Henrywood and Agarwal, Equation (3)

Percentage Accurate: 74.0% → 91.3%

Time: 8.9s

Alternatives: 10

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: 74.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{-201}:\\ \;\;\;\;c0 \cdot \frac{\frac{\sqrt{-A}}{\sqrt{-V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+301}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e-201)
   (* c0 (/ (/ (sqrt (- A)) (sqrt (- V))) (sqrt l)))
   (if (<= (* V l) 0.0)
     (* c0 (pow (/ V (/ A l)) -0.5))
     (if (<= (* V l) 4e+301)
       (* c0 (/ (sqrt A) (sqrt (* V l))))
       (/ c0 (sqrt (/ l (/ A V))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e-201) {
		tmp = c0 * ((sqrt(-A) / sqrt(-V)) / sqrt(l));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * pow((V / (A / l)), -0.5);
	} else if ((V * l) <= 4e+301) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 / sqrt((l / (A / 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 ((v * l) <= (-1d-201)) then
        tmp = c0 * ((sqrt(-a) / sqrt(-v)) / sqrt(l))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 * ((v / (a / l)) ** (-0.5d0))
    else if ((v * l) <= 4d+301) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 / sqrt((l / (a / 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 ((V * l) <= -1e-201) {
		tmp = c0 * ((Math.sqrt(-A) / Math.sqrt(-V)) / Math.sqrt(l));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * Math.pow((V / (A / l)), -0.5);
	} else if ((V * l) <= 4e+301) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 / Math.sqrt((l / (A / V)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e-201:
		tmp = c0 * ((math.sqrt(-A) / math.sqrt(-V)) / math.sqrt(l))
	elif (V * l) <= 0.0:
		tmp = c0 * math.pow((V / (A / l)), -0.5)
	elif (V * l) <= 4e+301:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 / math.sqrt((l / (A / V)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e-201)
		tmp = Float64(c0 * Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(-V))) / sqrt(l)));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * (Float64(V / Float64(A / l)) ^ -0.5));
	elseif (Float64(V * l) <= 4e+301)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 / sqrt(Float64(l / Float64(A / V))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e-201)
		tmp = c0 * ((sqrt(-A) / sqrt(-V)) / sqrt(l));
	elseif ((V * l) <= 0.0)
		tmp = c0 * ((V / (A / l)) ^ -0.5);
	elseif ((V * l) <= 4e+301)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 / sqrt((l / (A / 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[N[(V * l), $MachinePrecision], -1e-201], N[(c0 * N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[Power[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e+301], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -9.99999999999999946e-202

   1. Initial program 77.9%

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

      1. associate-/r* 73.9%

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

      2. sqrt-div 41.8%

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

   3. Applied egg-rr 41.8%

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

      1. frac-2neg 41.8%

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

      2. sqrt-div 51.2%

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

   5. Applied egg-rr 51.2%

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

   if -9.99999999999999946e-202 < (*.f64 V l) < 0.0

   1. Initial program 72.6%

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

      1. pow1/2 72.6%

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

      2. clear-num 72.6%

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

      3. inv-pow 72.6%

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

      4. pow-pow 72.6%

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

      5. associate-/l* 75.8%

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

      6. metadata-eval 75.8%

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

   3. Applied egg-rr 75.8%

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

   if 0.0 < (*.f64 V l) < 4.00000000000000021e301

   1. Initial program 90.5%

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

      1. sqrt-div 99.5%

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

      2. associate-*r/ 97.8%

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

   3. Applied egg-rr 97.8%

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

      1. *-commutative 97.8%

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

      2. associate-*l/ 99.5%

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

   5. Simplified 99.5%

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

   if 4.00000000000000021e301 < (*.f64 V l)

