Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.3% → 89.6%

Time: 13.4s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.3% 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: 89.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+249}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \frac{1}{\frac{\sqrt{V \cdot \left(-\ell\right)}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+293}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{\ell} \cdot \sqrt{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -2e+249)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* V l) -4e-319)
     (* c0 (/ 1.0 (/ (sqrt (* V (- l))) (sqrt (- A)))))
     (if (<= (* V l) 0.0)
       (* c0 (sqrt (/ (/ A V) l)))
       (if (<= (* V l) 2e+293)
         (* c0 (* (sqrt A) (pow (* V l) -0.5)))
         (/ (* c0 (sqrt A)) (* (sqrt l) (sqrt V))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+249) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= -4e-319) {
		tmp = c0 * (1.0 / (sqrt((V * -l)) / sqrt(-A)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if ((V * l) <= 2e+293) {
		tmp = c0 * (sqrt(A) * pow((V * l), -0.5));
	} else {
		tmp = (c0 * sqrt(A)) / (sqrt(l) * sqrt(V));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-2d+249)) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= (-4d-319)) then
        tmp = c0 * (1.0d0 / (sqrt((v * -l)) / sqrt(-a)))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if ((v * l) <= 2d+293) then
        tmp = c0 * (sqrt(a) * ((v * l) ** (-0.5d0)))
    else
        tmp = (c0 * sqrt(a)) / (sqrt(l) * sqrt(v))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+249) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= -4e-319) {
		tmp = c0 * (1.0 / (Math.sqrt((V * -l)) / Math.sqrt(-A)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if ((V * l) <= 2e+293) {
		tmp = c0 * (Math.sqrt(A) * Math.pow((V * l), -0.5));
	} else {
		tmp = (c0 * Math.sqrt(A)) / (Math.sqrt(l) * Math.sqrt(V));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -2e+249:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= -4e-319:
		tmp = c0 * (1.0 / (math.sqrt((V * -l)) / math.sqrt(-A)))
	elif (V * l) <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif (V * l) <= 2e+293:
		tmp = c0 * (math.sqrt(A) * math.pow((V * l), -0.5))
	else:
		tmp = (c0 * math.sqrt(A)) / (math.sqrt(l) * math.sqrt(V))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -2e+249)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= -4e-319)
		tmp = Float64(c0 * Float64(1.0 / Float64(sqrt(Float64(V * Float64(-l))) / sqrt(Float64(-A)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (Float64(V * l) <= 2e+293)
		tmp = Float64(c0 * Float64(sqrt(A) * (Float64(V * l) ^ -0.5)));
	else
		tmp = Float64(Float64(c0 * sqrt(A)) / Float64(sqrt(l) * sqrt(V)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -2e+249)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= -4e-319)
		tmp = c0 * (1.0 / (sqrt((V * -l)) / sqrt(-A)));
	elseif ((V * l) <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif ((V * l) <= 2e+293)
		tmp = c0 * (sqrt(A) * ((V * l) ^ -0.5));
	else
		tmp = (c0 * sqrt(A)) / (sqrt(l) * sqrt(V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -2e+249], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -4e-319], N[(c0 * N[(1.0 / N[(N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e+293], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * N[Sqrt[A], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[V], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -1.9999999999999998e249

   1. Initial program 54.5%

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

      1. associate-/r* 81.8%

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

      2. sqrt-div 40.3%

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

   3. Applied egg-rr 40.3%

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

   if -1.9999999999999998e249 < (*.f64 V l) < -4.0000049e-319

   1. Initial program 86.2%

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

      1. clear-num 86.2%

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

      2. sqrt-div 86.6%

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

      3. metadata-eval 86.6%

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

      4. associate-/l* 76.4%

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

   3. Applied egg-rr 76.4%

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

      1. associate-/l* 86.6%

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

   5. Simplified 86.6%

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

      1. frac-2neg 86.6%

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

      2. sqrt-div 98.9%

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

      3. distribute-rgt-neg-in 98.9%

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

   7. Applied egg-rr 98.9%

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

   if -4.0000049e-319 < (*.f64 V l) < -0.0

   1. Initial program 52.1%

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

      1. associate-/r* 69.2%

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

      2. div-inv 69.2%

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

   3. Applied egg-rr 69.2%

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

      1. un-div-inv 69.2%

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

   5. Applied egg-rr 69.2%

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

   if -0.0 < (*.f64 V l) < 1.9999999999999998e293

   1. Initial program 88.7%

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

      1. div-inv 88.6%

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

      2. sqrt-prod 99.3%

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

      3. pow1/2 99.3%

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

      4. inv-pow 99.3%

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

      5. pow-pow 99.4%

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

      6. metadata-eval 99.4%

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

   3. Applied egg-rr 99.4%

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

   if 1.9999999999999998e293 < (*.f64 V l)

