Henrywood and Agarwal, Equation (3)

Percentage Accurate: 74.2% → 89.6%

Time: 9.7s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.2% 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, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+167}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-260}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+206}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -2e+167)
   (* (sqrt (/ A V)) (/ c0 (sqrt l)))
   (if (<= (* V l) -5e-260)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 0.0)
       (* c0 (/ (sqrt (/ A (- l))) (sqrt (- V))))
       (if (<= (* V l) 1e+206)
         (* c0 (/ (sqrt A) (sqrt (* V l))))
         (* c0 (sqrt (/ (/ A V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+167) {
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	} else if ((V * l) <= -5e-260) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (sqrt((A / -l)) / sqrt(-V));
	} else if ((V * l) <= 1e+206) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, 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+167)) then
        tmp = sqrt((a / v)) * (c0 / sqrt(l))
    else if ((v * l) <= (-5d-260)) then
        tmp = c0 * (sqrt(-a) / sqrt((v * -l)))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 * (sqrt((a / -l)) / sqrt(-v))
    else if ((v * l) <= 1d+206) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 * sqrt(((a / v) / l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+167) {
		tmp = Math.sqrt((A / V)) * (c0 / Math.sqrt(l));
	} else if ((V * l) <= -5e-260) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (Math.sqrt((A / -l)) / Math.sqrt(-V));
	} else if ((V * l) <= 1e+206) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -2e+167:
		tmp = math.sqrt((A / V)) * (c0 / math.sqrt(l))
	elif (V * l) <= -5e-260:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = c0 * (math.sqrt((A / -l)) / math.sqrt(-V))
	elif (V * l) <= 1e+206:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -2e+167)
		tmp = Float64(sqrt(Float64(A / V)) * Float64(c0 / sqrt(l)));
	elseif (Float64(V * l) <= -5e-260)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / Float64(-l))) / sqrt(Float64(-V))));
	elseif (Float64(V * l) <= 1e+206)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -2e+167)
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	elseif ((V * l) <= -5e-260)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = c0 * (sqrt((A / -l)) / sqrt(-V));
	elseif ((V * l) <= 1e+206)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, 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+167], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-260], 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[(N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+206], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -2.0000000000000001e167

   1. Initial program 54.8%

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

      1. frac-2neg 54.8%

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

      2. div-inv 54.8%

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

      3. distribute-rgt-neg-in 54.8%

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

   4. Applied egg-rr 54.8%

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

      1. add-sqr-sqrt 54.7%

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

      2. associate-*r* 54.7%

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

      3. pow1/2 54.7%

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

      4. sqrt-pow1 54.7%

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

      5. associate-*r/ 54.7%

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

      6. distribute-rgt-neg-out 54.7%

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

      7. distribute-lft-neg-in 54.7%

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

      8. frac-times 54.7%

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

      9. frac-2neg 54.7%

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

      10. sqrt-pow1 54.7%

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

      11. pow1/2 54.7%

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

   6. Applied egg-rr 28.8%

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

      1. *-commutative 28.8%

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

      2. associate-/l* 28.9%

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

   8. Simplified 28.9%

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

   if -2.0000000000000001e167 < (*.f64 V l) < -5.0000000000000003e-260

   1. Initial program 85.1%

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

      1. frac-2neg 85.1%

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

      2. sqrt-div 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

   4. Applied egg-rr 99.4%

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

   if -5.0000000000000003e-260 < (*.f64 V l) < -0.0

   1. Initial program 46.8%

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

      1. frac-2neg 46.8%

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

      2. div-inv 46.8%

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

      3. distribute-rgt-neg-in 46.8%

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

   4. Applied egg-rr 46.8%

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

   5. Taylor expanded in c0 around 0 46.8%

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

      1. associate-/l/ 66.9%

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

   7. Simplified 66.9%

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

      1. frac-2neg 66.9%

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

      2. sqrt-div 52.7%

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

      3. distribute-neg-frac2 52.7%

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

   9. Applied egg-rr 52.7%

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

   if -0.0 < (*.f64 V l) < 1e206

   1. Initial program 88.4%

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

      1. sqrt-div 98.9%

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

      2. div-inv 98.8%

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

   4. Applied egg-rr 98.8%

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

      1. associate-*r/ 98.9%

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

      2. *-rgt-identity 98.9%

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

   6. Simplified 98.9%

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

   if 1e206 < (*.f64 V l)

