: 90.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if A < -3.999999999999988e-310

   1. Initial program 68.7%

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

      1. sqrt-div 0.0%

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

      2. associate-*r/ 0.0%

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

   3. Applied egg-rr 0.0%

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

      1. associate-*l/ 0.0%

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

   5. Simplified 0.0%

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

      1. associate-*l/ 0.0%

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

      2. sqrt-prod 0.0%

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

      3. associate-/r* 0.0%

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

   7. Applied egg-rr 0.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. *-un-lft-identity 0.0%

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

      3. times-frac 0.0%

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

      4. associate-/l* 0.0%

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

      5. sqrt-undiv 0.0%

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

      6. associate-/l* 0.0%

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

      7. sqrt-undiv 19.9%

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

   9. Applied egg-rr 19.9%

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

       1. /-rgt-identity 19.9%

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

       2. associate-*r/ 19.9%

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

       3. rem-square-sqrt 30.6%

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

       4. associate-/l/ 32.0%

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

   11. Simplified 32.0%

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

       1. frac-2neg 32.0%

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

       2. sqrt-div 41.7%

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

   13. Applied egg-rr 41.7%

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

   if -3.999999999999988e-310 < A

   1. Initial program 74.8%

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

      1. sqrt-div 91.2%

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

      2. associate-*r/ 89.4%

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

   3. Applied egg-rr 89.4%

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

      1. associate-/l* 91.3%

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

   5. Simplified 91.3%

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

4. Final simplification 66.3%

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

Alternative 3: 89.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-311}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 10^{-312}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell \cdot V}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (/ (sqrt (/ (- A) l)) (sqrt (- V))))))
   (if (<= (* l V) (- INFINITY))
     t_0
     (if (<= (* l V) -2e-311)
       (* c0 (/ (sqrt (- A)) (sqrt (* l (- V)))))
       (if (<= (* l V) 1e-312) t_0 (/ c0 (/ (sqrt (* l V)) (sqrt A))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (sqrt((-A / l)) / sqrt(-V));
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((l * V) <= -2e-311) {
		tmp = c0 * (sqrt(-A) / sqrt((l * -V)));
	} else if ((l * V) <= 1e-312) {
		tmp = t_0;
	} else {
		tmp = c0 / (sqrt((l * V)) / sqrt(A));
	}
	return tmp;
}

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (Math.sqrt((-A / l)) / Math.sqrt(-V));
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if ((l * V) <= -2e-311) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((l * -V)));
	} else if ((l * V) <= 1e-312) {
		tmp = t_0;
	} else {
		tmp = c0 / (Math.sqrt((l * V)) / Math.sqrt(A));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = c0 * (math.sqrt((-A / l)) / math.sqrt(-V))
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = t_0
	elif (l * V) <= -2e-311:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((l * -V)))
	elif (l * V) <= 1e-312:
		tmp = t_0
	else:
		tmp = c0 / (math.sqrt((l * V)) / math.sqrt(A))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * Float64(sqrt(Float64(Float64(-A) / l)) / sqrt(Float64(-V))))
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(l * V) <= -2e-311)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(l * Float64(-V)))));
	elseif (Float64(l * V) <= 1e-312)
		tmp = t_0;
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(l * V)) / sqrt(A)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * (sqrt((-A / l)) / sqrt(-V));
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = t_0;
	elseif ((l * V) <= -2e-311)
		tmp = c0 * (sqrt(-A) / sqrt((l * -V)));
	elseif ((l * V) <= 1e-312)
		tmp = t_0;
	else
		tmp = c0 / (sqrt((l * V)) / 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_] := Block[{t$95$0 = N[(c0 * N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(l * V), $MachinePrecision], -2e-311], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 1e-312], t$95$0, N[(c0 / N[(N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -inf.0 or -1.9999999999999e-311 < (*.f64 V l) < 9.9999999999847e-313

   1. Initial program 43.2%

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

      1. pow1/2 43.2%

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

      2. clear-num 43.2%

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

      3. inv-pow 43.2%

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

      4. pow-pow 43.2%

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

      5. associate-/l* 69.6%

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

      6. metadata-eval 69.6%

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

   3. Applied egg-rr 69.6%

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

      1. associate-/l* 43.2%

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

      2. *-lft-identity 43.2%

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

      3. times-frac 69.6%

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

      4. /-rgt-identity 69.6%

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

   5. Simplified 69.6%

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

   6. Taylor expanded in c0 around 0 43.2%

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

      1. associate-/l/ 68.7%

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

   8. Simplified 68.7%

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

      1. frac-2neg 68.7%

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

      2. sqrt-div 52.8%

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

   10. Applied egg-rr 52.8%

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

       1. distribute-neg-frac 52.8%

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

   12. Simplified 52.8%

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

   if -inf.0 < (*.f64 V l) < -1.9999999999999e-311

   1. Initial program 83.6%

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

      1. frac-2neg 83.6%

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

      2. sqrt-div 99.3%

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

      3. *-commutative 99.3%

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

      4. distribute-rgt-neg-in 99.3%

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

   3. Applied egg-rr 99.3%

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

   if 9.9999999999847e-313 < (*.f64 V l)