   1. Initial program 12.1%

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

      1. *-commutative 12.1%

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

      2. sqrt-div 12.1%

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

      3. associate-*l/ 11.9%

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

   3. Applied egg-rr 11.9%

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

      1. sqrt-prod 49.6%

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

   5. Applied egg-rr 49.6%

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

      1. *-commutative 49.6%

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

   7. Simplified 49.6%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+249}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \frac{1}{\frac{\sqrt{V \cdot \left(-\ell\right)}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+293}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{\ell} \cdot \sqrt{V}}\\ \end{array} \]

Alternative 2: 89.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+249}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \frac{1}{\frac{\sqrt{V \cdot \left(-\ell\right)}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+293}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell}} \cdot \frac{c0}{\sqrt{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -2e+249)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* V l) -4e-319)
     (* c0 (/ 1.0 (/ (sqrt (* V (- l))) (sqrt (- A)))))
     (if (<= (* V l) 0.0)
       (* c0 (sqrt (/ (/ A V) l)))
       (if (<= (* V l) 2e+293)
         (* c0 (* (sqrt A) (pow (* V l) -0.5)))
         (* (/ (sqrt A) (sqrt l)) (/ c0 (sqrt V))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+249) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= -4e-319) {
		tmp = c0 * (1.0 / (sqrt((V * -l)) / sqrt(-A)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if ((V * l) <= 2e+293) {
		tmp = c0 * (sqrt(A) * pow((V * l), -0.5));
	} else {
		tmp = (sqrt(A) / sqrt(l)) * (c0 / sqrt(V));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-2d+249)) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= (-4d-319)) then
        tmp = c0 * (1.0d0 / (sqrt((v * -l)) / sqrt(-a)))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if ((v * l) <= 2d+293) then
        tmp = c0 * (sqrt(a) * ((v * l) ** (-0.5d0)))
    else
        tmp = (sqrt(a) / sqrt(l)) * (c0 / sqrt(v))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+249) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= -4e-319) {
		tmp = c0 * (1.0 / (Math.sqrt((V * -l)) / Math.sqrt(-A)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if ((V * l) <= 2e+293) {
		tmp = c0 * (Math.sqrt(A) * Math.pow((V * l), -0.5));
	} else {
		tmp = (Math.sqrt(A) / Math.sqrt(l)) * (c0 / Math.sqrt(V));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -2e+249:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= -4e-319:
		tmp = c0 * (1.0 / (math.sqrt((V * -l)) / math.sqrt(-A)))
	elif (V * l) <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif (V * l) <= 2e+293:
		tmp = c0 * (math.sqrt(A) * math.pow((V * l), -0.5))
	else:
		tmp = (math.sqrt(A) / math.sqrt(l)) * (c0 / math.sqrt(V))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -2e+249)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= -4e-319)
		tmp = Float64(c0 * Float64(1.0 / Float64(sqrt(Float64(V * Float64(-l))) / sqrt(Float64(-A)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (Float64(V * l) <= 2e+293)
		tmp = Float64(c0 * Float64(sqrt(A) * (Float64(V * l) ^ -0.5)));
	else
		tmp = Float64(Float64(sqrt(A) / sqrt(l)) * Float64(c0 / sqrt(V)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -2e+249)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= -4e-319)
		tmp = c0 * (1.0 / (sqrt((V * -l)) / sqrt(-A)));
	elseif ((V * l) <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif ((V * l) <= 2e+293)
		tmp = c0 * (sqrt(A) * ((V * l) ^ -0.5));
	else
		tmp = (sqrt(A) / sqrt(l)) * (c0 / sqrt(V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -2e+249], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -4e-319], N[(c0 * N[(1.0 / N[(N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e+293], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(c0 / N[Sqrt[V], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -1.9999999999999998e249

