Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.8% → 89.4%

Time: 12.9s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\frac{\sqrt{-V}}{\sqrt{\frac{-A}{\ell}}}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+248}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-282}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ c0 (/ (sqrt (- V)) (sqrt (/ (- A) l))))))
   (if (<= (* V l) -5e+248)
     t_0
     (if (<= (* V l) -2e-282)
       (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
       (if (<= (* V l) 0.0)
         t_0
         (if (<= (* V l) 2e+296)
           (/ (* c0 (sqrt A)) (sqrt (* V l)))
           (/ c0 (sqrt (* V (/ l A))))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 / (sqrt(-V) / sqrt((-A / l)));
	double tmp;
	if ((V * l) <= -5e+248) {
		tmp = t_0;
	} else if ((V * l) <= -2e-282) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 2e+296) {
		tmp = (c0 * sqrt(A)) / sqrt((V * l));
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 / (sqrt(-v) / sqrt((-a / l)))
    if ((v * l) <= (-5d+248)) then
        tmp = t_0
    else if ((v * l) <= (-2d-282)) then
        tmp = c0 * (sqrt(-a) / sqrt((v * -l)))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else if ((v * l) <= 2d+296) then
        tmp = (c0 * sqrt(a)) / sqrt((v * l))
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 / (Math.sqrt(-V) / Math.sqrt((-A / l)));
	double tmp;
	if ((V * l) <= -5e+248) {
		tmp = t_0;
	} else if ((V * l) <= -2e-282) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 2e+296) {
		tmp = (c0 * Math.sqrt(A)) / Math.sqrt((V * l));
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = c0 / (math.sqrt(-V) / math.sqrt((-A / l)))
	tmp = 0
	if (V * l) <= -5e+248:
		tmp = t_0
	elif (V * l) <= -2e-282:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = t_0
	elif (V * l) <= 2e+296:
		tmp = (c0 * math.sqrt(A)) / math.sqrt((V * l))
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 / Float64(sqrt(Float64(-V)) / sqrt(Float64(Float64(-A) / l))))
	tmp = 0.0
	if (Float64(V * l) <= -5e+248)
		tmp = t_0;
	elseif (Float64(V * l) <= -2e-282)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	elseif (Float64(V * l) <= 2e+296)
		tmp = Float64(Float64(c0 * sqrt(A)) / sqrt(Float64(V * l)));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 / (sqrt(-V) / sqrt((-A / l)));
	tmp = 0.0;
	if ((V * l) <= -5e+248)
		tmp = t_0;
	elseif ((V * l) <= -2e-282)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	elseif ((V * l) <= 2e+296)
		tmp = (c0 * sqrt(A)) / sqrt((V * l));
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 / N[(N[Sqrt[(-V)], $MachinePrecision] / N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e+248], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -2e-282], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], 2e+296], N[(N[(c0 * N[Sqrt[A], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -4.9999999999999996e248 or -2e-282 < (*.f64 V l) < -0.0

   1. Initial program 53.6%

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

      1. pow1/2 53.6%

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

      2. clear-num 53.6%

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

      3. inv-pow 53.6%

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

      4. pow-pow 53.6%

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

      5. associate-/l* 74.0%

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

      6. metadata-eval 74.0%

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

   3. Applied egg-rr 74.0%

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

      1. div-inv 73.9%

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

      2. clear-num 73.9%

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

      3. metadata-eval 73.9%

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

      4. sqrt-pow2 74.0%

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

      5. inv-pow 74.0%

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

      6. un-div-inv 74.0%

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

   5. Applied egg-rr 74.0%

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

      1. clear-num 74.0%

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

      2. div-inv 74.0%

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

      3. frac-2neg 74.0%

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

      4. sqrt-div 46.8%

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

   7. Applied egg-rr 46.8%

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

      1. distribute-neg-frac 46.8%

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

   9. Simplified 46.8%

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

   if -4.9999999999999996e248 < (*.f64 V l) < -2e-282

   1. Initial program 88.4%

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

      1. frac-2neg 88.4%

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

      2. sqrt-div 99.6%

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

      3. *-commutative 99.6%

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

      4. distribute-rgt-neg-in 99.6%

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

   3. Applied egg-rr 99.6%

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

   if -0.0 < (*.f64 V l) < 1.99999999999999996e296

   1. Initial program 80.4%

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

      1. *-commutative 80.4%

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

      2. sqrt-div 99.1%

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

      3. associate-*l/ 97.0%

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

   3. Applied egg-rr 97.0%

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

   if 1.99999999999999996e296 < (*.f64 V l)