   1. Initial program 76.4%

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

      1. sqrt-div 94.5%

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

      2. associate-*r/ 92.5%

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

   3. Applied egg-rr 92.5%

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

      1. associate-/l* 94.5%

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

   5. Simplified 94.5%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-311}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 10^{-312}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell \cdot V}}{\sqrt{A}}}\\ \end{array} \]

Alternative 4: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-311}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 10^{-312}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell \cdot V}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) (- INFINITY))
   (/ c0 (* (sqrt l) (sqrt (/ V A))))
   (if (<= (* l V) -2e-311)
     (* c0 (/ (sqrt (- A)) (sqrt (* l (- V)))))
     (if (<= (* l V) 1e-312)
       (/ c0 (sqrt (* V (/ l A))))
       (/ c0 (/ (sqrt (* l V)) (sqrt A)))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	} else if ((l * V) <= -2e-311) {
		tmp = c0 * (sqrt(-A) / sqrt((l * -V)));
	} else if ((l * V) <= 1e-312) {
		tmp = c0 / sqrt((V * (l / A)));
	} else {
		tmp = c0 / (sqrt((l * V)) / sqrt(A));
	}
	return tmp;
}

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 / (Math.sqrt(l) * Math.sqrt((V / A)));
	} else if ((l * V) <= -2e-311) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((l * -V)));
	} else if ((l * V) <= 1e-312) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else {
		tmp = c0 / (Math.sqrt((l * V)) / Math.sqrt(A));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = c0 / (math.sqrt(l) * math.sqrt((V / A)))
	elif (l * V) <= -2e-311:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((l * -V)))
	elif (l * V) <= 1e-312:
		tmp = c0 / math.sqrt((V * (l / A)))
	else:
		tmp = c0 / (math.sqrt((l * V)) / math.sqrt(A))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))));
	elseif (Float64(l * V) <= -2e-311)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(l * Float64(-V)))));
	elseif (Float64(l * V) <= 1e-312)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(l * V)) / 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 ((l * V) <= -Inf)
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	elseif ((l * V) <= -2e-311)
		tmp = c0 * (sqrt(-A) / sqrt((l * -V)));
	elseif ((l * V) <= 1e-312)
		tmp = c0 / sqrt((V * (l / A)));
	else
		tmp = c0 / (sqrt((l * V)) / 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[(l * V), $MachinePrecision], (-Infinity)], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -2e-311], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 1e-312], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 63.3%

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

      1. sqrt-div 0.0%

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

      2. associate-*r/ 0.0%

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

   3. Applied egg-rr 0.0%

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

      1. associate-*l/ 0.0%

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

   5. Simplified 0.0%

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

      1. associate-*l/ 0.0%

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

      2. sqrt-prod 0.0%

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

      3. associate-/r* 0.0%

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

   7. Applied egg-rr 0.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. *-un-lft-identity 0.0%

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

      3. times-frac 0.0%

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

      4. associate-/l* 0.0%

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

      5. sqrt-undiv 0.0%

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

      6. associate-/l* 0.0%

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

      7. sqrt-undiv 28.7%

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

   9. Applied egg-rr 28.7%

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

       1. /-rgt-identity 28.7%

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

       2. associate-*r/ 28.7%

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

       3. rem-square-sqrt 34.3%

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

       4. associate-/l/ 34.3%

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

   11. Simplified 34.3%

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

   if -inf.0 < (*.f64 V l) < -1.9999999999999e-311

   1. Initial program 83.6%

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

      1. frac-2neg 83.6%

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

      2. sqrt-div 99.3%

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

      3. *-commutative 99.3%

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

      4. distribute-rgt-neg-in 99.3%

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

   3. Applied egg-rr 99.3%

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

   if -1.9999999999999e-311 < (*.f64 V l) < 9.9999999999847e-313

   1. Initial program 33.4%

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

      1. sqrt-div 19.5%

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

      2. associate-*r/ 19.5%

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

   3. Applied egg-rr 19.5%

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

      1. associate-*l/ 17.6%

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

   5. Simplified 17.6%

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

      1. expm1-log1p-u 12.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \ell}} \cdot \sqrt{A}\right)\right)} \]

