Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.9% → 88.9%

Time: 10.4s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.9% 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: 88.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-307}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-320}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* V l) -1e-307)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 5e-320)
       (* c0 (pow (* V (/ l A)) -0.5))
       (/ (* c0 (sqrt A)) (sqrt (* V l)))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= -1e-307) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 5e-320) {
		tmp = c0 * pow((V * (l / A)), -0.5);
	} else {
		tmp = (c0 * sqrt(A)) / sqrt((V * l));
	}
	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 / V)) / Math.sqrt(l));
	} else if ((V * l) <= -1e-307) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 5e-320) {
		tmp = c0 * Math.pow((V * (l / A)), -0.5);
	} else {
		tmp = (c0 * Math.sqrt(A)) / Math.sqrt((V * l));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= -1e-307:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 5e-320:
		tmp = c0 * math.pow((V * (l / A)), -0.5)
	else:
		tmp = (c0 * math.sqrt(A)) / math.sqrt((V * l))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= -1e-307)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 5e-320)
		tmp = Float64(c0 * (Float64(V * Float64(l / A)) ^ -0.5));
	else
		tmp = Float64(Float64(c0 * sqrt(A)) / sqrt(Float64(V * l)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= -1e-307)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 5e-320)
		tmp = c0 * ((V * (l / A)) ^ -0.5);
	else
		tmp = (c0 * sqrt(A)) / sqrt((V * l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-307], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-320], N[(c0 * N[Power[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(c0 * N[Sqrt[A], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 24.1%

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

      1. associate-/r* 42.6%

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

      2. sqrt-div 30.5%

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

   3. Applied egg-rr 30.5%

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

   if -inf.0 < (*.f64 V l) < -9.99999999999999909e-308

   1. Initial program 86.4%

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

      1. frac-2neg 86.4%

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

      2. sqrt-div 99.4%

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

      3. *-commutative 99.4%

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

      4. distribute-rgt-neg-in 99.4%

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

   3. Applied egg-rr 99.4%

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

   if -9.99999999999999909e-308 < (*.f64 V l) < 4.99994e-320

   1. Initial program 41.3%

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

      1. pow1/2 41.3%

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

      2. clear-num 41.3%

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

      3. inv-pow 41.3%

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

      4. pow-pow 41.3%

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

      5. associate-/l* 70.4%

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

      6. metadata-eval 70.4%

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

   3. Applied egg-rr 70.4%

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

      1. associate-/l* 41.3%

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

      2. *-lft-identity 41.3%

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

      3. times-frac 70.4%

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

      4. /-rgt-identity 70.4%

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

   5. Simplified 70.4%

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

   if 4.99994e-320 < (*.f64 V l)

   1. Initial program 71.7%

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

      1. *-commutative 71.7%

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

      2. sqrt-div 91.7%

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

      3. associate-*l/ 90.7%

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

   3. Applied egg-rr 90.7%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-307}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-320}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]

Alternative 2: 80.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-322} \lor \neg \left(t_0 \leq 0\right) \land t_0 \leq 10^{+304}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (or (<= t_0 -5e-322) (and (not (<= t_0 0.0)) (<= t_0 1e+304)))
     t_0
     (sqrt (* A (* (/ c0 l) (/ c0 V)))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= -5e-322) || (!(t_0 <= 0.0) && (t_0 <= 1e+304))) {
		tmp = t_0;
	} else {
		tmp = sqrt((A * ((c0 / l) * (c0 / 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) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if ((t_0 <= (-5d-322)) .or. (.not. (t_0 <= 0.0d0)) .and. (t_0 <= 1d+304)) then
        tmp = t_0
    else
        tmp = sqrt((a * ((c0 / l) * (c0 / v))))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= -5e-322) || (!(t_0 <= 0.0) && (t_0 <= 1e+304))) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt((A * ((c0 / l) * (c0 / V))));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if (t_0 <= -5e-322) or (not (t_0 <= 0.0) and (t_0 <= 1e+304)):
		tmp = t_0
	else:
		tmp = math.sqrt((A * ((c0 / l) * (c0 / V))))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if ((t_0 <= -5e-322) || (!(t_0 <= 0.0) && (t_0 <= 1e+304)))
		tmp = t_0;
	else
		tmp = sqrt(Float64(A * Float64(Float64(c0 / l) * Float64(c0 / V))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if ((t_0 <= -5e-322) || (~((t_0 <= 0.0)) && (t_0 <= 1e+304)))
		tmp = t_0;
	else
		tmp = sqrt((A * ((c0 / l) * (c0 / 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_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-322], And[N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision], LessEqual[t$95$0, 1e+304]]], t$95$0, N[Sqrt[N[(A * N[(N[(c0 / l), $MachinePrecision] * N[(c0 / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < -4.99006e-322 or 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 9.9999999999999994e303