   1. Initial program 54.5%

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

      1. associate-/r* 81.8%

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

      2. sqrt-div 40.3%

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

   3. Applied egg-rr 40.3%

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

   if -1.9999999999999998e249 < (*.f64 V l) < -4.0000049e-319

   1. Initial program 86.2%

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

      1. clear-num 86.2%

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

      2. sqrt-div 86.6%

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

      3. metadata-eval 86.6%

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

      4. associate-/l* 76.4%

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

   3. Applied egg-rr 76.4%

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

      1. associate-/l* 86.6%

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

   5. Simplified 86.6%

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

      1. frac-2neg 86.6%

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

      2. sqrt-div 98.9%

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

      3. distribute-rgt-neg-in 98.9%

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

   7. Applied egg-rr 98.9%

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

   if -4.0000049e-319 < (*.f64 V l) < -0.0

   1. Initial program 52.1%

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

      1. associate-/r* 69.2%

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

      2. div-inv 69.2%

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

   3. Applied egg-rr 69.2%

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

      1. un-div-inv 69.2%

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

   5. Applied egg-rr 69.2%

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

   if -0.0 < (*.f64 V l) < 1.9999999999999998e293

   1. Initial program 88.7%

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

      1. div-inv 88.6%

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

      2. sqrt-prod 99.3%

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

      3. pow1/2 99.3%

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

      4. inv-pow 99.3%

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

      5. pow-pow 99.4%

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

      6. metadata-eval 99.4%

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

   3. Applied egg-rr 99.4%

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

   if 1.9999999999999998e293 < (*.f64 V l)

   1. Initial program 12.1%

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

      1. pow1/2 12.1%

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

      2. clear-num 12.1%

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

      3. inv-pow 12.1%

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

      4. pow-pow 12.1%

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

      5. associate-/l* 25.8%

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

      6. metadata-eval 25.8%

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

   3. Applied egg-rr 25.8%

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

      1. associate-/l* 12.1%

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

   5. Simplified 12.1%

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

      1. metadata-eval 12.1%

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

      2. sqrt-pow2 12.1%

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

      3. inv-pow 12.1%

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

      4. un-div-inv 12.1%

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

      5. sqrt-div 12.1%

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

      6. associate-/l* 11.9%

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

      7. sqrt-prod 49.6%

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

      8. *-commutative 49.6%

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

      9. associate-/r* 49.4%

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

      10. *-commutative 49.4%

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

   7. Applied egg-rr 49.4%

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

      1. associate-/r* 49.6%

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

      2. times-frac 49.5%

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

      3. *-commutative 49.5%

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

   9. Simplified 49.5%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+249}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \frac{1}{\frac{\sqrt{V \cdot \left(-\ell\right)}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+293}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell}} \cdot \frac{c0}{\sqrt{V}}\\ \end{array} \]

Alternative 3: 90.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\frac{\sqrt{-A}}{\sqrt{-V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -5e-310)
   (* c0 (/ (/ (sqrt (- A)) (sqrt (- V))) (sqrt l)))
   (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = c0 * ((sqrt(-A) / sqrt(-V)) / sqrt(l));
	} else {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (a <= (-5d-310)) then
        tmp = c0 * ((sqrt(-a) / sqrt(-v)) / sqrt(l))
    else
        tmp = c0 * (sqrt(a) * sqrt(((1.0d0 / v) / l)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = c0 * ((Math.sqrt(-A) / Math.sqrt(-V)) / Math.sqrt(l));
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -5e-310:
		tmp = c0 * ((math.sqrt(-A) / math.sqrt(-V)) / math.sqrt(l))
	else:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -5e-310)
		tmp = Float64(c0 * Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(-V))) / sqrt(l)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (A <= -5e-310)
		tmp = c0 * ((sqrt(-A) / sqrt(-V)) / sqrt(l));
	else
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -5e-310], N[(c0 * N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -4.999999999999985e-310

   1. Initial program 78.3%

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

      1. associate-/r* 78.6%

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

      2. sqrt-div 41.8%

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

   3. Applied egg-rr 41.8%

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

      1. frac-2neg 41.8%

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

      2. sqrt-div 49.0%

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

   5. Applied egg-rr 49.0%

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

   if -4.999999999999985e-310 < A

   1. Initial program 73.9%

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

      1. pow1/2 73.9%

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

      2. clear-num 73.5%

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

      3. inv-pow 73.5%

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

      4. pow-pow 74.2%

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

      5. associate-/l* 70.2%

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

      6. metadata-eval 70.2%

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

   3. Applied egg-rr 70.2%

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

      1. associate-/l* 74.2%

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

   5. Simplified 74.2%

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

      1. add-sqr-sqrt 74.0%

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

      2. sqrt-unprod 73.5%

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

      3. pow-prod-up 73.5%

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

      4. metadata-eval 73.5%

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

      5. inv-pow 73.5%

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

      6. clear-num 73.9%

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

      7. div-inv 73.9%

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

      8. sqrt-prod 81.7%

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

      9. associate-/r* 82.7%

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

   7. Applied egg-rr 82.7%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\frac{\sqrt{-A}}{\sqrt{-V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \]