      2. expm1-udef 9.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \ell}} \cdot \sqrt{A}\right)} - 1} \]

      3. associate-*l/ 11.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}}\right)} - 1 \]

      4. associate-/l* 11.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}}\right)} - 1 \]

      5. sqrt-div 17.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\frac{V \cdot \ell}{A}}}}\right)} - 1 \]

      6. associate-*r/ 24.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{V \cdot \frac{\ell}{A}}}}\right)} - 1 \]

   7. Applied egg-rr 24.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 33.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)\right)} \]

      2. expm1-log1p 65.0%

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

   9. Simplified 65.0%

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

   if 9.9999999999847e-313 < (*.f64 V l)

   1. Initial program 76.4%

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

      1. sqrt-div 94.5%

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

      2. associate-*r/ 92.5%

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

   3. Applied egg-rr 92.5%

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

      1. associate-/l* 94.5%

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

   5. Simplified 94.5%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-311}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 10^{-312}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell \cdot V}}{\sqrt{A}}}\\ \end{array} \]

Alternative 5: 83.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -4 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \left({\left(\frac{V}{A}\right)}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell \cdot V}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -4e-310)
   (* c0 (* (pow (/ V A) -0.5) (pow l -0.5)))
   (/ c0 (/ (sqrt (* l V)) (sqrt A)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -4e-310) {
		tmp = c0 * (pow((V / A), -0.5) * pow(l, -0.5));
	} else {
		tmp = c0 / (sqrt((l * V)) / 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 (a <= (-4d-310)) then
        tmp = c0 * (((v / a) ** (-0.5d0)) * (l ** (-0.5d0)))
    else
        tmp = c0 / (sqrt((l * v)) / 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 (A <= -4e-310) {
		tmp = c0 * (Math.pow((V / A), -0.5) * Math.pow(l, -0.5));
	} else {
		tmp = c0 / (Math.sqrt((l * V)) / Math.sqrt(A));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -4e-310:
		tmp = c0 * (math.pow((V / A), -0.5) * math.pow(l, -0.5))
	else:
		tmp = c0 / (math.sqrt((l * V)) / math.sqrt(A))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -4e-310)
		tmp = Float64(c0 * Float64((Float64(V / A) ^ -0.5) * (l ^ -0.5)));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(l * V)) / 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 (A <= -4e-310)
		tmp = c0 * (((V / A) ^ -0.5) * (l ^ -0.5));
	else
		tmp = c0 / (sqrt((l * V)) / 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[A, -4e-310], N[(c0 * N[(N[Power[N[(V / A), $MachinePrecision], -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -3.999999999999988e-310

   1. Initial program 68.7%

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

      1. pow1/2 68.7%

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

      2. clear-num 67.7%

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

      3. inv-pow 67.7%

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

      4. pow-pow 67.7%

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

      5. associate-/l* 70.6%

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

      6. metadata-eval 70.6%

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

   3. Applied egg-rr 70.6%

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

      1. associate-/l* 67.7%

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

      2. *-lft-identity 67.7%

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

      3. times-frac 70.8%

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

      4. /-rgt-identity 70.8%

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

   5. Simplified 70.8%

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

      1. clear-num 70.4%

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

      2. div-inv 70.6%

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

      3. associate-/r/ 68.1%

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

      4. unpow-prod-down 32.1%

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

   7. Applied egg-rr 32.1%

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

   if -3.999999999999988e-310 < A

   1. Initial program 74.8%

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

      1. sqrt-div 91.2%

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

      2. associate-*r/ 89.4%

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

   3. Applied egg-rr 89.4%

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

      1. associate-/l* 91.3%

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

   5. Simplified 91.3%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -4 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \left({\left(\frac{V}{A}\right)}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell \cdot V}}{\sqrt{A}}}\\ \end{array} \]

Alternative 6: 81.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-309}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= l 5e-309)
   (/ c0 (sqrt (* V (/ l A))))
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= 5e-309) {
		tmp = c0 / sqrt((V * (l / A)));
	} else {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= 5d-309) then
        tmp = c0 / sqrt((v * (l / a)))
    else
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= 5e-309) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if l <= 5e-309:
		tmp = c0 / math.sqrt((V * (l / A)))
	else:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (l <= 5e-309)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	else
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (l <= 5e-309)
		tmp = c0 / sqrt((V * (l / A)));
	else
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[l, 5e-309], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.9999999999999995e-309