   1. Initial program 52.3%

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

      1. associate-/r* 76.0%

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

   3. Simplified 76.0%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+167}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-260}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+206}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-228}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+273}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 2e-228)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 2e+273)
       (* c0 (sqrt (* A (/ (/ 1.0 V) l))))
       (/ c0 (sqrt (* V (/ l A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 2e-228) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 2e+273) {
		tmp = c0 * sqrt((A * ((1.0 / V) / l)));
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 2d-228) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 2d+273) then
        tmp = c0 * sqrt((a * ((1.0d0 / v) / l)))
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 2e-228) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 2e+273) {
		tmp = c0 * Math.sqrt((A * ((1.0 / V) / l)));
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 2e-228:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 2e+273:
		tmp = c0 * math.sqrt((A * ((1.0 / V) / l)))
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 2e-228)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 2e+273)
		tmp = Float64(c0 * sqrt(Float64(A * Float64(Float64(1.0 / V) / l))));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 2e-228)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 2e+273)
		tmp = c0 * sqrt((A * ((1.0 / V) / l)));
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-228], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+273], N[(c0 * N[Sqrt[N[(A * N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]), $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 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.00000000000000007e-228

   1. Initial program 68.4%

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

      1. associate-/r* 74.3%

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

   3. Simplified 74.3%

      \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}} \]
   4. Add Preprocessing

   if 2.00000000000000007e-228 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.99999999999999989e273

   1. Initial program 99.5%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.5%

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

      3. associate-/r* 99.6%

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

   4. Applied egg-rr 99.6%

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

   if 1.99999999999999989e273 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 50.8%

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

      1. *-commutative 50.8%

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

      2. associate-/r* 53.7%

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

      3. sqrt-div 45.9%

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

      4. associate-*l/ 46.0%

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

   4. Applied egg-rr 46.0%

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

      1. associate-/l* 43.4%

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

      2. *-commutative 43.4%

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

      3. clear-num 43.4%

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

      4. sqrt-div 43.4%

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

      5. metadata-eval 43.4%

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

      6. times-frac 46.0%

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

      7. sqrt-prod 58.1%

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

      8. associate-*r/ 53.8%

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

      9. *-commutative 53.8%

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

      10. associate-*r/ 61.1%

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

      11. *-rgt-identity 61.1%

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

      12. add-cube-cbrt 61.0%

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

      13. associate-/l* 61.0%

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

      14. pow2 61.0%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{c0}\right)}^{2}} \cdot \frac{\sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}} \]

   6. Applied egg-rr 61.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{c0}\right)}^{2} \cdot \frac{\sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 61.0%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{c0}\right)}^{2} \cdot \sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}}} \]

      2. unpow2 61.0%

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

      3. rem-3cbrt-lft 61.1%

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

   8. Simplified 61.1%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{-228}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{+273}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-228}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+273}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{1}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 2e-228)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 2e+273)
       (* c0 (sqrt (* A (/ 1.0 (* V l)))))
       (/ c0 (sqrt (* V (/ l A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 2e-228) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 2e+273) {
		tmp = c0 * sqrt((A * (1.0 / (V * l))));
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 2d-228) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 2d+273) then
        tmp = c0 * sqrt((a * (1.0d0 / (v * l))))
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 2e-228) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 2e+273) {
		tmp = c0 * Math.sqrt((A * (1.0 / (V * l))));
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 2e-228:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 2e+273:
		tmp = c0 * math.sqrt((A * (1.0 / (V * l))))
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 2e-228)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 2e+273)
		tmp = Float64(c0 * sqrt(Float64(A * Float64(1.0 / Float64(V * l)))));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 2e-228)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 2e+273)
		tmp = c0 * sqrt((A * (1.0 / (V * l))));
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-228], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+273], N[(c0 * N[Sqrt[N[(A * N[(1.0 / N[(V * l), $MachinePrecision]), $MachinePrecision]), $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 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.00000000000000007e-228

   1. Initial program 68.4%

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

      1. associate-/r* 74.3%

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

   3. Simplified 74.3%

      \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}} \]
   4. Add Preprocessing

   if 2.00000000000000007e-228 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.99999999999999989e273

   1. Initial program 99.5%

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

      1. associate-/r* 85.0%

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

   3. Simplified 85.0%

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

      1. clear-num 83.4%

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

      2. associate-/r/ 85.1%

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

   6. Applied egg-rr 85.1%

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

      1. frac-2neg 85.1%

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

      2. distribute-rgt-neg-in 85.1%

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

      3. associate-*l/ 99.6%

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

      4. associate-/r* 99.5%

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

      5. add-sqr-sqrt 41.4%

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

      6. add-sqr-sqrt 19.8%

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

      7. sqrt-unprod 10.6%

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

      8. sqr-neg 10.6%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 33.9%

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

      12. sqr-neg 33.9%

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

      13. sqrt-unprod 57.9%

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

      14. add-sqr-sqrt 99.5%

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

   8. Applied egg-rr 99.5%

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

   if 1.99999999999999989e273 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 50.8%