   1. Initial program 47.0%

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

      1. pow1/2 47.0%

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

      2. clear-num 47.0%

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

      3. inv-pow 47.0%

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

      4. pow-pow 47.0%

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

      5. associate-/l* 83.8%

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

      6. metadata-eval 83.8%

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

   3. Applied egg-rr 83.8%

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

      1. div-inv 83.7%

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

      2. clear-num 83.8%

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

      3. metadata-eval 83.8%

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

      4. sqrt-pow2 83.7%

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

      5. inv-pow 83.7%

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

      6. un-div-inv 83.8%

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

   5. Applied egg-rr 83.8%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+248}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{-V}}{\sqrt{\frac{-A}{\ell}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-282}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{-V}}{\sqrt{\frac{-A}{\ell}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]

Alternative 2: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+123}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-146}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-312}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -5e+123)
   (/ c0 (* (sqrt (/ V A)) (sqrt l)))
   (if (<= (* V l) -1e-146)
     (* c0 (pow (/ (* V l) A) -0.5))
     (if (<= (* V l) 2e-312)
       (/ c0 (sqrt (* l (/ V A))))
       (if (<= (* V l) 2e+296)
         (/ (* c0 (sqrt A)) (sqrt (* V l)))
         (/ c0 (sqrt (* V (/ l A)))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+123) {
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	} else if ((V * l) <= -1e-146) {
		tmp = c0 * pow(((V * l) / A), -0.5);
	} else if ((V * l) <= 2e-312) {
		tmp = c0 / sqrt((l * (V / A)));
	} else if ((V * l) <= 2e+296) {
		tmp = (c0 * sqrt(A)) / sqrt((V * l));
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-5d+123)) then
        tmp = c0 / (sqrt((v / a)) * sqrt(l))
    else if ((v * l) <= (-1d-146)) then
        tmp = c0 * (((v * l) / a) ** (-0.5d0))
    else if ((v * l) <= 2d-312) then
        tmp = c0 / sqrt((l * (v / a)))
    else if ((v * l) <= 2d+296) then
        tmp = (c0 * sqrt(a)) / sqrt((v * l))
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+123) {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	} else if ((V * l) <= -1e-146) {
		tmp = c0 * Math.pow(((V * l) / A), -0.5);
	} else if ((V * l) <= 2e-312) {
		tmp = c0 / Math.sqrt((l * (V / A)));
	} else if ((V * l) <= 2e+296) {
		tmp = (c0 * Math.sqrt(A)) / Math.sqrt((V * l));
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -5e+123:
		tmp = c0 / (math.sqrt((V / A)) * math.sqrt(l))
	elif (V * l) <= -1e-146:
		tmp = c0 * math.pow(((V * l) / A), -0.5)
	elif (V * l) <= 2e-312:
		tmp = c0 / math.sqrt((l * (V / A)))
	elif (V * l) <= 2e+296:
		tmp = (c0 * math.sqrt(A)) / math.sqrt((V * l))
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -5e+123)
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * sqrt(l)));
	elseif (Float64(V * l) <= -1e-146)
		tmp = Float64(c0 * (Float64(Float64(V * l) / A) ^ -0.5));
	elseif (Float64(V * l) <= 2e-312)
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	elseif (Float64(V * l) <= 2e+296)
		tmp = Float64(Float64(c0 * sqrt(A)) / sqrt(Float64(V * l)));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -5e+123)
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	elseif ((V * l) <= -1e-146)
		tmp = c0 * (((V * l) / A) ^ -0.5);
	elseif ((V * l) <= 2e-312)
		tmp = c0 / sqrt((l * (V / A)));
	elseif ((V * l) <= 2e+296)
		tmp = (c0 * sqrt(A)) / sqrt((V * l));
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -5e+123], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-146], N[(c0 * N[Power[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e-312], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e+296], N[(N[(c0 * N[Sqrt[A], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -4.99999999999999974e123