   1. Initial program 74.6%

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

      1. sqrt-div 43.4%

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

      2. associate-*r/ 42.5%

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

   3. Applied egg-rr 42.5%

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

      1. associate-*l/ 41.6%

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

   5. Simplified 41.6%

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

      1. expm1-log1p-u 30.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \ell}} \cdot \sqrt{A}\right)\right)} \]

      2. expm1-udef 14.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \ell}} \cdot \sqrt{A}\right)} - 1} \]

      3. associate-*l/ 14.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}}\right)} - 1 \]

      4. associate-/l* 14.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}}\right)} - 1 \]

      5. sqrt-div 28.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\frac{V \cdot \ell}{A}}}}\right)} - 1 \]

      6. associate-*r/ 28.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{V \cdot \frac{\ell}{A}}}}\right)} - 1 \]

   7. Applied egg-rr 28.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 56.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)\right)} \]

      2. expm1-log1p 76.9%

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

   9. Simplified 76.9%

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

   if 4.9999999999999995e-309 < l

   1. Initial program 68.1%

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

      1. associate-/r* 67.1%

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

      2. sqrt-div 74.4%

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

   3. Applied egg-rr 74.4%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-309}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 7: 82.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if A < -3.999999999999988e-310

   1. Initial program 68.7%

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

      1. associate-/r* 67.9%

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

      2. sqrt-div 31.6%

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

   3. Applied egg-rr 31.6%

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

   if -3.999999999999988e-310 < A

   1. Initial program 74.8%

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

      1. sqrt-div 91.2%

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

      2. associate-*r/ 89.4%

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

   3. Applied egg-rr 89.4%

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

      1. associate-*l/ 87.1%

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

   5. Simplified 87.1%

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

4. Final simplification 59.1%

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

Alternative 8: 82.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if A < -3.999999999999988e-310

   1. Initial program 68.7%

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

      1. sqrt-div 0.0%

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

      2. associate-*r/ 0.0%

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

   3. Applied egg-rr 0.0%

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

      1. associate-*l/ 0.0%

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

   5. Simplified 0.0%

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

      1. associate-*l/ 0.0%

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

      2. sqrt-prod 0.0%

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

      3. associate-/r* 0.0%

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

   7. Applied egg-rr 0.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. *-un-lft-identity 0.0%

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

      3. times-frac 0.0%

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

      4. associate-/l* 0.0%

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

      5. sqrt-undiv 0.0%

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

      6. associate-/l* 0.0%

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

      7. sqrt-undiv 19.9%

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

   9. Applied egg-rr 19.9%

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

       1. /-rgt-identity 19.9%

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

       2. associate-*r/ 19.9%

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

       3. rem-square-sqrt 30.6%

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

       4. associate-/l/ 32.0%

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

   11. Simplified 32.0%

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

   if -3.999999999999988e-310 < A

   1. Initial program 74.8%

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

      1. sqrt-div 91.2%

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

      2. associate-*r/ 89.4%

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

   3. Applied egg-rr 89.4%

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

      1. associate-*l/ 87.1%

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

   5. Simplified 87.1%

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

4. Final simplification 59.3%

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

Alternative 9: 83.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if A < -3.999999999999988e-310

   1. Initial program 68.7%

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

      1. sqrt-div 0.0%

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

      2. associate-*r/ 0.0%

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

   3. Applied egg-rr 0.0%

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

      1. associate-*l/ 0.0%

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

   5. Simplified 0.0%

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

      1. associate-*l/ 0.0%

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

      2. sqrt-prod 0.0%

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

      3. associate-/r* 0.0%

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

   7. Applied egg-rr 0.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. *-un-lft-identity 0.0%

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

      3. times-frac 0.0%

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

      4. associate-/l* 0.0%

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

      5. sqrt-undiv 0.0%

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

      6. associate-/l* 0.0%

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

      7. sqrt-undiv 19.9%

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

   9. Applied egg-rr 19.9%

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

       1. /-rgt-identity 19.9%

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

       2. associate-*r/ 19.9%

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

       3. rem-square-sqrt 30.6%

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

       4. associate-/l/ 32.0%

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

   11. Simplified 32.0%

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

   if -3.999999999999988e-310 < A

   1. Initial program 74.8%

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

      1. sqrt-div 91.2%

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

      2. associate-*r/ 89.4%

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

   3. Applied egg-rr 89.4%

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

      1. associate-/l* 91.3%

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

   5. Simplified 91.3%

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

4. Final simplification 61.4%

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

Alternative 10: 79.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \end{array} \]