   1. Initial program 90.6%

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

   if -4.99006e-322 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 0.0 or 9.9999999999999994e303 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 40.5%

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

      1. add-sqr-sqrt 40.5%

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

      2. pow2 40.5%

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

      3. pow1/2 40.5%

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

      4. sqrt-pow1 40.5%

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

      5. metadata-eval 40.5%

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

   3. Applied egg-rr 40.5%

      \[\leadsto c0 \cdot \color{blue}{{\left({\left(\frac{A}{V \cdot \ell}\right)}^{0.25}\right)}^{2}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 40.5%

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

      2. sqrt-unprod 40.5%

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

      3. *-commutative 40.5%

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

      4. *-commutative 40.5%

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

      5. swap-sqr 40.0%

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

      6. pow-pow 40.0%

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

      7. metadata-eval 40.0%

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

      8. metadata-eval 40.0%

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

      9. pow-pow 40.0%

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

      10. metadata-eval 40.0%

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

      11. metadata-eval 40.0%

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

      12. sqr-pow 40.0%

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

      13. pow1 40.0%

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

   5. Applied egg-rr 40.0%

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

      1. associate-*l/ 52.5%

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

      2. associate-*r/ 52.5%

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

      3. times-frac 65.1%

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

   7. Simplified 65.1%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq -5 \cdot 10^{-322} \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0\right) \land c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{+304}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \end{array} \]

Alternative 3: 80.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-322}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{c0}{V} \cdot \frac{c0 \cdot A}{\ell}}\\ \mathbf{elif}\;t_0 \leq 10^{+304}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 -5e-322)
     t_0
     (if (<= t_0 0.0)
       (sqrt (* (/ c0 V) (/ (* c0 A) l)))
       (if (<= t_0 1e+304) t_0 (sqrt (* A (* (/ c0 l) (/ c0 V)))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= -5e-322) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = sqrt(((c0 / V) * ((c0 * A) / l)));
	} else if (t_0 <= 1e+304) {
		tmp = t_0;
	} else {
		tmp = sqrt((A * ((c0 / l) * (c0 / 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) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= (-5d-322)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = sqrt(((c0 / v) * ((c0 * a) / l)))
    else if (t_0 <= 1d+304) then
        tmp = t_0
    else
        tmp = sqrt((a * ((c0 / l) * (c0 / v))))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= -5e-322) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = Math.sqrt(((c0 / V) * ((c0 * A) / l)));
	} else if (t_0 <= 1e+304) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt((A * ((c0 / l) * (c0 / V))));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= -5e-322:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = math.sqrt(((c0 / V) * ((c0 * A) / l)))
	elif t_0 <= 1e+304:
		tmp = t_0
	else:
		tmp = math.sqrt((A * ((c0 / l) * (c0 / V))))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= -5e-322)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = sqrt(Float64(Float64(c0 / V) * Float64(Float64(c0 * A) / l)));
	elseif (t_0 <= 1e+304)
		tmp = t_0;
	else
		tmp = sqrt(Float64(A * Float64(Float64(c0 / l) * Float64(c0 / V))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= -5e-322)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = sqrt(((c0 / V) * ((c0 * A) / l)));
	elseif (t_0 <= 1e+304)
		tmp = t_0;
	else
		tmp = sqrt((A * ((c0 / l) * (c0 / 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_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-322], t$95$0, If[LessEqual[t$95$0, 0.0], N[Sqrt[N[(N[(c0 / V), $MachinePrecision] * N[(N[(c0 * A), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 1e+304], t$95$0, N[Sqrt[N[(A * N[(N[(c0 / l), $MachinePrecision] * N[(c0 / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < -4.99006e-322 or 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 9.9999999999999994e303