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

      1. *-commutative 50.8%

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

      2. associate-/r* 53.7%

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

      3. sqrt-div 45.9%

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

      4. associate-*l/ 46.0%

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

   4. Applied egg-rr 46.0%

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

      1. associate-/l* 43.4%

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

      2. *-commutative 43.4%

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

      3. clear-num 43.4%

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

      4. sqrt-div 43.4%

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

      5. metadata-eval 43.4%

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

      6. times-frac 46.0%

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

      7. sqrt-prod 58.1%

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

      8. associate-*r/ 53.8%

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

      9. *-commutative 53.8%

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

      10. associate-*r/ 61.1%

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

      11. *-rgt-identity 61.1%

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

      12. add-cube-cbrt 61.0%

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

      13. associate-/l* 61.0%

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

      14. pow2 61.0%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{c0}\right)}^{2}} \cdot \frac{\sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}} \]

   6. Applied egg-rr 61.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{c0}\right)}^{2} \cdot \frac{\sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 61.0%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{c0}\right)}^{2} \cdot \sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}}} \]

      2. unpow2 61.0%

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

      3. rem-3cbrt-lft 61.1%

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

   8. Simplified 61.1%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{-228}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{+273}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{1}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-228} \lor \neg \left(t\_0 \leq 4 \cdot 10^{+302}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (or (<= t_0 2e-228) (not (<= t_0 4e+302)))
     (* c0 (sqrt (/ (/ A V) l)))
     t_0)))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 2e-228) || !(t_0 <= 4e+302)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if ((t_0 <= 2d-228) .or. (.not. (t_0 <= 4d+302))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 2e-228) || !(t_0 <= 4e+302)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if (t_0 <= 2e-228) or not (t_0 <= 4e+302):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if ((t_0 <= 2e-228) || !(t_0 <= 4e+302))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if ((t_0 <= 2e-228) || ~((t_0 <= 4e+302)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 2e-228], N[Not[LessEqual[t$95$0, 4e+302]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.00000000000000007e-228 or 4.0000000000000003e302 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 64.9%

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

      1. associate-/r* 70.8%

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

   3. Simplified 70.8%

      \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}} \]
   4. Add Preprocessing

   if 2.00000000000000007e-228 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.0000000000000003e302

   1. Initial program 99.5%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{-228} \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 4 \cdot 10^{+302}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-228}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+273}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 2e-228)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 2e+273) t_0 (/ c0 (sqrt (* V (/ l A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 2e-228) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 2e+273) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 2d-228) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 2d+273) then
        tmp = t_0
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 2e-228) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 2e+273) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 2e-228:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 2e+273:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 2e-228)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 2e+273)
		tmp = t_0;
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 2e-228)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 2e+273)
		tmp = t_0;
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-228], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+273], t$95$0, N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.00000000000000007e-228

   1. Initial program 68.4%

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

      1. associate-/r* 74.3%

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

   3. Simplified 74.3%

      \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}} \]
   4. Add Preprocessing

   if 2.00000000000000007e-228 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.99999999999999989e273

   1. Initial program 99.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing

   if 1.99999999999999989e273 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 50.8%

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

      1. *-commutative 50.8%

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

      2. associate-/r* 53.7%

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

      3. sqrt-div 45.9%

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

      4. associate-*l/ 46.0%

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

   4. Applied egg-rr 46.0%

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

      1. associate-/l* 43.4%

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

      2. *-commutative 43.4%

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

      3. clear-num 43.4%

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

      4. sqrt-div 43.4%

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

      5. metadata-eval 43.4%

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

      6. times-frac 46.0%

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

      7. sqrt-prod 58.1%

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

      8. associate-*r/ 53.8%

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

      9. *-commutative 53.8%

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

      10. associate-*r/ 61.1%

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

      11. *-rgt-identity 61.1%

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

      12. add-cube-cbrt 61.0%

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

      13. associate-/l* 61.0%

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

      14. pow2 61.0%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{c0}\right)}^{2}} \cdot \frac{\sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}} \]