   1. Initial program 16.9%

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

      1. *-un-lft-identity 16.9%

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

      2. times-frac 78.5%

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

   3. Applied egg-rr 78.5%

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

      1. frac-times 16.9%

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

      2. *-un-lft-identity 16.9%

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

      3. associate-/r* 78.7%

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

      4. sqrt-undiv 39.0%

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

      5. associate-*r/ 39.1%

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

      6. associate-/l* 39.0%

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

      7. sqrt-undiv 78.7%

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

   5. Applied egg-rr 78.7%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{-201}:\\ \;\;\;\;c0 \cdot \frac{\frac{\sqrt{-A}}{\sqrt{-V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+301}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]

Alternative 2: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \left({\left(\frac{V}{A}\right)}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-282}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-261}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+301}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (* c0 (* (pow (/ V A) -0.5) (pow l -0.5)))
   (if (<= (* V l) -5e-282)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 5e-261)
       (* c0 (/ 1.0 (sqrt (/ V (/ A l)))))
       (if (<= (* V l) 4e+301)
         (* c0 (/ (sqrt A) (sqrt (* V l))))
         (/ c0 (sqrt (/ l (/ A V)))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 * (pow((V / A), -0.5) * pow(l, -0.5));
	} else if ((V * l) <= -5e-282) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 5e-261) {
		tmp = c0 * (1.0 / sqrt((V / (A / l))));
	} else if ((V * l) <= 4e+301) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 / sqrt((l / (A / V)));
	}
	return tmp;
}

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 * (Math.pow((V / A), -0.5) * Math.pow(l, -0.5));
	} else if ((V * l) <= -5e-282) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 5e-261) {
		tmp = c0 * (1.0 / Math.sqrt((V / (A / l))));
	} else if ((V * l) <= 4e+301) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 / Math.sqrt((l / (A / V)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 * (math.pow((V / A), -0.5) * math.pow(l, -0.5))
	elif (V * l) <= -5e-282:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 5e-261:
		tmp = c0 * (1.0 / math.sqrt((V / (A / l))))
	elif (V * l) <= 4e+301:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 / math.sqrt((l / (A / V)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 * Float64((Float64(V / A) ^ -0.5) * (l ^ -0.5)));
	elseif (Float64(V * l) <= -5e-282)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 5e-261)
		tmp = Float64(c0 * Float64(1.0 / sqrt(Float64(V / Float64(A / l)))));
	elseif (Float64(V * l) <= 4e+301)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 / sqrt(Float64(l / Float64(A / V))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 * (((V / A) ^ -0.5) * (l ^ -0.5));
	elseif ((V * l) <= -5e-282)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 5e-261)
		tmp = c0 * (1.0 / sqrt((V / (A / l))));
	elseif ((V * l) <= 4e+301)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 / sqrt((l / (A / 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[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 * N[(N[Power[N[(V / A), $MachinePrecision], -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-282], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-261], N[(c0 * N[(1.0 / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e+301], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 28.6%

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

      1. *-un-lft-identity 28.6%

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

      2. times-frac 66.1%

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

   3. Applied egg-rr 66.1%

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

      1. frac-times 28.6%

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

      2. *-un-lft-identity 28.6%

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

      3. associate-/l/ 66.2%

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

      4. clear-num 66.1%

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

      5. unpow-1 66.1%

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

      6. sqrt-pow1 66.2%

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

      7. associate-/r/ 66.2%

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

      8. metadata-eval 66.2%

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

      9. unpow-prod-down 33.9%

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

   5. Applied egg-rr 33.9%

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

   if -inf.0 < (*.f64 V l) < -5.0000000000000001e-282

   1. Initial program 87.5%

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

      1. frac-2neg 87.5%

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

      2. sqrt-div 99.5%

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

      3. *-commutative 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

   3. Applied egg-rr 99.5%

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

   if -5.0000000000000001e-282 < (*.f64 V l) < 4.99999999999999981e-261

   1. Initial program 71.1%

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

      1. clear-num 71.1%

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

      2. sqrt-div 71.1%

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

      3. metadata-eval 71.1%

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

      4. associate-/l* 77.1%

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

   3. Applied egg-rr 77.1%

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

   if 4.99999999999999981e-261 < (*.f64 V l) < 4.00000000000000021e301

   1. Initial program 90.0%

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

      1. sqrt-div 99.5%

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

      2. associate-*r/ 97.7%

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

   3. Applied egg-rr 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-*l/ 99.5%

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

   5. Simplified 99.5%

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

   if 4.00000000000000021e301 < (*.f64 V l)