Alternative 4: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+249}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -2e+249)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* V l) -4e-319)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 0.0)
       (* c0 (sqrt (/ (/ A V) l)))
       (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+249) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= -4e-319) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-2d+249)) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= (-4d-319)) then
        tmp = c0 * (sqrt(-a) / sqrt((v * -l)))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = c0 * (sqrt(a) * sqrt(((1.0d0 / v) / l)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+249) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= -4e-319) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -2e+249:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= -4e-319:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -2e+249)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= -4e-319)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -2e+249)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= -4e-319)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -2e+249], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -4e-319], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -1.9999999999999998e249

   1. Initial program 54.5%

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

      1. associate-/r* 81.8%

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

      2. sqrt-div 40.3%

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

   3. Applied egg-rr 40.3%

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

   if -1.9999999999999998e249 < (*.f64 V l) < -4.0000049e-319

   1. Initial program 86.2%

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

      1. frac-2neg 86.2%

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

      2. sqrt-div 98.9%

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

      3. *-commutative 98.9%

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

      4. distribute-rgt-neg-in 98.9%

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

   3. Applied egg-rr 98.9%

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

   if -4.0000049e-319 < (*.f64 V l) < -0.0

   1. Initial program 52.1%

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

      1. associate-/r* 69.2%

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

      2. div-inv 69.2%

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

   3. Applied egg-rr 69.2%

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

      1. un-div-inv 69.2%

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

   5. Applied egg-rr 69.2%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 78.4%

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

      1. pow1/2 78.3%

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

      2. clear-num 77.8%

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

      3. inv-pow 77.8%

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

      4. pow-pow 78.7%

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

      5. associate-/l* 72.2%

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

      6. metadata-eval 72.2%

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

   3. Applied egg-rr 72.2%

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

      1. associate-/l* 78.7%

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

   5. Simplified 78.7%

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

      1. add-sqr-sqrt 78.4%

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

      2. sqrt-unprod 77.9%

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

      3. pow-prod-up 77.9%

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

      4. metadata-eval 77.9%

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

      5. inv-pow 77.9%

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

      6. clear-num 78.4%

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

      7. div-inv 78.3%

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

      8. sqrt-prod 87.6%

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

      9. associate-/r* 88.8%

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

   7. Applied egg-rr 88.8%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+249}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \]

Alternative 5: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+249}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \frac{1}{\frac{\sqrt{V \cdot \left(-\ell\right)}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -2e+249)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* V l) -4e-319)
     (* c0 (/ 1.0 (/ (sqrt (* V (- l))) (sqrt (- A)))))
     (if (<= (* V l) 0.0)
       (* c0 (sqrt (/ (/ A V) l)))
       (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+249) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= -4e-319) {
		tmp = c0 * (1.0 / (sqrt((V * -l)) / sqrt(-A)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-2d+249)) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= (-4d-319)) then
        tmp = c0 * (1.0d0 / (sqrt((v * -l)) / sqrt(-a)))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = c0 * (sqrt(a) * sqrt(((1.0d0 / v) / l)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+249) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= -4e-319) {
		tmp = c0 * (1.0 / (Math.sqrt((V * -l)) / Math.sqrt(-A)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -2e+249:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= -4e-319:
		tmp = c0 * (1.0 / (math.sqrt((V * -l)) / math.sqrt(-A)))
	elif (V * l) <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -2e+249)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= -4e-319)
		tmp = Float64(c0 * Float64(1.0 / Float64(sqrt(Float64(V * Float64(-l))) / sqrt(Float64(-A)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -2e+249)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= -4e-319)
		tmp = c0 * (1.0 / (sqrt((V * -l)) / sqrt(-A)));
	elseif ((V * l) <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -2e+249], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -4e-319], N[(c0 * N[(1.0 / N[(N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -1.9999999999999998e249