   6. Applied egg-rr 61.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{c0}\right)}^{2} \cdot \frac{\sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 61.0%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{c0}\right)}^{2} \cdot \sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}}} \]

      2. unpow2 61.0%

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

      3. rem-3cbrt-lft 61.1%

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

   8. Simplified 61.1%

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

Alternative 6: 89.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+167}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-260}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-315}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+206}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -2e+167)
   (* (sqrt (/ A V)) (/ c0 (sqrt l)))
   (if (<= (* V l) -5e-260)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 5e-315)
       (/ c0 (sqrt (* V (/ l A))))
       (if (<= (* V l) 1e+206)
         (* c0 (/ (sqrt A) (sqrt (* V l))))
         (* c0 (sqrt (/ (/ A V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+167) {
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	} else if ((V * l) <= -5e-260) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 5e-315) {
		tmp = c0 / sqrt((V * (l / A)));
	} else if ((V * l) <= 1e+206) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, 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+167)) then
        tmp = sqrt((a / v)) * (c0 / sqrt(l))
    else if ((v * l) <= (-5d-260)) then
        tmp = c0 * (sqrt(-a) / sqrt((v * -l)))
    else if ((v * l) <= 5d-315) then
        tmp = c0 / sqrt((v * (l / a)))
    else if ((v * l) <= 1d+206) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 * sqrt(((a / v) / l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+167) {
		tmp = Math.sqrt((A / V)) * (c0 / Math.sqrt(l));
	} else if ((V * l) <= -5e-260) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 5e-315) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else if ((V * l) <= 1e+206) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -2e+167:
		tmp = math.sqrt((A / V)) * (c0 / math.sqrt(l))
	elif (V * l) <= -5e-260:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 5e-315:
		tmp = c0 / math.sqrt((V * (l / A)))
	elif (V * l) <= 1e+206:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -2e+167)
		tmp = Float64(sqrt(Float64(A / V)) * Float64(c0 / sqrt(l)));
	elseif (Float64(V * l) <= -5e-260)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 5e-315)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	elseif (Float64(V * l) <= 1e+206)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -2e+167)
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	elseif ((V * l) <= -5e-260)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 5e-315)
		tmp = c0 / sqrt((V * (l / A)));
	elseif ((V * l) <= 1e+206)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, 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+167], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-260], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-315], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+206], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -2.0000000000000001e167

   1. Initial program 54.8%

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

      1. frac-2neg 54.8%

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

      2. div-inv 54.8%

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

      3. distribute-rgt-neg-in 54.8%

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

   4. Applied egg-rr 54.8%

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

      1. add-sqr-sqrt 54.7%

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

      2. associate-*r* 54.7%

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

      3. pow1/2 54.7%

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

      4. sqrt-pow1 54.7%

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

      5. associate-*r/ 54.7%

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

      6. distribute-rgt-neg-out 54.7%

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

      7. distribute-lft-neg-in 54.7%

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

      8. frac-times 54.7%

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

      9. frac-2neg 54.7%

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

      10. sqrt-pow1 54.7%

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

      11. pow1/2 54.7%

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

   6. Applied egg-rr 28.8%

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

      1. *-commutative 28.8%

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

      2. associate-/l* 28.9%

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

   8. Simplified 28.9%

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

   if -2.0000000000000001e167 < (*.f64 V l) < -5.0000000000000003e-260

   1. Initial program 85.1%

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

      1. frac-2neg 85.1%

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

      2. sqrt-div 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

   4. Applied egg-rr 99.4%

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

   if -5.0000000000000003e-260 < (*.f64 V l) < 5.0000000023e-315

   1. Initial program 47.4%

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

      1. *-commutative 47.4%

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

      2. associate-/r* 67.8%

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

      3. sqrt-div 35.8%

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

      4. associate-*l/ 31.0%

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

   4. Applied egg-rr 31.0%

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

      1. associate-/l* 35.9%

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

      2. *-commutative 35.9%

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

      3. clear-num 35.9%

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

      4. sqrt-div 35.7%

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

      5. metadata-eval 35.7%

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

      6. times-frac 35.8%

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

      7. sqrt-prod 69.4%

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

      8. associate-*r/ 47.4%

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

      9. *-commutative 47.4%

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

      10. associate-*r/ 69.4%

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

      11. *-rgt-identity 69.4%

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

      12. add-cube-cbrt 69.2%

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

      13. associate-/l* 69.2%

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

      14. pow2 69.2%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{c0}\right)}^{2}} \cdot \frac{\sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}} \]