   1. Initial program 16.9%

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

      1. *-un-lft-identity 16.9%

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

      2. times-frac 78.5%

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

   3. Applied egg-rr 78.5%

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

      1. frac-times 16.9%

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

      2. *-un-lft-identity 16.9%

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

      3. associate-/r* 78.7%

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

      4. sqrt-undiv 39.0%

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

      5. associate-*r/ 39.1%

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

      6. associate-/l* 39.0%

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

      7. sqrt-undiv 78.7%

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

   5. Applied egg-rr 78.7%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \left({\left(\frac{V}{A}\right)}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-282}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-261}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+301}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]

Alternative 3: 91.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-\frac{A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-282}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-261}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+301}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (* c0 (/ (sqrt (- (/ A l))) (sqrt (- V))))
   (if (<= (* V l) -5e-282)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 5e-261)
       (* c0 (/ 1.0 (sqrt (/ V (/ A l)))))
       (if (<= (* V l) 4e+301)
         (* c0 (/ (sqrt A) (sqrt (* V l))))
         (/ c0 (sqrt (/ l (/ A V)))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 * (sqrt(-(A / l)) / sqrt(-V));
	} else if ((V * l) <= -5e-282) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 5e-261) {
		tmp = c0 * (1.0 / sqrt((V / (A / l))));
	} else if ((V * l) <= 4e+301) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 / sqrt((l / (A / V)));
	}
	return tmp;
}

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 * (Math.sqrt(-(A / l)) / Math.sqrt(-V));
	} else if ((V * l) <= -5e-282) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 5e-261) {
		tmp = c0 * (1.0 / Math.sqrt((V / (A / l))));
	} else if ((V * l) <= 4e+301) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 / Math.sqrt((l / (A / V)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 * (math.sqrt(-(A / l)) / math.sqrt(-V))
	elif (V * l) <= -5e-282:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 5e-261:
		tmp = c0 * (1.0 / math.sqrt((V / (A / l))))
	elif (V * l) <= 4e+301:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 / math.sqrt((l / (A / V)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 * Float64(sqrt(Float64(-Float64(A / l))) / sqrt(Float64(-V))));
	elseif (Float64(V * l) <= -5e-282)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 5e-261)
		tmp = Float64(c0 * Float64(1.0 / sqrt(Float64(V / Float64(A / l)))));
	elseif (Float64(V * l) <= 4e+301)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 / sqrt(Float64(l / Float64(A / V))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 * (sqrt(-(A / l)) / sqrt(-V));
	elseif ((V * l) <= -5e-282)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 5e-261)
		tmp = c0 * (1.0 / sqrt((V / (A / l))));
	elseif ((V * l) <= 4e+301)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 / sqrt((l / (A / 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[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 * N[(N[Sqrt[(-N[(A / l), $MachinePrecision])], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-282], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-261], N[(c0 * N[(1.0 / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e+301], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 28.6%