   1. Initial program 90.6%

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

   if -4.99006e-322 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 0.0

   1. Initial program 39.6%

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

      1. sqrt-div 45.6%

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

      2. associate-*r/ 45.7%

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

   3. Applied egg-rr 45.7%

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

      1. associate-*l/ 45.7%

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

   5. Simplified 45.7%

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

      1. add-sqr-sqrt 34.0%

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

      2. sqrt-unprod 34.5%

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

      3. *-commutative 34.5%

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

      4. *-commutative 34.5%

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

      5. swap-sqr 32.9%

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

      6. add-sqr-sqrt 32.9%

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

      7. frac-times 32.7%

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

      8. add-sqr-sqrt 50.3%

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

   7. Applied egg-rr 50.3%

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

      1. frac-times 58.5%

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

      2. *-commutative 58.5%

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

   9. Applied egg-rr 58.5%

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

   10. Taylor expanded in A around 0 50.3%

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

       1. associate-*r/ 50.3%

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

       2. unpow2 50.3%

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

       3. *-commutative 50.3%

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

       4. times-frac 58.5%

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

       5. *-commutative 58.5%

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

       6. *-commutative 58.5%

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

       7. associate-*l* 61.7%

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

       8. *-commutative 61.7%

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

       9. associate-*r/ 61.7%

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

   12. Simplified 61.7%

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

   if 9.9999999999999994e303 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 42.2%

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

      1. add-sqr-sqrt 42.2%

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

      2. pow2 42.2%

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

      3. pow1/2 42.2%

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

      4. sqrt-pow1 42.2%

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

      5. metadata-eval 42.2%

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

   3. Applied egg-rr 42.2%

      \[\leadsto c0 \cdot \color{blue}{{\left({\left(\frac{A}{V \cdot \ell}\right)}^{0.25}\right)}^{2}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 42.2%

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

      2. sqrt-unprod 42.2%

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

      3. *-commutative 42.2%

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

      4. *-commutative 42.2%

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

      5. swap-sqr 41.6%

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

      6. pow-pow 41.6%

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

      7. metadata-eval 41.6%

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

      8. metadata-eval 41.6%

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

      9. pow-pow 41.6%

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

      10. metadata-eval 41.6%

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

      11. metadata-eval 41.6%

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

      12. sqr-pow 41.6%

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

      13. pow1 41.6%

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

   5. Applied egg-rr 41.6%

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

      1. associate-*l/ 56.4%

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

      2. associate-*r/ 56.4%

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

      3. times-frac 77.1%

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

   7. Simplified 77.1%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq -5 \cdot 10^{-322}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0:\\ \;\;\;\;\sqrt{\frac{c0}{V} \cdot \frac{c0 \cdot A}{\ell}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{+304}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \end{array} \]

Alternative 4: 79.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-322}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{A \cdot \left(c0 \cdot \frac{c0}{\ell}\right)}{V}}\\ \mathbf{elif}\;t_0 \leq 10^{+304}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 -5e-322)
     t_0
     (if (<= t_0 0.0)
       (sqrt (/ (* A (* c0 (/ c0 l))) V))
       (if (<= t_0 1e+304) t_0 (sqrt (* A (* (/ c0 l) (/ c0 V)))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= -5e-322) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = sqrt(((A * (c0 * (c0 / l))) / V));
	} else if (t_0 <= 1e+304) {
		tmp = t_0;
	} else {
		tmp = sqrt((A * ((c0 / l) * (c0 / 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) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= (-5d-322)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = sqrt(((a * (c0 * (c0 / l))) / v))
    else if (t_0 <= 1d+304) then
        tmp = t_0
    else
        tmp = sqrt((a * ((c0 / l) * (c0 / v))))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= -5e-322) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = Math.sqrt(((A * (c0 * (c0 / l))) / V));
	} else if (t_0 <= 1e+304) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt((A * ((c0 / l) * (c0 / V))));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= -5e-322:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = math.sqrt(((A * (c0 * (c0 / l))) / V))
	elif t_0 <= 1e+304:
		tmp = t_0
	else:
		tmp = math.sqrt((A * ((c0 / l) * (c0 / V))))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= -5e-322)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = sqrt(Float64(Float64(A * Float64(c0 * Float64(c0 / l))) / V));
	elseif (t_0 <= 1e+304)
		tmp = t_0;
	else
		tmp = sqrt(Float64(A * Float64(Float64(c0 / l) * Float64(c0 / V))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= -5e-322)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = sqrt(((A * (c0 * (c0 / l))) / V));
	elseif (t_0 <= 1e+304)
		tmp = t_0;
	else
		tmp = sqrt((A * ((c0 / l) * (c0 / 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_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-322], t$95$0, If[LessEqual[t$95$0, 0.0], N[Sqrt[N[(N[(A * N[(c0 * N[(c0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 1e+304], t$95$0, N[Sqrt[N[(A * N[(N[(c0 / l), $MachinePrecision] * N[(c0 / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < -4.99006e-322 or 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 9.9999999999999994e303