   6. Applied egg-rr 69.2%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{c0}\right)}^{2} \cdot \frac{\sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 69.2%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{c0}\right)}^{2} \cdot \sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}}} \]

      2. unpow2 69.2%

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

      3. rem-3cbrt-lft 69.4%

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

   8. Simplified 69.4%

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

   if 5.0000000023e-315 < (*.f64 V l) < 1e206

   1. Initial program 88.6%

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

      1. sqrt-div 99.3%

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

      2. div-inv 99.1%

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

   4. Applied egg-rr 99.1%

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

      1. associate-*r/ 99.3%

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

      2. *-rgt-identity 99.3%

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

   6. Simplified 99.3%

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

   if 1e206 < (*.f64 V l)

   1. Initial program 52.3%

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

      1. associate-/r* 76.0%

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

   3. Simplified 76.0%

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

Alternative 7: 86.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+163}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-230}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-315}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+206}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e+163)
   (* (sqrt (/ A V)) (/ c0 (sqrt l)))
   (if (<= (* V l) -2e-230)
     (/ c0 (sqrt (/ (* V l) A)))
     (if (<= (* V l) 5e-315)
       (/ c0 (sqrt (* V (/ l A))))
       (if (<= (* V l) 1e+206)
         (* c0 (/ (sqrt A) (sqrt (* V l))))
         (* c0 (sqrt (/ (/ A V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+163) {
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	} else if ((V * l) <= -2e-230) {
		tmp = c0 / sqrt(((V * l) / A));
	} else if ((V * l) <= 5e-315) {
		tmp = c0 / sqrt((V * (l / A)));
	} else if ((V * l) <= 1e+206) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-1d+163)) then
        tmp = sqrt((a / v)) * (c0 / sqrt(l))
    else if ((v * l) <= (-2d-230)) then
        tmp = c0 / sqrt(((v * l) / a))
    else if ((v * l) <= 5d-315) then
        tmp = c0 / sqrt((v * (l / a)))
    else if ((v * l) <= 1d+206) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 * sqrt(((a / v) / l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+163) {
		tmp = Math.sqrt((A / V)) * (c0 / Math.sqrt(l));
	} else if ((V * l) <= -2e-230) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else if ((V * l) <= 5e-315) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else if ((V * l) <= 1e+206) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e+163:
		tmp = math.sqrt((A / V)) * (c0 / math.sqrt(l))
	elif (V * l) <= -2e-230:
		tmp = c0 / math.sqrt(((V * l) / A))
	elif (V * l) <= 5e-315:
		tmp = c0 / math.sqrt((V * (l / A)))
	elif (V * l) <= 1e+206:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e+163)
		tmp = Float64(sqrt(Float64(A / V)) * Float64(c0 / sqrt(l)));
	elseif (Float64(V * l) <= -2e-230)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	elseif (Float64(V * l) <= 5e-315)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	elseif (Float64(V * l) <= 1e+206)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e+163)
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	elseif ((V * l) <= -2e-230)
		tmp = c0 / sqrt(((V * l) / A));
	elseif ((V * l) <= 5e-315)
		tmp = c0 / sqrt((V * (l / A)));
	elseif ((V * l) <= 1e+206)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -1e+163], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-230], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-315], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+206], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -9.9999999999999994e162

   1. Initial program 53.4%

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

      1. frac-2neg 53.4%

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

      2. div-inv 53.4%

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

      3. distribute-rgt-neg-in 53.4%

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

   4. Applied egg-rr 53.4%

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

      1. add-sqr-sqrt 53.3%

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

      2. associate-*r* 53.3%

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

      3. pow1/2 53.3%

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

      4. sqrt-pow1 53.3%

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

      5. associate-*r/ 53.4%

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

      6. distribute-rgt-neg-out 53.4%

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

      7. distribute-lft-neg-in 53.4%

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

      8. frac-times 53.4%

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

      9. frac-2neg 53.4%

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

      10. sqrt-pow1 53.3%

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

      11. pow1/2 53.3%

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

   6. Applied egg-rr 28.2%

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

      1. *-commutative 28.2%

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

      2. associate-/l* 28.2%

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

   8. Simplified 28.2%

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

   if -9.9999999999999994e162 < (*.f64 V l) < -2.00000000000000009e-230

   1. Initial program 87.3%

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

      1. *-commutative 87.3%

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

      2. associate-/r* 72.8%

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

      3. sqrt-div 41.8%

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

      4. associate-*l/ 41.7%

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

   4. Applied egg-rr 41.7%

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

      1. associate-/l* 39.6%

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

      2. *-commutative 39.6%

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

      3. clear-num 38.6%

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

      4. sqrt-div 38.6%

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

      5. metadata-eval 38.6%

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

      6. times-frac 40.8%

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

      7. sqrt-prod 73.3%

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

      8. associate-*r/ 88.9%

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

      9. *-commutative 88.9%

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

      10. associate-*r/ 71.4%

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

      11. *-rgt-identity 71.4%

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

      12. add-cube-cbrt 70.4%

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

      13. associate-/l* 70.3%

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

      14. pow2 70.3%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{c0}\right)}^{2}} \cdot \frac{\sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}} \]