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

      1. *-un-lft-identity 28.6%

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

      2. times-frac 66.1%

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

   3. Applied egg-rr 66.1%

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

      1. associate-*l/ 66.2%

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

      2. *-un-lft-identity 66.2%

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

   5. Applied egg-rr 66.2%

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

      1. frac-2neg 66.2%

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

      2. sqrt-div 39.7%

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

   7. Applied egg-rr 39.7%

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

      1. distribute-neg-frac 39.7%

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

   9. Simplified 39.7%

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

   if -inf.0 < (*.f64 V l) < -5.0000000000000001e-282

   1. Initial program 87.5%

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

      1. frac-2neg 87.5%

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

      2. sqrt-div 99.5%

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

      3. *-commutative 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

   3. Applied egg-rr 99.5%

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

   if -5.0000000000000001e-282 < (*.f64 V l) < 4.99999999999999981e-261

   1. Initial program 71.1%

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

      1. clear-num 71.1%

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

      2. sqrt-div 71.1%

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

      3. metadata-eval 71.1%

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

      4. associate-/l* 77.1%

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

   3. Applied egg-rr 77.1%

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

   if 4.99999999999999981e-261 < (*.f64 V l) < 4.00000000000000021e301

   1. Initial program 90.0%

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

      1. sqrt-div 99.5%

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

      2. associate-*r/ 97.7%

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

   3. Applied egg-rr 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-*l/ 99.5%

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

   5. Simplified 99.5%

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

   if 4.00000000000000021e301 < (*.f64 V l)

   1. Initial program 16.9%

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

      1. *-un-lft-identity 16.9%

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

      2. times-frac 78.5%

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

   3. Applied egg-rr 78.5%

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

      1. frac-times 16.9%

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

      2. *-un-lft-identity 16.9%

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

      3. associate-/r* 78.7%

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

      4. sqrt-undiv 39.0%

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

      5. associate-*r/ 39.1%

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

      6. associate-/l* 39.0%

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

      7. sqrt-undiv 78.7%

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

   5. Applied egg-rr 78.7%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-\frac{A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-282}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-261}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+301}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]

Alternative 4: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\frac{-\ell}{A}}}{\sqrt{\frac{-1}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-282}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-261}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+301}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (/ c0 (/ (sqrt (/ (- l) A)) (sqrt (/ -1.0 V))))
   (if (<= (* V l) -5e-282)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 5e-261)
       (* c0 (/ 1.0 (sqrt (/ V (/ A l)))))
       (if (<= (* V l) 4e+301)
         (* c0 (/ (sqrt A) (sqrt (* V l))))
         (/ c0 (sqrt (/ l (/ A V)))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 / (sqrt((-l / A)) / sqrt((-1.0 / V)));
	} else if ((V * l) <= -5e-282) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 5e-261) {
		tmp = c0 * (1.0 / sqrt((V / (A / l))));
	} else if ((V * l) <= 4e+301) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 / sqrt((l / (A / V)));
	}
	return tmp;
}

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 / (Math.sqrt((-l / A)) / Math.sqrt((-1.0 / V)));
	} else if ((V * l) <= -5e-282) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 5e-261) {
		tmp = c0 * (1.0 / Math.sqrt((V / (A / l))));
	} else if ((V * l) <= 4e+301) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 / Math.sqrt((l / (A / V)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 / (math.sqrt((-l / A)) / math.sqrt((-1.0 / V)))
	elif (V * l) <= -5e-282:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 5e-261:
		tmp = c0 * (1.0 / math.sqrt((V / (A / l))))
	elif (V * l) <= 4e+301:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 / math.sqrt((l / (A / V)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 / Float64(sqrt(Float64(Float64(-l) / A)) / sqrt(Float64(-1.0 / V))));
	elseif (Float64(V * l) <= -5e-282)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 5e-261)
		tmp = Float64(c0 * Float64(1.0 / sqrt(Float64(V / Float64(A / l)))));
	elseif (Float64(V * l) <= 4e+301)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 / sqrt(Float64(l / Float64(A / V))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 / (sqrt((-l / A)) / sqrt((-1.0 / V)));
	elseif ((V * l) <= -5e-282)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 5e-261)
		tmp = c0 * (1.0 / sqrt((V / (A / l))));
	elseif ((V * l) <= 4e+301)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 / sqrt((l / (A / 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[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 / N[(N[Sqrt[N[((-l) / A), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(-1.0 / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-282], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-261], N[(c0 * N[(1.0 / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e+301], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 28.6%