   1. Initial program 90.6%

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

   if -4.99006e-322 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 0.0

   1. Initial program 39.6%

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

      1. sqrt-div 45.6%

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

      2. associate-*r/ 45.7%

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

   3. Applied egg-rr 45.7%

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

      1. associate-*l/ 45.7%

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

   5. Simplified 45.7%

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

      1. add-sqr-sqrt 34.0%

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

      2. sqrt-unprod 34.5%

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

      3. *-commutative 34.5%

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

      4. *-commutative 34.5%

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

      5. swap-sqr 32.9%

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

      6. add-sqr-sqrt 32.9%

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

      7. frac-times 32.7%

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

      8. add-sqr-sqrt 50.3%

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

   7. Applied egg-rr 50.3%

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

      1. frac-times 58.5%

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

      2. *-commutative 58.5%

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

      3. associate-*l/ 56.9%

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

      4. associate-*l/ 59.9%

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

   9. Applied egg-rr 59.9%

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

   if 9.9999999999999994e303 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 42.2%

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

      1. add-sqr-sqrt 42.2%

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

      2. pow2 42.2%

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

      3. pow1/2 42.2%

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

      4. sqrt-pow1 42.2%

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

      5. metadata-eval 42.2%

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

   3. Applied egg-rr 42.2%

      \[\leadsto c0 \cdot \color{blue}{{\left({\left(\frac{A}{V \cdot \ell}\right)}^{0.25}\right)}^{2}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 42.2%

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

      2. sqrt-unprod 42.2%

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

      3. *-commutative 42.2%

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

      4. *-commutative 42.2%

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

      5. swap-sqr 41.6%

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

      6. pow-pow 41.6%

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

      7. metadata-eval 41.6%

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

      8. metadata-eval 41.6%

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

      9. pow-pow 41.6%

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

      10. metadata-eval 41.6%

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

      11. metadata-eval 41.6%

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

      12. sqr-pow 41.6%

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

      13. pow1 41.6%

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

   5. Applied egg-rr 41.6%

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

      1. associate-*l/ 56.4%

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

      2. associate-*r/ 56.4%

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

      3. times-frac 77.1%

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

   7. Simplified 77.1%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq -5 \cdot 10^{-322}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0:\\ \;\;\;\;\sqrt{\frac{A \cdot \left(c0 \cdot \frac{c0}{\ell}\right)}{V}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{+304}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \end{array} \]