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 (* l V))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 5e+302) (* c0 (sqrt t_0)) (* c0 (pow (* V (/ l A)) -0.5))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 5e+302) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 * pow((V * (l / A)), -0.5);
	}
	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 / (l * v)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 5d+302) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 * ((v * (l / a)) ** (-0.5d0))
    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 / (l * V);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 5e+302) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 * Math.pow((V * (l / A)), -0.5);
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (l * V)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 5e+302:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 * math.pow((V * (l / A)), -0.5)
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(l * V))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 5e+302)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 * (Float64(V * Float64(l / A)) ^ -0.5));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (l * V);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 5e+302)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 * ((V * (l / A)) ^ -0.5);
	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[(l * V), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+302], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Power[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 48.1%

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

      1. associate-/r* 57.1%

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

      2. div-inv 57.1%

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

   3. Applied egg-rr 57.1%

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

      1. un-div-inv 57.1%

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

   5. Applied egg-rr 57.1%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 5e302

   1. Initial program 98.0%

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

   if 5e302 < (/.f64 A (*.f64 V l))

   1. Initial program 32.1%

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

      1. pow1/2 32.1%

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

      2. clear-num 32.1%

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

      3. inv-pow 32.1%

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

      4. pow-pow 33.3%

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

      5. associate-/l* 49.3%

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

      6. metadata-eval 49.3%

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

   3. Applied egg-rr 49.3%

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

      1. associate-/l* 33.3%

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

      2. *-lft-identity 33.3%

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

      3. times-frac 49.3%

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

      4. /-rgt-identity 49.3%

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

   5. Simplified 49.3%

      \[\leadsto c0 \cdot \color{blue}{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{\ell \cdot V} \leq 5 \cdot 10^{+302}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \]

Alternative 11: 78.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{V} \cdot \frac{c0}{\ell}\right)}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \end{array} \]

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 (* l V))))
   (if (<= t_0 2e-318)
     (sqrt (* A (* (/ c0 V) (/ c0 l))))
     (if (<= t_0 5e+302) (* c0 (sqrt t_0)) (* c0 (pow (* V (/ l A)) -0.5))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double tmp;
	if (t_0 <= 2e-318) {
		tmp = sqrt((A * ((c0 / V) * (c0 / l))));
	} else if (t_0 <= 5e+302) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 * pow((V * (l / A)), -0.5);
	}
	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 / (l * v)
    if (t_0 <= 2d-318) then
        tmp = sqrt((a * ((c0 / v) * (c0 / l))))
    else if (t_0 <= 5d+302) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 * ((v * (l / a)) ** (-0.5d0))
    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 / (l * V);
	double tmp;
	if (t_0 <= 2e-318) {
		tmp = Math.sqrt((A * ((c0 / V) * (c0 / l))));
	} else if (t_0 <= 5e+302) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 * Math.pow((V * (l / A)), -0.5);
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (l * V)
	tmp = 0
	if t_0 <= 2e-318:
		tmp = math.sqrt((A * ((c0 / V) * (c0 / l))))
	elif t_0 <= 5e+302:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 * math.pow((V * (l / A)), -0.5)
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(l * V))
	tmp = 0.0
	if (t_0 <= 2e-318)
		tmp = sqrt(Float64(A * Float64(Float64(c0 / V) * Float64(c0 / l))));
	elseif (t_0 <= 5e+302)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 * (Float64(V * Float64(l / A)) ^ -0.5));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (l * V);
	tmp = 0.0;
	if (t_0 <= 2e-318)
		tmp = sqrt((A * ((c0 / V) * (c0 / l))));
	elseif (t_0 <= 5e+302)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 * ((V * (l / A)) ^ -0.5);
	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[(l * V), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-318], N[Sqrt[N[(A * N[(N[(c0 / V), $MachinePrecision] * N[(c0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 5e+302], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Power[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 2.0000024e-318