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

      1. *-un-lft-identity 28.6%

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

      2. times-frac 66.1%

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

   3. Applied egg-rr 66.1%

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

      1. add-sqr-sqrt 31.9%

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

      2. sqrt-prod 40.5%

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

      3. sqrt-prod 28.2%

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

      4. *-commutative 28.2%

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

      5. frac-times 28.2%

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

      6. *-un-lft-identity 28.2%

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

      7. expm1-log1p-u 28.2%

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

      8. expm1-udef 28.2%

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

   5. Applied egg-rr 28.4%

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

      1. expm1-def 59.5%

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

      2. expm1-log1p 66.1%

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

      3. associate-/r/ 66.1%

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

      4. *-commutative 66.1%

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

   7. Simplified 66.1%

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

      1. pow1 66.1%

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

      2. metadata-eval 66.1%

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

      3. pow-flip 66.1%

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

      4. inv-pow 66.1%

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

      5. times-frac 66.1%

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

      6. *-un-lft-identity 66.1%

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

      7. associate-/l/ 66.1%

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

      8. frac-2neg 66.1%

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

      9. sqrt-div 39.7%

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

   9. Applied egg-rr 39.7%

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

       1. distribute-neg-frac 39.7%

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

       2. distribute-neg-frac 39.7%

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

       3. metadata-eval 39.7%

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

   11. Simplified 39.7%

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

   if -inf.0 < (*.f64 V l) < -5.0000000000000001e-282

   1. Initial program 87.5%

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

      1. frac-2neg 87.5%

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

      2. sqrt-div 99.5%

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

      3. *-commutative 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

   3. Applied egg-rr 99.5%

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

   if -5.0000000000000001e-282 < (*.f64 V l) < 4.99999999999999981e-261

   1. Initial program 71.1%

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

      1. clear-num 71.1%

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

      2. sqrt-div 71.1%

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

      3. metadata-eval 71.1%

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

      4. associate-/l* 77.1%

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

   3. Applied egg-rr 77.1%

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

   if 4.99999999999999981e-261 < (*.f64 V l) < 4.00000000000000021e301

   1. Initial program 90.0%

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

      1. sqrt-div 99.5%

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

      2. associate-*r/ 97.7%

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

   3. Applied egg-rr 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-*l/ 99.5%

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

   5. Simplified 99.5%

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

   if 4.00000000000000021e301 < (*.f64 V l)

   1. Initial program 16.9%

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

      1. *-un-lft-identity 16.9%

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

      2. times-frac 78.5%

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

   3. Applied egg-rr 78.5%

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

      1. frac-times 16.9%

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

      2. *-un-lft-identity 16.9%

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

      3. associate-/r* 78.7%

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

      4. sqrt-undiv 39.0%

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

      5. associate-*r/ 39.1%

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

      6. associate-/l* 39.0%

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

      7. sqrt-undiv 78.7%

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

   5. Applied egg-rr 78.7%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\frac{-\ell}{A}}}{\sqrt{\frac{-1}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-282}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-261}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+301}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]

Alternative 5: 83.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+263}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-261}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+301}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -2e+263)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* V l) 5e-261)
     (/ c0 (sqrt (/ (* V l) A)))
     (if (<= (* V l) 4e+301)
       (* c0 (/ (sqrt A) (sqrt (* V l))))
       (/ c0 (sqrt (/ l (/ A V))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+263) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= 5e-261) {
		tmp = c0 / sqrt(((V * l) / A));
	} else if ((V * l) <= 4e+301) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 / sqrt((l / (A / 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 ((v * l) <= (-2d+263)) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= 5d-261) then
        tmp = c0 / sqrt(((v * l) / a))
    else if ((v * l) <= 4d+301) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 / sqrt((l / (a / 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 ((V * l) <= -2e+263) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= 5e-261) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else if ((V * l) <= 4e+301) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 / Math.sqrt((l / (A / V)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -2e+263:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= 5e-261:
		tmp = c0 / math.sqrt(((V * l) / A))
	elif (V * l) <= 4e+301:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 / math.sqrt((l / (A / V)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -2e+263)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= 5e-261)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	elseif (Float64(V * l) <= 4e+301)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 / sqrt(Float64(l / Float64(A / V))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -2e+263)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= 5e-261)
		tmp = c0 / sqrt(((V * l) / A));
	elseif ((V * l) <= 4e+301)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 / sqrt((l / (A / 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[N[(V * l), $MachinePrecision], -2e+263], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-261], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e+301], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -2.00000000000000003e263