Alternative 5: 84.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+176}:\\ \;\;\;\;c0 \cdot \frac{t_0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-38}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-320}:\\ \;\;\;\;c0 \cdot \left(t_0 \cdot {\ell}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (/ A V))))
   (if (<= (* V l) -2e+176)
     (* c0 (/ t_0 (sqrt l)))
     (if (<= (* V l) -2e-38)
       (* c0 (sqrt (/ A (* V l))))
       (if (<= (* V l) 5e-320)
         (* c0 (* t_0 (pow l -0.5)))
         (/ (* c0 (sqrt A)) (sqrt (* V l))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / V));
	double tmp;
	if ((V * l) <= -2e+176) {
		tmp = c0 * (t_0 / sqrt(l));
	} else if ((V * l) <= -2e-38) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 5e-320) {
		tmp = c0 * (t_0 * pow(l, -0.5));
	} else {
		tmp = (c0 * sqrt(A)) / sqrt((V * l));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a / v))
    if ((v * l) <= (-2d+176)) then
        tmp = c0 * (t_0 / sqrt(l))
    else if ((v * l) <= (-2d-38)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 5d-320) then
        tmp = c0 * (t_0 * (l ** (-0.5d0)))
    else
        tmp = (c0 * sqrt(a)) / sqrt((v * l))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / V));
	double tmp;
	if ((V * l) <= -2e+176) {
		tmp = c0 * (t_0 / Math.sqrt(l));
	} else if ((V * l) <= -2e-38) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 5e-320) {
		tmp = c0 * (t_0 * Math.pow(l, -0.5));
	} else {
		tmp = (c0 * Math.sqrt(A)) / Math.sqrt((V * l));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / V))
	tmp = 0
	if (V * l) <= -2e+176:
		tmp = c0 * (t_0 / math.sqrt(l))
	elif (V * l) <= -2e-38:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 5e-320:
		tmp = c0 * (t_0 * math.pow(l, -0.5))
	else:
		tmp = (c0 * math.sqrt(A)) / math.sqrt((V * l))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(A / V))
	tmp = 0.0
	if (Float64(V * l) <= -2e+176)
		tmp = Float64(c0 * Float64(t_0 / sqrt(l)));
	elseif (Float64(V * l) <= -2e-38)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 5e-320)
		tmp = Float64(c0 * Float64(t_0 * (l ^ -0.5)));
	else
		tmp = Float64(Float64(c0 * sqrt(A)) / sqrt(Float64(V * l)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((A / V));
	tmp = 0.0;
	if ((V * l) <= -2e+176)
		tmp = c0 * (t_0 / sqrt(l));
	elseif ((V * l) <= -2e-38)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 5e-320)
		tmp = c0 * (t_0 * (l ^ -0.5));
	else
		tmp = (c0 * sqrt(A)) / sqrt((V * l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -2e+176], N[(c0 * N[(t$95$0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-38], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-320], N[(c0 * N[(t$95$0 * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * N[Sqrt[A], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -2e176

   1. Initial program 48.8%

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

      1. associate-/r* 55.9%

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

      2. sqrt-div 34.9%

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

   3. Applied egg-rr 34.9%

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

   if -2e176 < (*.f64 V l) < -1.9999999999999999e-38

   1. Initial program 96.4%

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

   if -1.9999999999999999e-38 < (*.f64 V l) < 4.99994e-320

   1. Initial program 63.5%

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

      1. add-sqr-sqrt 63.5%

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

      2. pow2 63.5%

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

      3. pow1/2 63.5%

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

      4. sqrt-pow1 63.5%

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

      5. metadata-eval 63.5%

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

   3. Applied egg-rr 63.5%

      \[\leadsto c0 \cdot \color{blue}{{\left({\left(\frac{A}{V \cdot \ell}\right)}^{0.25}\right)}^{2}} \]
   4. Step-by-step derivation

      1. pow-pow 63.5%

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

      2. metadata-eval 63.5%

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

      3. pow1/2 63.5%

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

      4. associate-/r* 71.7%

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

      5. un-div-inv 71.7%

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

      6. sqrt-prod 48.4%

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

      7. *-commutative 48.4%

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

      8. inv-pow 48.4%

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

      9. sqrt-pow1 48.4%

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

      10. metadata-eval 48.4%

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

   5. Applied egg-rr 48.4%

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

   if 4.99994e-320 < (*.f64 V l)

   1. Initial program 71.7%

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

      1. *-commutative 71.7%

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

      2. sqrt-div 91.7%

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

      3. associate-*l/ 90.7%

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

   3. Applied egg-rr 90.7%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+176}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-38}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-320}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot {\ell}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]