   6. Applied egg-rr 70.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{c0}\right)}^{2} \cdot \frac{\sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 70.4%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{c0}\right)}^{2} \cdot \sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}}} \]

      2. unpow2 70.4%

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

      3. rem-3cbrt-lft 71.4%

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

   8. Simplified 71.4%

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

   9. Taylor expanded in V around 0 88.9%

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

   if -2.00000000000000009e-230 < (*.f64 V l) < 5.0000000023e-315

   1. Initial program 48.9%

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

      1. *-commutative 48.9%

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

      2. associate-/r* 67.8%

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

      3. sqrt-div 35.6%

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

      4. associate-*l/ 29.0%

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

   4. Applied egg-rr 29.0%

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

      1. associate-/l* 35.7%

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

      2. *-commutative 35.7%

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

      3. clear-num 35.7%

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

      4. sqrt-div 35.6%

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

      5. metadata-eval 35.6%

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

      6. times-frac 35.7%

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

      7. sqrt-prod 69.4%

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

      8. associate-*r/ 48.9%

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

      9. *-commutative 48.9%

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

      10. associate-*r/ 69.4%

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

      11. *-rgt-identity 69.4%

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

      12. add-cube-cbrt 69.1%

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

      13. associate-/l* 69.0%

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

      14. pow2 69.0%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{c0}\right)}^{2}} \cdot \frac{\sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}} \]

   6. Applied egg-rr 69.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{c0}\right)}^{2} \cdot \frac{\sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 69.1%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{c0}\right)}^{2} \cdot \sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}}} \]

      2. unpow2 69.1%

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

      3. rem-3cbrt-lft 69.4%

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

   8. Simplified 69.4%

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

   if 5.0000000023e-315 < (*.f64 V l) < 1e206

   1. Initial program 88.6%

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

      1. sqrt-div 99.3%

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

      2. div-inv 99.1%

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

   4. Applied egg-rr 99.1%

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

      1. associate-*r/ 99.3%

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

      2. *-rgt-identity 99.3%

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

   6. Simplified 99.3%

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

   if 1e206 < (*.f64 V l)

   1. Initial program 52.3%

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

      1. associate-/r* 76.0%

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

   3. Simplified 76.0%

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

Alternative 8: 86.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+163}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-230}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-315}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+206}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e+163)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* V l) -2e-230)
     (/ c0 (sqrt (/ (* V l) A)))
     (if (<= (* V l) 5e-315)
       (/ c0 (sqrt (* V (/ l A))))
       (if (<= (* V l) 1e+206)
         (* c0 (/ (sqrt A) (sqrt (* V l))))
         (* c0 (sqrt (/ (/ A V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+163) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= -2e-230) {
		tmp = c0 / sqrt(((V * l) / A));
	} else if ((V * l) <= 5e-315) {
		tmp = c0 / sqrt((V * (l / A)));
	} else if ((V * l) <= 1e+206) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-1d+163)) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= (-2d-230)) then
        tmp = c0 / sqrt(((v * l) / a))
    else if ((v * l) <= 5d-315) then
        tmp = c0 / sqrt((v * (l / a)))
    else if ((v * l) <= 1d+206) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 * sqrt(((a / v) / l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+163) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= -2e-230) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else if ((V * l) <= 5e-315) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else if ((V * l) <= 1e+206) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e+163:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= -2e-230:
		tmp = c0 / math.sqrt(((V * l) / A))
	elif (V * l) <= 5e-315:
		tmp = c0 / math.sqrt((V * (l / A)))
	elif (V * l) <= 1e+206:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e+163)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= -2e-230)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	elseif (Float64(V * l) <= 5e-315)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	elseif (Float64(V * l) <= 1e+206)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e+163)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= -2e-230)
		tmp = c0 / sqrt(((V * l) / A));
	elseif ((V * l) <= 5e-315)
		tmp = c0 / sqrt((V * (l / A)));
	elseif ((V * l) <= 1e+206)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -1e+163], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-230], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-315], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+206], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -9.9999999999999994e162