   1. Initial program 38.5%

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

      1. associate-/r* 68.2%

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

      2. sqrt-div 37.2%

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

   3. Applied egg-rr 37.2%

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

   if -2.00000000000000003e263 < (*.f64 V l) < 4.99999999999999981e-261

   1. Initial program 83.0%

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

      1. *-un-lft-identity 83.0%

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

      2. times-frac 74.9%

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

   3. Applied egg-rr 74.9%

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

      1. add-sqr-sqrt 34.3%

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

      2. sqrt-prod 29.5%

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

      3. sqrt-prod 27.8%

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

      4. *-commutative 27.8%

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

      5. frac-times 30.7%

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

      6. *-un-lft-identity 30.7%

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

      7. expm1-log1p-u 30.4%

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

      8. expm1-udef 23.8%

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

   5. Applied egg-rr 30.3%

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

      1. expm1-def 47.9%

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

      2. expm1-log1p 77.0%

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

      3. associate-/r/ 74.6%

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

      4. *-commutative 74.6%

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

   7. Simplified 74.6%

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

   8. Taylor expanded in V around 0 83.1%

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

   if 4.99999999999999981e-261 < (*.f64 V l) < 4.00000000000000021e301

   1. Initial program 90.0%

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

      1. sqrt-div 99.5%

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

      2. associate-*r/ 97.7%

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

   3. Applied egg-rr 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-*l/ 99.5%

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

   5. Simplified 99.5%

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

   if 4.00000000000000021e301 < (*.f64 V l)

   1. Initial program 16.9%

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

      1. *-un-lft-identity 16.9%

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

      2. times-frac 78.5%

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

   3. Applied egg-rr 78.5%

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

      1. frac-times 16.9%

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

      2. *-un-lft-identity 16.9%

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

      3. associate-/r* 78.7%

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

      4. sqrt-undiv 39.0%

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

      5. associate-*r/ 39.1%

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

      6. associate-/l* 39.0%

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

      7. sqrt-undiv 78.7%

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

   5. Applied egg-rr 78.7%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+263}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-261}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+301}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]

Alternative 6: 83.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+307}:\\ \;\;\;\;c0 \cdot \left({\left(\frac{V}{A}\right)}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-261}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+301}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -5e+307)
   (* c0 (* (pow (/ V A) -0.5) (pow l -0.5)))
   (if (<= (* V l) 5e-261)
     (/ c0 (sqrt (/ (* V l) A)))
     (if (<= (* V l) 4e+301)
       (* c0 (/ (sqrt A) (sqrt (* V l))))
       (/ c0 (sqrt (/ l (/ A V))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+307) {
		tmp = c0 * (pow((V / A), -0.5) * pow(l, -0.5));
	} else if ((V * l) <= 5e-261) {
		tmp = c0 / sqrt(((V * l) / A));
	} else if ((V * l) <= 4e+301) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 / sqrt((l / (A / 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 ((v * l) <= (-5d+307)) then
        tmp = c0 * (((v / a) ** (-0.5d0)) * (l ** (-0.5d0)))
    else if ((v * l) <= 5d-261) then
        tmp = c0 / sqrt(((v * l) / a))
    else if ((v * l) <= 4d+301) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 / sqrt((l / (a / 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 ((V * l) <= -5e+307) {
		tmp = c0 * (Math.pow((V / A), -0.5) * Math.pow(l, -0.5));
	} else if ((V * l) <= 5e-261) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else if ((V * l) <= 4e+301) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 / Math.sqrt((l / (A / V)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -5e+307:
		tmp = c0 * (math.pow((V / A), -0.5) * math.pow(l, -0.5))
	elif (V * l) <= 5e-261:
		tmp = c0 / math.sqrt(((V * l) / A))
	elif (V * l) <= 4e+301:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 / math.sqrt((l / (A / V)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -5e+307)
		tmp = Float64(c0 * Float64((Float64(V / A) ^ -0.5) * (l ^ -0.5)));
	elseif (Float64(V * l) <= 5e-261)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	elseif (Float64(V * l) <= 4e+301)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 / sqrt(Float64(l / Float64(A / V))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -5e+307)
		tmp = c0 * (((V / A) ^ -0.5) * (l ^ -0.5));
	elseif ((V * l) <= 5e-261)
		tmp = c0 / sqrt(((V * l) / A));
	elseif ((V * l) <= 4e+301)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 / sqrt((l / (A / 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[N[(V * l), $MachinePrecision], -5e+307], N[(c0 * N[(N[Power[N[(V / A), $MachinePrecision], -0.5], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-261], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e+301], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -5e307