   1. Initial program 61.8%

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

      1. pow1/2 61.8%

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

      2. clear-num 61.0%

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

      3. inv-pow 61.0%

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

      4. pow-pow 61.0%

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

      5. associate-/l* 73.6%

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

      6. metadata-eval 73.6%

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

   3. Applied egg-rr 73.6%

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

      1. div-inv 73.6%

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

      2. clear-num 73.6%

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

      3. metadata-eval 73.6%

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

      4. sqrt-pow2 73.6%

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

      5. inv-pow 73.6%

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

      6. un-div-inv 73.6%

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

   5. Applied egg-rr 73.6%

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

      1. clear-num 73.6%

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

      2. div-inv 73.6%

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

      3. associate-/r/ 73.6%

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

      4. sqrt-prod 35.4%

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

   7. Applied egg-rr 35.4%

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

   if -4.99999999999999974e123 < (*.f64 V l) < -1.00000000000000003e-146

   1. Initial program 93.5%

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

      1. pow1/2 93.5%

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

      2. clear-num 93.5%

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

      3. inv-pow 93.5%

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

      4. pow-pow 95.3%

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

      5. associate-/l* 80.7%

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

      6. metadata-eval 80.7%

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

   3. Applied egg-rr 80.7%

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

      1. associate-/l* 95.3%

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

   5. Simplified 95.3%

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

   if -1.00000000000000003e-146 < (*.f64 V l) < 2.0000000000019e-312

   1. Initial program 61.2%

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

      1. pow1/2 61.2%

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

      2. clear-num 61.2%

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

      3. inv-pow 61.2%

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

      4. pow-pow 61.3%

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

      5. associate-/l* 76.2%

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

      6. metadata-eval 76.2%

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

   3. Applied egg-rr 76.2%

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

      1. div-inv 76.3%

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

      2. clear-num 76.3%

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

      3. metadata-eval 76.3%

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

      4. sqrt-pow2 76.3%

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

      5. inv-pow 76.3%

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

      6. un-div-inv 76.4%

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

   5. Applied egg-rr 76.4%

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

      1. *-commutative 76.4%

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

      2. associate-*l/ 61.4%

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

      3. associate-*r/ 76.4%

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

   7. Simplified 76.4%

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

   if 2.0000000000019e-312 < (*.f64 V l) < 1.99999999999999996e296

   1. Initial program 80.3%

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

      1. *-commutative 80.3%

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

      2. sqrt-div 99.2%

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

      3. associate-*l/ 97.2%

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

   3. Applied egg-rr 97.2%

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

   if 1.99999999999999996e296 < (*.f64 V l)

   1. Initial program 47.0%

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

      1. pow1/2 47.0%

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

      2. clear-num 47.0%

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

      3. inv-pow 47.0%

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

      4. pow-pow 47.0%

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

      5. associate-/l* 83.8%

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

      6. metadata-eval 83.8%

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

   3. Applied egg-rr 83.8%

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

      1. div-inv 83.7%

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

      2. clear-num 83.8%

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

      3. metadata-eval 83.8%

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

      4. sqrt-pow2 83.7%

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

      5. inv-pow 83.7%

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

      6. un-div-inv 83.8%

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

   5. Applied egg-rr 83.8%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+123}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-146}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-312}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]