   1. Initial program 48.2%

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

      1. pow1/2 48.2%

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

      2. clear-num 47.3%

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

      3. inv-pow 47.3%

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

      4. pow-pow 47.3%

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

      5. associate-/l* 56.2%

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

      6. metadata-eval 56.2%

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

   3. Applied egg-rr 56.2%

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

      1. associate-/l* 47.3%

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

      2. *-lft-identity 47.3%

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

      3. times-frac 56.1%

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

      4. /-rgt-identity 56.1%

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

   5. Simplified 56.1%

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

      1. add-sqr-sqrt 52.4%

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

      2. sqrt-unprod 49.0%

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

      3. *-commutative 49.0%

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

      4. *-commutative 49.0%

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

      5. swap-sqr 48.6%

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

      6. pow-prod-up 48.6%

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

      7. metadata-eval 48.6%

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

   7. Applied egg-rr 48.6%

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

      1. unpow-1 48.6%

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

      2. associate-*r/ 46.8%

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

      3. associate-/r/ 47.7%

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

      4. associate-*l/ 47.7%

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

      5. metadata-eval 47.7%

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

      6. distribute-lft-neg-in 47.7%

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

      7. neg-mul-1 47.7%

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

      8. remove-double-neg 47.7%

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

      9. associate-*l/ 54.0%

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

      10. associate-*r/ 54.0%

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

      11. times-frac 59.5%

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

   9. Simplified 59.5%

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

   if 2.0000024e-318 < (/.f64 A (*.f64 V l)) < 5e302

   1. Initial program 98.3%

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

   if 5e302 < (/.f64 A (*.f64 V l))

   1. Initial program 32.1%

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

      1. pow1/2 32.1%

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

      2. clear-num 32.1%

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

      3. inv-pow 32.1%

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

      4. pow-pow 33.3%

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

      5. associate-/l* 49.3%

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

      6. metadata-eval 49.3%

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

   3. Applied egg-rr 49.3%

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

      1. associate-/l* 33.3%

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

      2. *-lft-identity 33.3%

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

      3. times-frac 49.3%

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

      4. /-rgt-identity 49.3%

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

   5. Simplified 49.3%

      \[\leadsto c0 \cdot \color{blue}{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 2 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{V} \cdot \frac{c0}{\ell}\right)}\\ \mathbf{elif}\;\frac{A}{\ell \cdot V} \leq 5 \cdot 10^{+302}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \]

Alternative 12: 79.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{\frac{c0}{V} \cdot \left(A \cdot \frac{c0}{\ell}\right)}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \end{array} \]

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 (* l V))))
   (if (<= t_0 2e-318)
     (sqrt (* (/ c0 V) (* A (/ c0 l))))
     (if (<= t_0 5e+302) (* c0 (sqrt t_0)) (* c0 (pow (* V (/ l A)) -0.5))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double tmp;
	if (t_0 <= 2e-318) {
		tmp = sqrt(((c0 / V) * (A * (c0 / l))));
	} else if (t_0 <= 5e+302) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 * pow((V * (l / A)), -0.5);
	}
	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 / (l * v)
    if (t_0 <= 2d-318) then
        tmp = sqrt(((c0 / v) * (a * (c0 / l))))
    else if (t_0 <= 5d+302) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 * ((v * (l / a)) ** (-0.5d0))
    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 / (l * V);
	double tmp;
	if (t_0 <= 2e-318) {
		tmp = Math.sqrt(((c0 / V) * (A * (c0 / l))));
	} else if (t_0 <= 5e+302) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 * Math.pow((V * (l / A)), -0.5);
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (l * V)
	tmp = 0
	if t_0 <= 2e-318:
		tmp = math.sqrt(((c0 / V) * (A * (c0 / l))))
	elif t_0 <= 5e+302:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 * math.pow((V * (l / A)), -0.5)
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(l * V))
	tmp = 0.0
	if (t_0 <= 2e-318)
		tmp = sqrt(Float64(Float64(c0 / V) * Float64(A * Float64(c0 / l))));
	elseif (t_0 <= 5e+302)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 * (Float64(V * Float64(l / A)) ^ -0.5));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (l * V);
	tmp = 0.0;
	if (t_0 <= 2e-318)
		tmp = sqrt(((c0 / V) * (A * (c0 / l))));
	elseif (t_0 <= 5e+302)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 * ((V * (l / A)) ^ -0.5);
	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[(l * V), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-318], N[Sqrt[N[(N[(c0 / V), $MachinePrecision] * N[(A * N[(c0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 5e+302], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Power[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 2.0000024e-318

   1. Initial program 48.2%

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

      1. sqrt-div 37.3%

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

      2. associate-*r/ 37.2%

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

   3. Applied egg-rr 37.2%

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

      1. associate-*l/ 37.4%

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

   5. Simplified 37.4%

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

      1. expm1-log1p-u 37.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \ell}} \cdot \sqrt{A}\right)\right)} \]

      2. expm1-udef 25.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \ell}} \cdot \sqrt{A}\right)} - 1} \]

      3. associate-*l/ 25.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}}\right)} - 1 \]

      4. associate-/l* 25.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}}\right)} - 1 \]

      5. sqrt-div 47.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\frac{V \cdot \ell}{A}}}}\right)} - 1 \]

      6. associate-*r/ 48.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{V \cdot \frac{\ell}{A}}}}\right)} - 1 \]