   1. Initial program 27.0%

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

      1. *-un-lft-identity 27.0%

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

      2. times-frac 62.2%

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

   3. Applied egg-rr 62.2%

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

      1. frac-times 27.0%

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

      2. *-un-lft-identity 27.0%

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

      3. associate-/l/ 62.3%

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

      4. clear-num 62.2%

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

      5. unpow-1 62.2%

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

      6. sqrt-pow1 62.3%

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

      7. associate-/r/ 62.3%

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

      8. metadata-eval 62.3%

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

      9. unpow-prod-down 31.8%

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

   5. Applied egg-rr 31.8%

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

   if -5e307 < (*.f64 V l) < 4.99999999999999981e-261

   1. Initial program 83.4%

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

      1. *-un-lft-identity 83.4%

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

      2. times-frac 75.5%

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

   3. Applied egg-rr 75.5%

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

      1. add-sqr-sqrt 35.1%

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

      2. sqrt-prod 30.4%

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

      3. sqrt-prod 28.1%

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

      4. *-commutative 28.1%

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

      5. frac-times 30.9%

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

      6. *-un-lft-identity 30.9%

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

      7. expm1-log1p-u 30.6%

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

      8. expm1-udef 24.1%

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

   5. Applied egg-rr 31.1%

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

      1. expm1-def 49.0%

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

      2. expm1-log1p 77.6%

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

      3. associate-/r/ 75.2%

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

      4. *-commutative 75.2%

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

   7. Simplified 75.2%

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

   8. Taylor expanded in V around 0 83.4%

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

   if 4.99999999999999981e-261 < (*.f64 V l) < 4.00000000000000021e301

   1. Initial program 90.0%

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

      1. sqrt-div 99.5%

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

      2. associate-*r/ 97.7%

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

   3. Applied egg-rr 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-*l/ 99.5%

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

   5. Simplified 99.5%

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

   if 4.00000000000000021e301 < (*.f64 V l)

   1. Initial program 16.9%

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

      1. *-un-lft-identity 16.9%

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

      2. times-frac 78.5%

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

   3. Applied egg-rr 78.5%

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

      1. frac-times 16.9%

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

      2. *-un-lft-identity 16.9%

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

      3. associate-/r* 78.7%

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

      4. sqrt-undiv 39.0%

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

      5. associate-*r/ 39.1%

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

      6. associate-/l* 39.0%

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

      7. sqrt-undiv 78.7%

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

   5. Applied egg-rr 78.7%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+307}:\\ \;\;\;\;c0 \cdot \left({\left(\frac{V}{A}\right)}^{-0.5} \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-261}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+301}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]

Alternative 7: 78.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+263}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+55}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -2e+263)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* V l) 5e+55)
     (/ c0 (sqrt (/ (* V l) A)))
     (* c0 (sqrt (/ (/ A l) V))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+263) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= 5e+55) {
		tmp = c0 / sqrt(((V * l) / A));
	} else {
		tmp = c0 * sqrt(((A / 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 ((v * l) <= (-2d+263)) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= 5d+55) then
        tmp = c0 / sqrt(((v * l) / a))
    else
        tmp = c0 * sqrt(((a / 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 ((V * l) <= -2e+263) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= 5e+55) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -2e+263:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= 5e+55:
		tmp = c0 / math.sqrt(((V * l) / A))
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -2e+263)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= 5e+55)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / 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 ((V * l) <= -2e+263)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= 5e+55)
		tmp = c0 / sqrt(((V * l) / A));
	else
		tmp = c0 * sqrt(((A / 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[N[(V * l), $MachinePrecision], -2e+263], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+55], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -2.00000000000000003e263