Alternative 3: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+123}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-146}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-312}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -5e+123)
   (* c0 (/ 1.0 (* (sqrt (/ V A)) (sqrt l))))
   (if (<= (* V l) -1e-146)
     (* c0 (pow (/ (* V l) A) -0.5))
     (if (<= (* V l) 2e-312)
       (/ c0 (sqrt (* l (/ V A))))
       (if (<= (* V l) 2e+296)
         (/ (* c0 (sqrt A)) (sqrt (* V l)))
         (/ c0 (sqrt (* V (/ l A)))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+123) {
		tmp = c0 * (1.0 / (sqrt((V / A)) * sqrt(l)));
	} else if ((V * l) <= -1e-146) {
		tmp = c0 * pow(((V * l) / A), -0.5);
	} else if ((V * l) <= 2e-312) {
		tmp = c0 / sqrt((l * (V / A)));
	} else if ((V * l) <= 2e+296) {
		tmp = (c0 * sqrt(A)) / sqrt((V * l));
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-5d+123)) then
        tmp = c0 * (1.0d0 / (sqrt((v / a)) * sqrt(l)))
    else if ((v * l) <= (-1d-146)) then
        tmp = c0 * (((v * l) / a) ** (-0.5d0))
    else if ((v * l) <= 2d-312) then
        tmp = c0 / sqrt((l * (v / a)))
    else if ((v * l) <= 2d+296) then
        tmp = (c0 * sqrt(a)) / sqrt((v * l))
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+123) {
		tmp = c0 * (1.0 / (Math.sqrt((V / A)) * Math.sqrt(l)));
	} else if ((V * l) <= -1e-146) {
		tmp = c0 * Math.pow(((V * l) / A), -0.5);
	} else if ((V * l) <= 2e-312) {
		tmp = c0 / Math.sqrt((l * (V / A)));
	} else if ((V * l) <= 2e+296) {
		tmp = (c0 * Math.sqrt(A)) / Math.sqrt((V * l));
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -5e+123:
		tmp = c0 * (1.0 / (math.sqrt((V / A)) * math.sqrt(l)))
	elif (V * l) <= -1e-146:
		tmp = c0 * math.pow(((V * l) / A), -0.5)
	elif (V * l) <= 2e-312:
		tmp = c0 / math.sqrt((l * (V / A)))
	elif (V * l) <= 2e+296:
		tmp = (c0 * math.sqrt(A)) / math.sqrt((V * l))
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -5e+123)
		tmp = Float64(c0 * Float64(1.0 / Float64(sqrt(Float64(V / A)) * sqrt(l))));
	elseif (Float64(V * l) <= -1e-146)
		tmp = Float64(c0 * (Float64(Float64(V * l) / A) ^ -0.5));
	elseif (Float64(V * l) <= 2e-312)
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	elseif (Float64(V * l) <= 2e+296)
		tmp = Float64(Float64(c0 * sqrt(A)) / sqrt(Float64(V * l)));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -5e+123)
		tmp = c0 * (1.0 / (sqrt((V / A)) * sqrt(l)));
	elseif ((V * l) <= -1e-146)
		tmp = c0 * (((V * l) / A) ^ -0.5);
	elseif ((V * l) <= 2e-312)
		tmp = c0 / sqrt((l * (V / A)));
	elseif ((V * l) <= 2e+296)
		tmp = (c0 * sqrt(A)) / sqrt((V * l));
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -5e+123], N[(c0 * N[(1.0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-146], N[(c0 * N[Power[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e-312], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e+296], N[(N[(c0 * N[Sqrt[A], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -4.99999999999999974e123