   1. Initial program 53.4%

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

      1. associate-/r* 68.7%

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

      2. sqrt-div 28.2%

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

      3. div-inv 28.2%

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

   4. Applied egg-rr 28.2%

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

      1. associate-*r/ 28.2%

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

      2. *-rgt-identity 28.2%

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

   6. Simplified 28.2%

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

   if -9.9999999999999994e162 < (*.f64 V l) < -2.00000000000000009e-230

   1. Initial program 87.3%

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

      1. *-commutative 87.3%

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

      2. associate-/r* 72.8%

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

      3. sqrt-div 41.8%

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

      4. associate-*l/ 41.7%

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

   4. Applied egg-rr 41.7%

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

      1. associate-/l* 39.6%

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

      2. *-commutative 39.6%

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

      3. clear-num 38.6%

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

      4. sqrt-div 38.6%

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

      5. metadata-eval 38.6%

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

      6. times-frac 40.8%

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

      7. sqrt-prod 73.3%

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

      8. associate-*r/ 88.9%

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

      9. *-commutative 88.9%

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

      10. associate-*r/ 71.4%

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

      11. *-rgt-identity 71.4%

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

      12. add-cube-cbrt 70.4%

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

      13. associate-/l* 70.3%

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

      14. pow2 70.3%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{c0}\right)}^{2}} \cdot \frac{\sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}} \]

   6. Applied egg-rr 70.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{c0}\right)}^{2} \cdot \frac{\sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 70.4%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{c0}\right)}^{2} \cdot \sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}}} \]

      2. unpow2 70.4%

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

      3. rem-3cbrt-lft 71.4%

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

   8. Simplified 71.4%

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

   9. Taylor expanded in V around 0 88.9%

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

   if -2.00000000000000009e-230 < (*.f64 V l) < 5.0000000023e-315

   1. Initial program 48.9%

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

      1. *-commutative 48.9%

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

      2. associate-/r* 67.8%

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

      3. sqrt-div 35.6%

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

      4. associate-*l/ 29.0%

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

   4. Applied egg-rr 29.0%

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

      1. associate-/l* 35.7%

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

      2. *-commutative 35.7%

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

      3. clear-num 35.7%

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

      4. sqrt-div 35.6%

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

      5. metadata-eval 35.6%

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

      6. times-frac 35.7%

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

      7. sqrt-prod 69.4%

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

      8. associate-*r/ 48.9%

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

      9. *-commutative 48.9%

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

      10. associate-*r/ 69.4%

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

      11. *-rgt-identity 69.4%

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

      12. add-cube-cbrt 69.1%

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

      13. associate-/l* 69.0%

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

      14. pow2 69.0%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{c0}\right)}^{2}} \cdot \frac{\sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}} \]

   6. Applied egg-rr 69.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{c0}\right)}^{2} \cdot \frac{\sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 69.1%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{c0}\right)}^{2} \cdot \sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}}} \]

      2. unpow2 69.1%

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

      3. rem-3cbrt-lft 69.4%

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

   8. Simplified 69.4%

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

   if 5.0000000023e-315 < (*.f64 V l) < 1e206

   1. Initial program 88.6%

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

      1. sqrt-div 99.3%

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

      2. div-inv 99.1%

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

   4. Applied egg-rr 99.1%

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

      1. associate-*r/ 99.3%

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

      2. *-rgt-identity 99.3%

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

   6. Simplified 99.3%

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

   if 1e206 < (*.f64 V l)