   1. Initial program 38.5%

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

      1. associate-/r* 68.2%

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

      2. sqrt-div 37.2%

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

   3. Applied egg-rr 37.2%

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

   if -2.00000000000000003e263 < (*.f64 V l) < 5.00000000000000046e55

   1. Initial program 85.3%

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

      1. *-un-lft-identity 85.3%

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

      2. times-frac 78.1%

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

   3. Applied egg-rr 78.1%

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

      1. add-sqr-sqrt 36.9%

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

      2. sqrt-prod 29.4%

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

      3. sqrt-prod 27.3%

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

      4. *-commutative 27.3%

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

      5. frac-times 29.3%

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

      6. *-un-lft-identity 29.3%

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

      7. expm1-log1p-u 28.8%

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

      8. expm1-udef 22.2%

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

   5. Applied egg-rr 28.4%

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

      1. expm1-def 47.4%

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

      2. expm1-log1p 76.0%

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

      3. associate-/r/ 78.0%

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

      4. *-commutative 78.0%

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

   7. Simplified 78.0%

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

   8. Taylor expanded in V around 0 85.3%

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

   if 5.00000000000000046e55 < (*.f64 V l)

   1. Initial program 64.7%

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

      1. *-un-lft-identity 64.7%

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

      2. times-frac 86.0%

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

   3. Applied egg-rr 86.0%

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

      1. associate-*l/ 86.1%

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

      2. *-un-lft-identity 86.1%

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

   5. Applied egg-rr 86.1%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+263}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+55}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 8: 77.3% accurate, 0.9× speedup

Math

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

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0) (* c0 (pow (* V (/ l A)) -0.5)) (* c0 (sqrt t_0)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * pow((V * (l / A)), -0.5);
	} else {
		tmp = c0 * sqrt(t_0);
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * ((v * (l / a)) ** (-0.5d0))
    else
        tmp = c0 * sqrt(t_0)
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.pow((V * (l / A)), -0.5);
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * ((V * (l / A)) ^ -0.5);
	else
		tmp = c0 * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Power[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision], -0.5], $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

   1. Initial program 25.4%

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

      1. pow1/2 25.4%

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

      2. clear-num 25.4%

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

      3. inv-pow 25.4%

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

      4. pow-pow 25.4%

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

      5. associate-/l* 65.3%

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

      6. metadata-eval 65.3%

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

   3. Applied egg-rr 65.3%

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

      1. associate-/l* 25.4%

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

      2. *-lft-identity 25.4%

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

      3. times-frac 65.3%

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

      4. /-rgt-identity 65.3%

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

   5. Simplified 65.3%

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

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

   1. Initial program 87.8%

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

4. Final simplification 84.2%

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

Alternative 9: 77.4% accurate, 0.9× speedup

Math

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

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0) (* c0 (sqrt (/ (/ A l) V))) (* c0 (sqrt t_0)))))

C

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

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / l) / v))
    else
        tmp = c0 * sqrt(t_0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $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

   1. Initial program 25.4%

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

      1. *-un-lft-identity 25.4%

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

      2. times-frac 65.2%

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

   3. Applied egg-rr 65.2%

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

      1. associate-*l/ 65.3%

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

      2. *-un-lft-identity 65.3%

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

   5. Applied egg-rr 65.3%

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

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

   1. Initial program 87.8%

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

4. Final simplification 84.2%

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

Alternative 10: 74.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    code = c0 * sqrt((a / (v * l)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

2. Final simplification 77.6%

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

Reproduce

herbie shell --seed 2023254 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))