   7. Applied egg-rr 48.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 54.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)\right)} \]

      2. expm1-log1p 56.1%

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

   9. Simplified 56.1%

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

       1. add-sqr-sqrt 52.4%

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

       2. sqrt-unprod 49.0%

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

       3. frac-times 48.6%

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

       4. add-sqr-sqrt 48.6%

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

   11. Applied egg-rr 48.6%

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

       1. times-frac 63.1%

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

       2. associate-/r/ 63.0%

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

   13. Simplified 63.0%

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

   if 2.0000024e-318 < (/.f64 A (*.f64 V l)) < 5e302

   1. Initial program 98.3%

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

   if 5e302 < (/.f64 A (*.f64 V l))

   1. Initial program 32.1%

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

      1. pow1/2 32.1%

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

      2. clear-num 32.1%

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

      3. inv-pow 32.1%

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

      4. pow-pow 33.3%

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

      5. associate-/l* 49.3%

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

      6. metadata-eval 49.3%

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

   3. Applied egg-rr 49.3%

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

      1. associate-/l* 33.3%

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

      2. *-lft-identity 33.3%

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

      3. times-frac 49.3%

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

      4. /-rgt-identity 49.3%

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

   5. Simplified 49.3%

      \[\leadsto c0 \cdot \color{blue}{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 2 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{\frac{c0}{V} \cdot \left(A \cdot \frac{c0}{\ell}\right)}\\ \mathbf{elif}\;\frac{A}{\ell \cdot V} \leq 5 \cdot 10^{+302}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \]

Alternative 13: 78.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{\frac{c0}{V} \cdot \frac{A \cdot c0}{\ell}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \end{array} \]

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 (* l V))))
   (if (<= t_0 2e-318)
     (sqrt (* (/ c0 V) (/ (* A c0) l)))
     (if (<= t_0 5e+302) (* c0 (sqrt t_0)) (* c0 (pow (* V (/ l A)) -0.5))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double tmp;
	if (t_0 <= 2e-318) {
		tmp = sqrt(((c0 / V) * ((A * c0) / l)));
	} else if (t_0 <= 5e+302) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 * pow((V * (l / A)), -0.5);
	}
	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 / (l * v)
    if (t_0 <= 2d-318) then
        tmp = sqrt(((c0 / v) * ((a * c0) / l)))
    else if (t_0 <= 5d+302) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 * ((v * (l / a)) ** (-0.5d0))
    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 / (l * V);
	double tmp;
	if (t_0 <= 2e-318) {
		tmp = Math.sqrt(((c0 / V) * ((A * c0) / l)));
	} else if (t_0 <= 5e+302) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 * Math.pow((V * (l / A)), -0.5);
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (l * V)
	tmp = 0
	if t_0 <= 2e-318:
		tmp = math.sqrt(((c0 / V) * ((A * c0) / l)))
	elif t_0 <= 5e+302:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 * math.pow((V * (l / A)), -0.5)
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(l * V))
	tmp = 0.0
	if (t_0 <= 2e-318)
		tmp = sqrt(Float64(Float64(c0 / V) * Float64(Float64(A * c0) / l)));
	elseif (t_0 <= 5e+302)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 * (Float64(V * Float64(l / A)) ^ -0.5));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (l * V);
	tmp = 0.0;
	if (t_0 <= 2e-318)
		tmp = sqrt(((c0 / V) * ((A * c0) / l)));
	elseif (t_0 <= 5e+302)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 * ((V * (l / A)) ^ -0.5);
	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[(l * V), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-318], N[Sqrt[N[(N[(c0 / V), $MachinePrecision] * N[(N[(A * c0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 5e+302], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Power[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 2.0000024e-318

   1. Initial program 48.2%

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

      1. sqrt-div 37.3%

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

      2. associate-*r/ 37.2%

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

   3. Applied egg-rr 37.2%

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

      1. associate-*l/ 37.4%

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

   5. Simplified 37.4%

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

      1. expm1-log1p-u 37.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \ell}} \cdot \sqrt{A}\right)\right)} \]

      2. expm1-udef 25.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \ell}} \cdot \sqrt{A}\right)} - 1} \]

      3. associate-*l/ 25.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}}\right)} - 1 \]

      4. associate-/l* 25.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}}\right)} - 1 \]

      5. sqrt-div 47.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\frac{V \cdot \ell}{A}}}}\right)} - 1 \]

      6. associate-*r/ 48.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{V \cdot \frac{\ell}{A}}}}\right)} - 1 \]