   1. Initial program 61.8%

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

      1. clear-num 61.0%

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

      2. sqrt-div 61.0%

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

      3. metadata-eval 61.0%

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

      4. associate-/l* 73.6%

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

   3. Applied egg-rr 73.6%

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

      1. associate-/l* 61.0%

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

      2. *-lft-identity 61.0%

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

      3. times-frac 73.6%

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

      4. /-rgt-identity 73.6%

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

   5. Simplified 73.6%

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

      1. clear-num 73.6%

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

      2. div-inv 73.6%

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

      3. associate-/r/ 73.6%

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

      4. sqrt-prod 35.4%

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

   7. Applied egg-rr 35.4%

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

   if -4.99999999999999974e123 < (*.f64 V l) < -1.00000000000000003e-146

   1. Initial program 93.5%

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

      1. pow1/2 93.5%

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

      2. clear-num 93.5%

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

      3. inv-pow 93.5%

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

      4. pow-pow 95.3%

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

      5. associate-/l* 80.7%

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

      6. metadata-eval 80.7%

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

   3. Applied egg-rr 80.7%

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

      1. associate-/l* 95.3%

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

   5. Simplified 95.3%

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

   if -1.00000000000000003e-146 < (*.f64 V l) < 2.0000000000019e-312

   1. Initial program 61.2%

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

      1. pow1/2 61.2%

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

      2. clear-num 61.2%

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

      3. inv-pow 61.2%

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

      4. pow-pow 61.3%

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

      5. associate-/l* 76.2%

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

      6. metadata-eval 76.2%

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

   3. Applied egg-rr 76.2%

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

      1. div-inv 76.3%

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

      2. clear-num 76.3%

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

      3. metadata-eval 76.3%

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

      4. sqrt-pow2 76.3%

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

      5. inv-pow 76.3%

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

      6. un-div-inv 76.4%

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

   5. Applied egg-rr 76.4%

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

      1. *-commutative 76.4%

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

      2. associate-*l/ 61.4%

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

      3. associate-*r/ 76.4%

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

   7. Simplified 76.4%

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

   if 2.0000000000019e-312 < (*.f64 V l) < 1.99999999999999996e296

   1. Initial program 80.3%

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

      1. *-commutative 80.3%

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

      2. sqrt-div 99.2%

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

      3. associate-*l/ 97.2%

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

   3. Applied egg-rr 97.2%

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

   if 1.99999999999999996e296 < (*.f64 V l)

   1. Initial program 47.0%

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

      1. pow1/2 47.0%

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

      2. clear-num 47.0%

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

      3. inv-pow 47.0%

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

      4. pow-pow 47.0%

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

      5. associate-/l* 83.8%

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

      6. metadata-eval 83.8%

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

   3. Applied egg-rr 83.8%

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

      1. div-inv 83.7%

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

      2. clear-num 83.8%

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

      3. metadata-eval 83.8%

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

      4. sqrt-pow2 83.7%

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

      5. inv-pow 83.7%

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

      6. un-div-inv 83.8%

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

   5. Applied egg-rr 83.8%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+123}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-146}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-312}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]

Alternative 4: 89.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+294}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-282}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-312}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -2e+294)
   (* c0 (/ 1.0 (* (sqrt (/ V A)) (sqrt l))))
   (if (<= (* V l) -2e-282)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 2e-312)
       (/ c0 (sqrt (* l (/ V A))))
       (if (<= (* V l) 2e+296)
         (/ (* c0 (sqrt A)) (sqrt (* V l)))
         (/ c0 (sqrt (* V (/ l A)))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+294) {
		tmp = c0 * (1.0 / (sqrt((V / A)) * sqrt(l)));
	} else if ((V * l) <= -2e-282) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 2e-312) {
		tmp = c0 / sqrt((l * (V / A)));
	} else if ((V * l) <= 2e+296) {
		tmp = (c0 * sqrt(A)) / sqrt((V * l));
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-2d+294)) then
        tmp = c0 * (1.0d0 / (sqrt((v / a)) * sqrt(l)))
    else if ((v * l) <= (-2d-282)) then
        tmp = c0 * (sqrt(-a) / sqrt((v * -l)))
    else if ((v * l) <= 2d-312) then
        tmp = c0 / sqrt((l * (v / a)))
    else if ((v * l) <= 2d+296) then
        tmp = (c0 * sqrt(a)) / sqrt((v * l))
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+294) {
		tmp = c0 * (1.0 / (Math.sqrt((V / A)) * Math.sqrt(l)));
	} else if ((V * l) <= -2e-282) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 2e-312) {
		tmp = c0 / Math.sqrt((l * (V / A)));
	} else if ((V * l) <= 2e+296) {
		tmp = (c0 * Math.sqrt(A)) / Math.sqrt((V * l));
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -2e+294:
		tmp = c0 * (1.0 / (math.sqrt((V / A)) * math.sqrt(l)))
	elif (V * l) <= -2e-282:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 2e-312:
		tmp = c0 / math.sqrt((l * (V / A)))
	elif (V * l) <= 2e+296:
		tmp = (c0 * math.sqrt(A)) / math.sqrt((V * l))
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -2e+294)
		tmp = Float64(c0 * Float64(1.0 / Float64(sqrt(Float64(V / A)) * sqrt(l))));
	elseif (Float64(V * l) <= -2e-282)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 2e-312)
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	elseif (Float64(V * l) <= 2e+296)
		tmp = Float64(Float64(c0 * sqrt(A)) / sqrt(Float64(V * l)));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -2e+294)
		tmp = c0 * (1.0 / (sqrt((V / A)) * sqrt(l)));
	elseif ((V * l) <= -2e-282)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 2e-312)
		tmp = c0 / sqrt((l * (V / A)));
	elseif ((V * l) <= 2e+296)
		tmp = (c0 * sqrt(A)) / sqrt((V * l));
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -2e+294], N[(c0 * N[(1.0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-282], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e-312], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e+296], N[(N[(c0 * N[Sqrt[A], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -2.00000000000000013e294