   1. Initial program 54.5%

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

      1. associate-/r* 81.8%

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

      2. sqrt-div 40.3%

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

   3. Applied egg-rr 40.3%

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

   if -1.9999999999999998e249 < (*.f64 V l) < -4.0000049e-319

   1. Initial program 86.2%

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

      1. clear-num 86.2%

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

      2. sqrt-div 86.6%

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

      3. metadata-eval 86.6%

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

      4. associate-/l* 76.4%

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

   3. Applied egg-rr 76.4%

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

      1. associate-/l* 86.6%

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

   5. Simplified 86.6%

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

      1. frac-2neg 86.6%

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

      2. sqrt-div 98.9%

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

      3. distribute-rgt-neg-in 98.9%

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

   7. Applied egg-rr 98.9%

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

   if -4.0000049e-319 < (*.f64 V l) < -0.0

   1. Initial program 52.1%

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

      1. associate-/r* 69.2%

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

      2. div-inv 69.2%

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

   3. Applied egg-rr 69.2%

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

      1. un-div-inv 69.2%

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

   5. Applied egg-rr 69.2%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 78.4%

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

      1. pow1/2 78.3%

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

      2. clear-num 77.8%

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

      3. inv-pow 77.8%

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

      4. pow-pow 78.7%

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

      5. associate-/l* 72.2%

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

      6. metadata-eval 72.2%

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

   3. Applied egg-rr 72.2%

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

      1. associate-/l* 78.7%

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

   5. Simplified 78.7%

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

      1. add-sqr-sqrt 78.4%

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

      2. sqrt-unprod 77.9%

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

      3. pow-prod-up 77.9%

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

      4. metadata-eval 77.9%

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

      5. inv-pow 77.9%

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

      6. clear-num 78.4%

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

      7. div-inv 78.3%

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

      8. sqrt-prod 87.6%

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

      9. associate-/r* 88.8%

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

   7. Applied egg-rr 88.8%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+249}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \frac{1}{\frac{\sqrt{V \cdot \left(-\ell\right)}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \]

Alternative 6: 81.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{+281}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

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)) (sqrt l)))
     (if (<= t_0 4e+281) (* 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)) / sqrt(l));
	} else if (t_0 <= 4e+281) {
		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)) / sqrt(l))
    else if (t_0 <= 4d+281) 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)) / Math.sqrt(l));
	} else if (t_0 <= 4e+281) {
		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)) / math.sqrt(l))
	elif t_0 <= 4e+281:
		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 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (t_0 <= 4e+281)
		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)) / sqrt(l));
	elseif (t_0 <= 4e+281)
		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[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+281], 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 35.4%

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

      1. associate-/r* 51.9%

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

      2. sqrt-div 34.8%

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

   3. Applied egg-rr 34.8%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.0000000000000001e281

   1. Initial program 99.3%

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

   if 4.0000000000000001e281 < (/.f64 A (*.f64 V l))

   1. Initial program 50.8%

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

      1. pow1/2 50.8%

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

      2. clear-num 50.8%

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

      3. inv-pow 50.8%

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

      4. pow-pow 53.3%

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

      5. associate-/l* 61.6%

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

      6. metadata-eval 61.6%

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

   3. Applied egg-rr 61.6%

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

      1. associate-/l* 53.3%

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

   5. Simplified 53.3%

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

      1. metadata-eval 53.3%

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

      2. sqrt-pow2 53.3%

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

      3. inv-pow 53.3%

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

      4. un-div-inv 53.3%

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

      5. associate-*r/ 61.6%

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

   7. Applied egg-rr 61.6%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 4 \cdot 10^{+281}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]

Alternative 7: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -5e-310)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (a <= (-5d-310)) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else
        tmp = c0 * (sqrt(a) * sqrt(((1.0d0 / v) / l)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -5e-310:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	else:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -5e-310)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (A <= -5e-310)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	else
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -5e-310], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -4.999999999999985e-310