Alternative 6: 84.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+176}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-38}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-320}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (/ (sqrt (/ A V)) (sqrt l)))))
   (if (<= (* V l) -2e+176)
     t_0
     (if (<= (* V l) -2e-38)
       (* c0 (sqrt (/ A (* V l))))
       (if (<= (* V l) 5e-320) t_0 (/ (* c0 (sqrt A)) (sqrt (* V l))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (sqrt((A / V)) / sqrt(l));
	double tmp;
	if ((V * l) <= -2e+176) {
		tmp = t_0;
	} else if ((V * l) <= -2e-38) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 5e-320) {
		tmp = t_0;
	} else {
		tmp = (c0 * sqrt(A)) / sqrt((V * l));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * (sqrt((a / v)) / sqrt(l))
    if ((v * l) <= (-2d+176)) then
        tmp = t_0
    else if ((v * l) <= (-2d-38)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 5d-320) then
        tmp = t_0
    else
        tmp = (c0 * sqrt(a)) / sqrt((v * l))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	double tmp;
	if ((V * l) <= -2e+176) {
		tmp = t_0;
	} else if ((V * l) <= -2e-38) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 5e-320) {
		tmp = t_0;
	} else {
		tmp = (c0 * Math.sqrt(A)) / Math.sqrt((V * l));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	tmp = 0
	if (V * l) <= -2e+176:
		tmp = t_0
	elif (V * l) <= -2e-38:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 5e-320:
		tmp = t_0
	else:
		tmp = (c0 * math.sqrt(A)) / math.sqrt((V * l))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)))
	tmp = 0.0
	if (Float64(V * l) <= -2e+176)
		tmp = t_0;
	elseif (Float64(V * l) <= -2e-38)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 5e-320)
		tmp = t_0;
	else
		tmp = Float64(Float64(c0 * sqrt(A)) / sqrt(Float64(V * l)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * (sqrt((A / V)) / sqrt(l));
	tmp = 0.0;
	if ((V * l) <= -2e+176)
		tmp = t_0;
	elseif ((V * l) <= -2e-38)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 5e-320)
		tmp = t_0;
	else
		tmp = (c0 * sqrt(A)) / sqrt((V * l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -2e+176], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -2e-38], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-320], t$95$0, N[(N[(c0 * N[Sqrt[A], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -2e176 or -1.9999999999999999e-38 < (*.f64 V l) < 4.99994e-320

   1. Initial program 58.9%

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

      1. associate-/r* 66.8%

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

      2. sqrt-div 44.3%

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

   3. Applied egg-rr 44.3%

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

   if -2e176 < (*.f64 V l) < -1.9999999999999999e-38

   1. Initial program 96.4%

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

   if 4.99994e-320 < (*.f64 V l)

   1. Initial program 71.7%

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

      1. *-commutative 71.7%

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

      2. sqrt-div 91.7%

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

      3. associate-*l/ 90.7%

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

   3. Applied egg-rr 90.7%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+176}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-38}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-320}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]

Alternative 7: 81.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.2 \cdot 10^{-287}:\\ \;\;\;\;c0 \cdot {\left(\frac{1}{\frac{A}{V \cdot \ell}}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= l 6.2e-287)
   (* c0 (pow (/ 1.0 (/ A (* V l))) -0.5))
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= 6.2e-287) {
		tmp = c0 * pow((1.0 / (A / (V * l))), -0.5);
	} 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 <= 6.2d-287) then
        tmp = c0 * ((1.0d0 / (a / (v * l))) ** (-0.5d0))
    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 <= 6.2e-287) {
		tmp = c0 * Math.pow((1.0 / (A / (V * l))), -0.5);
	} 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 <= 6.2e-287:
		tmp = c0 * math.pow((1.0 / (A / (V * l))), -0.5)
	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 <= 6.2e-287)
		tmp = Float64(c0 * (Float64(1.0 / Float64(A / Float64(V * l))) ^ -0.5));
	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 <= 6.2e-287)
		tmp = c0 * ((1.0 / (A / (V * l))) ^ -0.5);
	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, 6.2e-287], N[(c0 * N[Power[N[(1.0 / N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $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 < 6.2000000000000001e-287

   1. Initial program 77.1%

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

      1. pow1/2 77.1%

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

      2. clear-num 76.1%

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

      3. inv-pow 76.1%

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

      4. pow-pow 76.1%

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

      5. associate-/l* 70.9%

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

      6. metadata-eval 70.9%

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

   3. Applied egg-rr 70.9%

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

      1. associate-/l* 76.1%

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

      2. *-lft-identity 76.1%

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

      3. times-frac 71.7%

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

      4. /-rgt-identity 71.7%

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

   5. Simplified 71.7%

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

      1. associate-*r/ 76.1%

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

      2. clear-num 76.1%

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

   7. Applied egg-rr 76.1%

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

   if 6.2000000000000001e-287 < l

   1. Initial program 67.5%

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

      1. associate-/r* 68.6%

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

      2. sqrt-div 81.9%

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

   3. Applied egg-rr 81.9%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.2 \cdot 10^{-287}:\\ \;\;\;\;c0 \cdot {\left(\frac{1}{\frac{A}{V \cdot \ell}}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 8: 83.2% accurate, 0.5× speedup