   1. Initial program 48.7%

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

      1. clear-num 48.7%

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

      2. sqrt-div 48.7%

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

      3. metadata-eval 48.7%

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

      4. associate-/l* 72.7%

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

   3. Applied egg-rr 72.7%

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

      1. associate-/l* 48.7%

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

      2. *-lft-identity 48.7%

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

      3. times-frac 72.7%

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

      4. /-rgt-identity 72.7%

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

   5. Simplified 72.7%

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

      1. clear-num 72.6%

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

      2. div-inv 72.6%

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

      3. associate-/r/ 72.6%

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

      4. sqrt-prod 29.2%

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

   7. Applied egg-rr 29.2%

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

   if -2.00000000000000013e294 < (*.f64 V l) < -2e-282

   1. Initial program 87.0%

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

      1. frac-2neg 87.0%

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

      2. sqrt-div 99.6%

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

      3. *-commutative 99.6%

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

      4. distribute-rgt-neg-in 99.6%

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

   3. Applied egg-rr 99.6%

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

   if -2e-282 < (*.f64 V l) < 2.0000000000019e-312

   1. Initial program 56.1%

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

      1. pow1/2 56.1%

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

      2. clear-num 56.1%

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

      3. inv-pow 56.1%

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

      4. pow-pow 56.1%

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

      5. associate-/l* 77.1%

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

      6. metadata-eval 77.1%

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

   3. Applied egg-rr 77.1%

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

      1. div-inv 77.1%

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

      2. clear-num 77.1%

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

      3. metadata-eval 77.1%

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

      4. sqrt-pow2 77.2%

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

      5. inv-pow 77.2%

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

      6. un-div-inv 77.2%

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

   5. Applied egg-rr 77.2%

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

      1. *-commutative 77.2%

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

      2. associate-*l/ 56.1%

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

      3. associate-*r/ 77.2%

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

   7. Simplified 77.2%

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

   if 2.0000000000019e-312 < (*.f64 V l) < 1.99999999999999996e296

   1. Initial program 80.3%

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

      1. *-commutative 80.3%

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

      2. sqrt-div 99.2%

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

      3. associate-*l/ 97.2%

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

   3. Applied egg-rr 97.2%

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

   if 1.99999999999999996e296 < (*.f64 V l)

   1. Initial program 47.0%

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

      1. pow1/2 47.0%

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

      2. clear-num 47.0%

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

      3. inv-pow 47.0%

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

      4. pow-pow 47.0%

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

      5. associate-/l* 83.8%

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

      6. metadata-eval 83.8%

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

   3. Applied egg-rr 83.8%

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

      1. div-inv 83.7%

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

      2. clear-num 83.8%

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

      3. metadata-eval 83.8%

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

      4. sqrt-pow2 83.7%

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

      5. inv-pow 83.7%

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

      6. un-div-inv 83.8%

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

   5. Applied egg-rr 83.8%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+294}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-282}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-312}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]

Alternative 5: 80.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6 \cdot 10^{-252}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= l 6e-252)
   (/ c0 (sqrt (* V (/ l A))))
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= 6e-252) {
		tmp = c0 / sqrt((V * (l / A)));
	} else {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= 6d-252) then
        tmp = c0 / sqrt((v * (l / a)))
    else
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= 6e-252) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if l <= 6e-252:
		tmp = c0 / math.sqrt((V * (l / A)))
	else:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (l <= 6e-252)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	else
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (l <= 6e-252)
		tmp = c0 / sqrt((V * (l / A)));
	else
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[l, 6e-252], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.9999999999999999e-252