   1. Initial program 78.3%

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

      1. associate-/r* 78.6%

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

      2. sqrt-div 41.8%

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

   3. Applied egg-rr 41.8%

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

   if -4.999999999999985e-310 < A

   1. Initial program 73.9%

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

      1. pow1/2 73.9%

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

      2. clear-num 73.5%

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

      3. inv-pow 73.5%

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

      4. pow-pow 74.2%

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

      5. associate-/l* 70.2%

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

      6. metadata-eval 70.2%

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

   3. Applied egg-rr 70.2%

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

      1. associate-/l* 74.2%

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

   5. Simplified 74.2%

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

      1. add-sqr-sqrt 74.0%

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

      2. sqrt-unprod 73.5%

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

      3. pow-prod-up 73.5%

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

      4. metadata-eval 73.5%

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

      5. inv-pow 73.5%

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

      6. clear-num 73.9%

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

      7. div-inv 73.9%

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

      8. sqrt-prod 81.7%

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

      9. associate-/r* 82.7%

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

   7. Applied egg-rr 82.7%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \]

Alternative 8: 84.2% accurate, 0.5× speedup

Math

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

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -5e-310)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (* c0 (* (sqrt A) (pow (* V l) -0.5)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else {
		tmp = c0 * (sqrt(A) * pow((V * l), -0.5));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (a <= (-5d-310)) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else
        tmp = c0 * (sqrt(a) * ((v * l) ** (-0.5d0)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.pow((V * l), -0.5));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -5e-310:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	else:
		tmp = c0 * (math.sqrt(A) * math.pow((V * l), -0.5))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -5e-310)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * (Float64(V * l) ^ -0.5)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (A <= -5e-310)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	else
		tmp = c0 * (sqrt(A) * ((V * l) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -5e-310], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -4.999999999999985e-310

   1. Initial program 78.3%

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

      1. associate-/r* 78.6%

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

      2. sqrt-div 41.8%

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

   3. Applied egg-rr 41.8%

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

   if -4.999999999999985e-310 < A

   1. Initial program 73.9%

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

      1. div-inv 73.9%

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

      2. sqrt-prod 81.7%

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

      3. pow1/2 81.7%

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

      4. inv-pow 81.7%

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

      5. pow-pow 81.8%

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

      6. metadata-eval 81.8%

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

   3. Applied egg-rr 81.8%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.0%

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

Alternative 9: 83.4% accurate, 0.5× speedup

Math

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

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= l -5e-310)
   (* (sqrt A) (/ c0 (sqrt (* V l))))
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -5e-310) {
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	} else {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-5d-310)) then
        tmp = sqrt(a) * (c0 / sqrt((v * l)))
    else
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -5e-310) {
		tmp = Math.sqrt(A) * (c0 / Math.sqrt((V * l)));
	} else {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if l <= -5e-310:
		tmp = math.sqrt(A) * (c0 / math.sqrt((V * l)))
	else:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(sqrt(A) * Float64(c0 / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (l <= -5e-310)
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	else
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[l, -5e-310], N[(N[Sqrt[A], $MachinePrecision] * N[(c0 / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 78.0%

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

      1. sqrt-div 39.3%

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

      2. associate-*r/ 38.5%

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

   3. Applied egg-rr 38.5%

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

      1. associate-*l/ 37.3%

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

   5. Simplified 37.3%

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

   if -4.999999999999985e-310 < l

   1. Initial program 74.4%

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

      1. associate-/r* 72.7%

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

      2. sqrt-div 82.3%

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

   3. Applied egg-rr 82.3%

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

4. Final simplification 59.8%

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

Alternative 10: 84.2% accurate, 0.5× speedup

Math

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

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -5e-310)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (* c0 (/ (sqrt A) (sqrt (* V l))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (a <= (-5d-310)) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -5e-310:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -5e-310)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (A <= -5e-310)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	else
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -5e-310], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -4.999999999999985e-310

   1. Initial program 78.3%

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

      1. associate-/r* 78.6%

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

      2. sqrt-div 41.8%

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

   3. Applied egg-rr 41.8%

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

   if -4.999999999999985e-310 < A

   1. Initial program 73.9%

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

      1. sqrt-div 81.7%

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

      2. associate-*r/ 80.3%

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

   3. Applied egg-rr 80.3%

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

      1. *-commutative 80.3%

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

      2. associate-*l/ 81.7%

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

   5. Simplified 81.7%

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

4. Final simplification 61.0%

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

Alternative 11: 79.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 5 \cdot 10^{+274}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \end{array} \end{array} \]

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 5e+274)))
     (* 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 <= 5e+274)) {
		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 <= 5d+274))) 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 <= 5e+274)) {
		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 <= 5e+274):
		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 <= 5e+274))
		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 <= 5e+274)))
		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, 5e+274]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 4.9999999999999998e274 < (/.f64 A (*.f64 V l))