Math

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

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-310)
   (* (sqrt A) (/ c0 (sqrt (* V l))))
   (* 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-310) {
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	} 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-310)) then
        tmp = sqrt(a) * (c0 / sqrt((v * l)))
    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-310) {
		tmp = Math.sqrt(A) * (c0 / Math.sqrt((V * l)));
	} 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-310:
		tmp = math.sqrt(A) * (c0 / math.sqrt((V * l)))
	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-310)
		tmp = Float64(sqrt(A) * Float64(c0 / sqrt(Float64(V * l))));
	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-310)
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	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-310], N[(N[Sqrt[A], $MachinePrecision] * N[(c0 / N[Sqrt[N[(V * l), $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 < -4.999999999999985e-310

   1. Initial program 77.7%

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

      1. sqrt-div 46.5%

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

      2. associate-*r/ 45.6%

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

   3. Applied egg-rr 45.6%

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

      1. associate-*l/ 45.1%

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

   5. Simplified 45.1%

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

   if -4.999999999999985e-310 < l

   1. Initial program 67.0%

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

      1. associate-/r* 68.8%

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

      2. sqrt-div 82.1%

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

   3. Applied egg-rr 82.1%

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

4. Final simplification 63.5%

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

Alternative 9: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \end{array} \]

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 (* V l))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 2e+301) (* 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 / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 2e+301) {
		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 / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 2d+301) 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 / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 2e+301) {
		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 / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 2e+301:
		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(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 2e+301)
		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 / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 2e+301)
		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[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+301], 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 32.9%

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

      1. add-sqr-sqrt 32.9%

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

      2. pow2 32.9%

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

      3. pow1/2 32.9%

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

      4. sqrt-pow1 32.9%

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

      5. metadata-eval 32.9%

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

   3. Applied egg-rr 32.9%

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

   4. Taylor expanded in c0 around 0 32.9%

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

      1. associate-/r* 42.4%

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

   6. Simplified 42.4%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 2.00000000000000011e301

   1. Initial program 98.2%

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

   if 2.00000000000000011e301 < (/.f64 A (*.f64 V l))

   1. Initial program 31.3%

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

      1. pow1/2 31.3%

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

      2. clear-num 31.3%

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

      3. inv-pow 31.3%

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

      4. pow-pow 32.5%

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

      5. associate-/l* 46.9%

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

      6. metadata-eval 46.9%

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

   3. Applied egg-rr 46.9%

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

      1. associate-/l* 32.5%

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

      2. *-lft-identity 32.5%

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

      3. times-frac 46.9%

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

      4. /-rgt-identity 46.9%

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

   5. Simplified 46.9%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+301}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \]

Alternative 10: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \end{array} \end{array} \]

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 (* V l))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 2e+299) (* c0 (sqrt t_0)) (* c0 (pow (* l (/ V A)) -0.5))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 2e+299) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 * pow((l * (V / 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 / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 2d+299) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 * ((l * (v / 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 / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 2e+299) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 * Math.pow((l * (V / A)), -0.5);
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 2e+299:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 * math.pow((l * (V / A)), -0.5)
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 2e+299)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 * (Float64(l * Float64(V / 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 / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 2e+299)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 * ((l * (V / 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[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+299], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Power[N[(l * N[(V / 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 32.9%

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

      1. add-sqr-sqrt 32.9%

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

      2. pow2 32.9%

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

      3. pow1/2 32.9%

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

      4. sqrt-pow1 32.9%

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

      5. metadata-eval 32.9%

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

   3. Applied egg-rr 32.9%

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

   4. Taylor expanded in c0 around 0 32.9%

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

      1. associate-/r* 42.4%

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

   6. Simplified 42.4%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 2.0000000000000001e299

   1. Initial program 98.2%

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

   if 2.0000000000000001e299 < (/.f64 A (*.f64 V l))

   1. Initial program 32.8%

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

      1. pow1/2 32.8%

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

      2. clear-num 32.8%

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

      3. inv-pow 32.8%

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

      4. pow-pow 33.9%

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

      5. associate-/l* 46.0%

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

      6. metadata-eval 46.0%

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

   3. Applied egg-rr 46.0%

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

      1. associate-/l* 33.9%

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

      2. *-lft-identity 33.9%

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

      3. times-frac 47.3%

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

      4. /-rgt-identity 47.3%

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

   5. Simplified 47.3%

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

   6. Taylor expanded in V around 0 36.6%

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

      1. exp-prod 36.6%

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

      2. log-prod 45.9%

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

      3. *-commutative 45.9%

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

      4. associate-*l/ 33.8%

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

      5. associate-*r/ 46.6%

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

      6. exp-prod 46.6%

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

      7. *-commutative 46.6%

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

      8. exp-to-pow 48.1%

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

   8. Simplified 48.1%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+299}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \end{array} \]