   1. Initial program 77.8%

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

      1. pow1/2 77.8%

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

      2. clear-num 77.0%

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

      3. inv-pow 77.0%

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

      4. pow-pow 77.6%

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

      5. associate-/l* 78.8%

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

      6. metadata-eval 78.8%

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

   3. Applied egg-rr 78.8%

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

      1. div-inv 77.8%

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

      2. clear-num 78.3%

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

      3. metadata-eval 78.3%

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

      4. sqrt-pow2 78.3%

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

      5. inv-pow 78.3%

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

      6. un-div-inv 78.3%

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

   5. Applied egg-rr 78.3%

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

   if 5.9999999999999999e-252 < l

   1. Initial program 69.7%

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

      1. associate-/r* 75.1%

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

      2. sqrt-div 82.9%

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

   3. Applied egg-rr 82.9%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6 \cdot 10^{-252}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 6: 80.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.4 \cdot 10^{-252}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= l 4.4e-252)
   (/ c0 (sqrt (* V (/ l A))))
   (/ c0 (* (sqrt (/ V A)) (sqrt l)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= 4.4e-252) {
		tmp = c0 / sqrt((V * (l / A)));
	} else {
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= 4.4d-252) then
        tmp = c0 / sqrt((v * (l / a)))
    else
        tmp = c0 / (sqrt((v / a)) * sqrt(l))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= 4.4e-252) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if l <= 4.4e-252:
		tmp = c0 / math.sqrt((V * (l / A)))
	else:
		tmp = c0 / (math.sqrt((V / A)) * math.sqrt(l))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (l <= 4.4e-252)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * sqrt(l)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (l <= 4.4e-252)
		tmp = c0 / sqrt((V * (l / A)));
	else
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[l, 4.4e-252], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.3999999999999998e-252

   1. Initial program 77.8%

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

      1. pow1/2 77.8%

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

      2. clear-num 77.0%

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

      3. inv-pow 77.0%

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

      4. pow-pow 77.6%

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

      5. associate-/l* 78.8%

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

      6. metadata-eval 78.8%

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

   3. Applied egg-rr 78.8%

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

      1. div-inv 77.8%

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

      2. clear-num 78.3%

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

      3. metadata-eval 78.3%

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

      4. sqrt-pow2 78.3%

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

      5. inv-pow 78.3%

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

      6. un-div-inv 78.3%

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

   5. Applied egg-rr 78.3%

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

   if 4.3999999999999998e-252 < l

   1. Initial program 69.7%

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

      1. pow1/2 69.7%

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

      2. clear-num 69.7%

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

      3. inv-pow 69.7%

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

      4. pow-pow 70.5%

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

      5. associate-/l* 69.0%

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

      6. metadata-eval 69.0%

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

   3. Applied egg-rr 69.0%

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

      1. div-inv 69.0%

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

      2. clear-num 69.9%

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

      3. metadata-eval 69.9%

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

      4. sqrt-pow2 69.8%

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

      5. inv-pow 69.8%

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

      6. un-div-inv 69.9%

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

   5. Applied egg-rr 69.9%

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

      1. clear-num 69.0%

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

      2. div-inv 69.0%

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

      3. associate-/r/ 74.7%

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

      4. sqrt-prod 82.5%

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

   7. Applied egg-rr 82.5%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.4 \cdot 10^{-252}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 7: 79.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{+206}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{V \cdot \frac{\ell}{A}}}{c0}}\\ \end{array} \end{array} \]

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 4e+206)
       (* c0 (sqrt t_0))
       (/ 1.0 (/ (sqrt (* V (/ l A))) c0))))))

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 <= 4e+206) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = 1.0 / (sqrt((V * (l / A))) / c0);
	}
	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 <= 4d+206) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = 1.0d0 / (sqrt((v * (l / a))) / c0)
    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 <= 4e+206) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = 1.0 / (Math.sqrt((V * (l / A))) / c0);
	}
	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 <= 4e+206:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = 1.0 / (math.sqrt((V * (l / A))) / c0)
	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 <= 4e+206)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(V * Float64(l / A))) / c0));
	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 <= 4e+206)
		tmp = c0 * sqrt(t_0);
	else
		tmp = 1.0 / (sqrt((V * (l / A))) / c0);
	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, 4e+206], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 45.6%

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

      1. add-sqr-sqrt 45.6%

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

      2. sqrt-unprod 45.6%

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

      3. pow1/2 45.6%

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

      4. *-commutative 45.6%

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

      5. *-commutative 45.6%

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

      6. swap-sqr 45.0%

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

      7. add-sqr-sqrt 45.0%

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

   3. Applied egg-rr 45.0%

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

      1. unpow1/2 45.0%

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

   5. Simplified 45.0%

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

      1. sqrt-prod 45.0%

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

      2. sqrt-prod 20.8%

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

      3. add-sqr-sqrt 45.6%

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

      4. associate-/r* 66.9%

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

   7. Applied egg-rr 66.9%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.0000000000000002e206

   1. Initial program 98.2%

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

   if 4.0000000000000002e206 < (/.f64 A (*.f64 V l))