   1. Initial program 43.6%

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

      1. associate-/r* 56.3%

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

      2. div-inv 56.3%

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

   3. Applied egg-rr 56.3%

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

      1. un-div-inv 56.3%

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

   5. Applied egg-rr 56.3%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.9999999999999998e274

   1. Initial program 99.3%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+274}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]

Alternative 12: 79.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{+281}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

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 4e+281) (* 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 <= 4e+281) {
		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 <= 4d+281) 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 <= 4e+281) {
		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 <= 4e+281:
		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 <= 4e+281)
		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 <= 4e+281)
		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, 4e+281], 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 35.4%

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

      1. associate-/r* 51.9%

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

      2. div-inv 51.9%

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

   3. Applied egg-rr 51.9%

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

      1. un-div-inv 51.9%

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

   5. Applied egg-rr 51.9%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.0000000000000001e281

   1. Initial program 99.3%

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

   if 4.0000000000000001e281 < (/.f64 A (*.f64 V l))

   1. Initial program 50.8%

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

      1. pow1/2 50.8%

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

      2. clear-num 50.8%

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

      3. inv-pow 50.8%

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

      4. pow-pow 53.3%

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

      5. associate-/l* 61.6%

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

      6. metadata-eval 61.6%

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

   3. Applied egg-rr 61.6%

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

      1. associate-/l* 53.3%

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

   5. Simplified 53.3%

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

      1. metadata-eval 53.3%

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

      2. sqrt-pow2 53.3%

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

      3. inv-pow 53.3%

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

      4. un-div-inv 53.3%

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

      5. associate-*r/ 61.6%

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

   7. Applied egg-rr 61.6%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 4 \cdot 10^{+281}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]

Alternative 13: 79.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 10^{-317}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{+281}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

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 1e-317)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_0 4e+281)
       (/ c0 (sqrt (/ (* V l) A)))
       (/ 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 <= 1e-317) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_0 <= 4e+281) {
		tmp = c0 / sqrt(((V * l) / A));
	} 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 <= 1d-317) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_0 <= 4d+281) then
        tmp = c0 / sqrt(((v * l) / a))
    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 <= 1e-317) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 4e+281) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} 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 <= 1e-317:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_0 <= 4e+281:
		tmp = c0 / math.sqrt(((V * l) / A))
	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 <= 1e-317)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 4e+281)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	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 <= 1e-317)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_0 <= 4e+281)
		tmp = c0 / sqrt(((V * l) / A));
	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, 1e-317], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+281], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $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)) < 1.00000023e-317

   1. Initial program 35.8%

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

      1. associate-/r* 51.0%

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

      2. div-inv 51.0%

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

   3. Applied egg-rr 51.0%

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

      1. associate-*l/ 52.0%

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

      2. un-div-inv 52.0%

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

   5. Applied egg-rr 52.0%

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

   if 1.00000023e-317 < (/.f64 A (*.f64 V l)) < 4.0000000000000001e281

   1. Initial program 99.6%

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

      1. associate-/r* 86.5%

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

      2. div-inv 86.4%

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

   3. Applied egg-rr 86.4%

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

      1. un-div-inv 86.5%

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

      2. sqrt-undiv 43.8%

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

      3. clear-num 43.7%

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

      4. un-div-inv 43.8%

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

      5. sqrt-undiv 86.4%

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

      6. un-div-inv 85.1%

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

      7. clear-num 85.7%

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

      8. *-commutative 85.7%

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

      9. associate-*l/ 99.6%

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

   5. Applied egg-rr 99.6%

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

   if 4.0000000000000001e281 < (/.f64 A (*.f64 V l))

   1. Initial program 50.8%

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

      1. pow1/2 50.8%

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

      2. clear-num 50.8%

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

      3. inv-pow 50.8%

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

      4. pow-pow 53.3%

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

      5. associate-/l* 61.6%

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

      6. metadata-eval 61.6%

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

   3. Applied egg-rr 61.6%

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

      1. associate-/l* 53.3%

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

   5. Simplified 53.3%

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

      1. metadata-eval 53.3%

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

      2. sqrt-pow2 53.3%

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

      3. inv-pow 53.3%

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

      4. un-div-inv 53.3%

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

      5. associate-*r/ 61.6%

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

   7. Applied egg-rr 61.6%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 10^{-317}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 4 \cdot 10^{+281}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]

Alternative 14: 73.3% 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 76.2%

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

2. Final simplification 76.2%

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

Reproduce

herbie shell --seed 2023287 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))