Alternative 11: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 2 \cdot 10^{+301}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \end{array} \end{array} \]

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 (or (<= t_0 0.0) (not (<= t_0 2e+301)))
     (* 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) || !(t_0 <= 2e+301)) {
		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) .or. (.not. (t_0 <= 2d+301))) 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) || !(t_0 <= 2e+301)) {
		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) or not (t_0 <= 2e+301):
		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) || !(t_0 <= 2e+301))
		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) || ~((t_0 <= 2e+301)))
		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[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 2e+301]], $MachinePrecision]], 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 or 2.00000000000000011e301 < (/.f64 A (*.f64 V l))

   1. Initial program 32.2%

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

      1. associate-/r* 43.9%

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

      2. div-inv 43.9%

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

   3. Applied egg-rr 43.9%

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

      1. associate-*l/ 44.0%

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

      2. div-inv 44.0%

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

   5. Applied egg-rr 44.0%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 2.00000000000000011e301

   1. Initial program 98.2%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+301}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]

Alternative 12: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 2 \cdot 10^{+299}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (or (<= t_0 0.0) (not (<= t_0 2e+299)))
     (* c0 (sqrt (/ (/ A V) l)))
     (* c0 (sqrt t_0)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 2e+299)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = c0 * sqrt(t_0);
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 2d+299))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = c0 * sqrt(t_0)
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 2e+299)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 2e+299):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 2e+299))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(c0 * sqrt(t_0));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 2e+299)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = c0 * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 2e+299]], $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 2.0000000000000001e299 < (/.f64 A (*.f64 V l))

   1. Initial program 32.8%

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

      1. add-sqr-sqrt 32.8%

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

      2. pow2 32.8%

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

      3. pow1/2 32.8%

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

      4. sqrt-pow1 32.8%

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

      5. metadata-eval 32.8%

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

   3. Applied egg-rr 32.8%

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

   4. Taylor expanded in c0 around 0 32.8%

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

      1. associate-/r* 44.5%

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

   6. Simplified 44.5%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 2.0000000000000001e299

   1. Initial program 98.2%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+299}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]

Alternative 13: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 2e+299) (* c0 (sqrt t_0)) (/ c0 (sqrt (* l (/ V A))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 2e+299) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((l * (V / A)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 2d+299) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((l * (v / a)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 2e+299) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((l * (V / A)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 2e+299:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((l * (V / A)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 2e+299)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 2e+299)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((l * (V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+299], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 32.9%

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

      1. add-sqr-sqrt 32.9%

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

      2. pow2 32.9%

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

      3. pow1/2 32.9%

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

      4. sqrt-pow1 32.9%

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

      5. metadata-eval 32.9%

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

   3. Applied egg-rr 32.9%

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

   4. Taylor expanded in c0 around 0 32.9%

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

      1. associate-/r* 42.4%

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

   6. Simplified 42.4%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 2.0000000000000001e299

   1. Initial program 98.2%

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

   if 2.0000000000000001e299 < (/.f64 A (*.f64 V l))

   1. Initial program 32.8%

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

      1. sqrt-div 45.3%

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

      2. associate-*r/ 45.2%

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

   3. Applied egg-rr 45.2%

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

      1. associate-*l/ 45.3%

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

   5. Simplified 45.3%

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

      1. expm1-log1p-u 28.8%

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

      2. expm1-udef 24.4%

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

      3. associate-*l/ 24.4%

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

      4. associate-/l* 24.4%

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

      5. sqrt-div 13.7%

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

      6. associate-*r/ 19.2%

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

      7. associate-*r/ 13.7%

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

      8. associate-*l/ 19.2%

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

      9. *-commutative 19.2%

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

   7. Applied egg-rr 19.2%

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

      1. expm1-def 23.3%

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

      2. expm1-log1p 48.0%

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

   9. Simplified 48.0%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+299}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]

Alternative 14: 73.9% 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 72.4%

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

2. Final simplification 72.4%

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

Reproduce

herbie shell --seed 2023256 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))