   7. Applied egg-rr 48.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 54.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)\right)} \]

      2. expm1-log1p 56.1%

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

   9. Simplified 56.1%

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

       1. add-sqr-sqrt 52.4%

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

       2. sqrt-unprod 49.0%

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

       3. frac-times 48.6%

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

       4. add-sqr-sqrt 48.6%

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

   11. Applied egg-rr 48.6%

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

       1. times-frac 63.1%

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

   13. Simplified 63.1%

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

   14. Taylor expanded in c0 around 0 63.1%

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

   if 2.0000024e-318 < (/.f64 A (*.f64 V l)) < 5e302

   1. Initial program 98.3%

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

   if 5e302 < (/.f64 A (*.f64 V l))

   1. Initial program 32.1%

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

      1. pow1/2 32.1%

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

      2. clear-num 32.1%

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

      3. inv-pow 32.1%

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

      4. pow-pow 33.3%

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

      5. associate-/l* 49.3%

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

      6. metadata-eval 49.3%

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

   3. Applied egg-rr 49.3%

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

      1. associate-/l* 33.3%

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

      2. *-lft-identity 33.3%

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

      3. times-frac 49.3%

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

      4. /-rgt-identity 49.3%

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

   5. Simplified 49.3%

      \[\leadsto c0 \cdot \color{blue}{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 2 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{\frac{c0}{V} \cdot \frac{A \cdot c0}{\ell}}\\ \mathbf{elif}\;\frac{A}{\ell \cdot V} \leq 5 \cdot 10^{+302}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \]

Alternative 14: 78.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 5 \cdot 10^{+272}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* l V))))
   (if (or (<= t_0 0.0) (not (<= t_0 5e+272)))
     (* c0 (sqrt (/ (/ A V) l)))
     (* c0 (sqrt t_0)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+272)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = c0 * sqrt(t_0);
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (l * v)
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 5d+272))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = c0 * sqrt(t_0)
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+272)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (l * V)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 5e+272):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(l * V))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 5e+272))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(c0 * sqrt(t_0));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (l * V);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 5e+272)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = c0 * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 5e+272]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 4.99999999999999973e272 < (/.f64 A (*.f64 V l))

   1. Initial program 40.0%

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

      1. associate-/r* 52.3%

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

      2. div-inv 52.3%

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

   3. Applied egg-rr 52.3%

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

      1. un-div-inv 52.3%

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

   5. Applied egg-rr 52.3%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.99999999999999973e272

   1. Initial program 98.0%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 0 \lor \neg \left(\frac{A}{\ell \cdot V} \leq 5 \cdot 10^{+272}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \end{array} \]

Alternative 15: 79.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

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 (* l V))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 5e+302) (* 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 / (l * V);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 5e+302) {
		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 / (l * v)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 5d+302) 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 / (l * V);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 5e+302) {
		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 / (l * V)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 5e+302:
		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(l * V))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 5e+302)
		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 / (l * V);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 5e+302)
		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[(l * V), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+302], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 48.1%

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

      1. associate-/r* 57.1%

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

      2. div-inv 57.1%

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

   3. Applied egg-rr 57.1%

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

      1. un-div-inv 57.1%

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

   5. Applied egg-rr 57.1%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 5e302

   1. Initial program 98.0%

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

   if 5e302 < (/.f64 A (*.f64 V l))

   1. Initial program 32.1%

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

      1. sqrt-div 42.3%

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

      2. associate-*r/ 42.2%

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

   3. Applied egg-rr 42.2%

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

      1. associate-*l/ 42.4%

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

   5. Simplified 42.4%

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

      1. expm1-log1p-u 25.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \ell}} \cdot \sqrt{A}\right)\right)} \]

      2. expm1-udef 18.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \ell}} \cdot \sqrt{A}\right)} - 1} \]

      3. associate-*l/ 18.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}}\right)} - 1 \]

      4. associate-/l* 18.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}}\right)} - 1 \]

      5. sqrt-div 17.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\frac{V \cdot \ell}{A}}}}\right)} - 1 \]

      6. associate-*r/ 22.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{V \cdot \frac{\ell}{A}}}}\right)} - 1 \]

   7. Applied egg-rr 22.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 26.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)\right)} \]

      2. expm1-log1p 49.3%

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

   9. Simplified 49.3%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{\ell \cdot V} \leq 5 \cdot 10^{+302}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]

Alternative 16: 72.7% accurate, 1.0× speedup

Math

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

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 (* l V)))))

C

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

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 / (l * v)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 71.7%

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

2. Final simplification 71.7%

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

Reproduce

herbie shell --seed 2023242 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))