   1. Initial program 53.5%

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

      1. pow1/2 53.5%

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

      2. clear-num 53.5%

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

      3. inv-pow 53.5%

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

      4. pow-pow 55.8%

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

      5. associate-/l* 63.3%

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

      6. metadata-eval 63.3%

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

   3. Applied egg-rr 63.3%

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

      1. div-inv 63.3%

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

      2. clear-num 63.3%

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

      3. metadata-eval 63.3%

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

      4. sqrt-pow2 63.4%

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

      5. inv-pow 63.4%

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

      6. un-div-inv 63.4%

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

      7. clear-num 63.4%

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

   5. Applied egg-rr 63.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{V \cdot \frac{\ell}{A}}}{c0}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 4 \cdot 10^{+206}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{V \cdot \frac{\ell}{A}}}{c0}}\\ \end{array} \]

Alternative 8: 79.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 10^{+296}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \end{array} \end{array} \]

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 1e+296)))
     (* 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 <= 1e+296)) {
		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 <= 1d+296))) 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 <= 1e+296)) {
		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 <= 1e+296):
		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 <= 1e+296))
		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 <= 1e+296)))
		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, 1e+296]], $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 9.99999999999999981e295 < (/.f64 A (*.f64 V l))

   1. Initial program 44.1%

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

      1. add-sqr-sqrt 32.8%

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

      2. sqrt-unprod 32.9%

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

      3. pow1/2 32.9%

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

      4. *-commutative 32.9%

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

      5. *-commutative 32.9%

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

      6. swap-sqr 32.4%

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

      7. add-sqr-sqrt 32.4%

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

   3. Applied egg-rr 32.4%

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

      1. unpow1/2 32.4%

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

   5. Simplified 32.4%

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

      1. sqrt-prod 32.4%

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

      2. sqrt-prod 20.7%

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

      3. add-sqr-sqrt 44.1%

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

      4. associate-/r* 61.2%

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

   7. Applied egg-rr 61.2%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 9.99999999999999981e295

   1. Initial program 98.2%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 10^{+296}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]

Alternative 9: 78.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+179}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 2e+179) (* c0 (sqrt t_0)) (/ c0 (sqrt (* V (/ l A))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 2e+179) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 2d+179) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 2e+179) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 2e+179:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 2e+179)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 2e+179)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+179], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 45.6%

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

      1. add-sqr-sqrt 45.6%

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

      2. sqrt-unprod 45.6%

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

      3. pow1/2 45.6%

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

      4. *-commutative 45.6%

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

      5. *-commutative 45.6%

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

      6. swap-sqr 45.0%

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

      7. add-sqr-sqrt 45.0%

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

   3. Applied egg-rr 45.0%

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

      1. unpow1/2 45.0%

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

   5. Simplified 45.0%

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

      1. sqrt-prod 45.0%

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

      2. sqrt-prod 20.8%

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

      3. add-sqr-sqrt 45.6%

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

      4. associate-/r* 66.9%

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

   7. Applied egg-rr 66.9%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 1.99999999999999996e179

   1. Initial program 98.1%

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

   if 1.99999999999999996e179 < (/.f64 A (*.f64 V l))

   1. Initial program 57.6%

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

      1. pow1/2 57.6%

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

      2. clear-num 57.6%

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

      3. inv-pow 57.6%

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

      4. pow-pow 59.7%

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

      5. associate-/l* 66.5%

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

      6. metadata-eval 66.5%

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

   3. Applied egg-rr 66.5%

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

      1. div-inv 66.5%

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

      2. clear-num 66.6%

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

      3. metadata-eval 66.6%

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

      4. sqrt-pow2 66.6%

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

      5. inv-pow 66.6%

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

      6. un-div-inv 66.6%

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

   5. Applied egg-rr 66.6%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+179}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]

Alternative 10: 73.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.4%

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

2. Final simplification 74.4%

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

Reproduce

herbie shell --seed 2023285 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))