   1. Initial program 52.3%

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

      1. associate-/r* 76.0%

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

   3. Simplified 76.0%

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

Alternative 9: 84.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+147}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-230}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-315}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+206}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e+147)
   (/ c0 (sqrt (/ l (/ A V))))
   (if (<= (* V l) -2e-230)
     (/ c0 (sqrt (/ (* V l) A)))
     (if (<= (* V l) 5e-315)
       (/ c0 (sqrt (* V (/ l A))))
       (if (<= (* V l) 1e+206)
         (* c0 (/ (sqrt A) (sqrt (* V l))))
         (* c0 (sqrt (/ (/ A V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+147) {
		tmp = c0 / sqrt((l / (A / V)));
	} else if ((V * l) <= -2e-230) {
		tmp = c0 / sqrt(((V * l) / A));
	} else if ((V * l) <= 5e-315) {
		tmp = c0 / sqrt((V * (l / A)));
	} else if ((V * l) <= 1e+206) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-1d+147)) then
        tmp = c0 / sqrt((l / (a / v)))
    else if ((v * l) <= (-2d-230)) then
        tmp = c0 / sqrt(((v * l) / a))
    else if ((v * l) <= 5d-315) then
        tmp = c0 / sqrt((v * (l / a)))
    else if ((v * l) <= 1d+206) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 * sqrt(((a / v) / l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+147) {
		tmp = c0 / Math.sqrt((l / (A / V)));
	} else if ((V * l) <= -2e-230) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else if ((V * l) <= 5e-315) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else if ((V * l) <= 1e+206) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e+147:
		tmp = c0 / math.sqrt((l / (A / V)))
	elif (V * l) <= -2e-230:
		tmp = c0 / math.sqrt(((V * l) / A))
	elif (V * l) <= 5e-315:
		tmp = c0 / math.sqrt((V * (l / A)))
	elif (V * l) <= 1e+206:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e+147)
		tmp = Float64(c0 / sqrt(Float64(l / Float64(A / V))));
	elseif (Float64(V * l) <= -2e-230)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	elseif (Float64(V * l) <= 5e-315)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	elseif (Float64(V * l) <= 1e+206)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e+147)
		tmp = c0 / sqrt((l / (A / V)));
	elseif ((V * l) <= -2e-230)
		tmp = c0 / sqrt(((V * l) / A));
	elseif ((V * l) <= 5e-315)
		tmp = c0 / sqrt((V * (l / A)));
	elseif ((V * l) <= 1e+206)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -1e+147], N[(c0 / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-230], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-315], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+206], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -9.9999999999999998e146

   1. Initial program 57.9%

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

      1. frac-2neg 57.9%

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

      2. div-inv 57.9%

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

      3. distribute-rgt-neg-in 57.9%

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

   4. Applied egg-rr 57.9%

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

   5. Taylor expanded in c0 around 0 57.9%

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

      1. associate-/l/ 69.5%

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

   7. Simplified 69.5%

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

      1. clear-num 69.4%

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

      2. associate-/l/ 69.4%

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

      3. sqrt-div 69.3%

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

      4. metadata-eval 69.3%

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

      5. inv-pow 69.3%

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

      6. sqrt-pow2 69.4%

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

      7. metadata-eval 69.4%

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

   9. Applied egg-rr 69.4%

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

       1. metadata-eval 69.4%

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

       2. pow-flip 69.3%

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

       3. pow1/2 69.3%

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

       4. associate-/r/ 69.5%

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

       5. clear-num 69.5%

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

       6. *-commutative 69.5%

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

       7. sqrt-prod 53.5%

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

       8. sqrt-div 0.0%

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

       9. associate-/r/ 0.0%

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

       10. sqrt-div 35.1%

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

       11. sqrt-undiv 71.8%

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

   11. Applied egg-rr 71.8%

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

   if -9.9999999999999998e146 < (*.f64 V l) < -2.00000000000000009e-230

   1. Initial program 86.4%

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

      1. *-commutative 86.4%

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

      2. associate-/r* 71.0%

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

      3. sqrt-div 37.9%

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

      4. associate-*l/ 37.8%

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

   4. Applied egg-rr 37.8%

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

      1. associate-/l* 35.4%

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

      2. *-commutative 35.4%

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

      3. clear-num 34.4%

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

      4. sqrt-div 34.4%

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

      5. metadata-eval 34.4%

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

      6. times-frac 36.7%

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

      7. sqrt-prod 71.4%

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

      8. associate-*r/ 88.1%

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

      9. *-commutative 88.1%

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

      10. associate-*r/ 71.1%

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

      11. *-rgt-identity 71.1%

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

      12. add-cube-cbrt 70.0%

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

      13. associate-/l* 70.0%

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

      14. pow2 70.0%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{c0}\right)}^{2}} \cdot \frac{\sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}} \]

   6. Applied egg-rr 70.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{c0}\right)}^{2} \cdot \frac{\sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 70.0%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{c0}\right)}^{2} \cdot \sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}}} \]

      2. unpow2 70.0%

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

      3. rem-3cbrt-lft 71.1%

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

   8. Simplified 71.1%

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

   9. Taylor expanded in V around 0 88.1%

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

   if -2.00000000000000009e-230 < (*.f64 V l) < 5.0000000023e-315

   1. Initial program 48.9%

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

      1. *-commutative 48.9%

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

      2. associate-/r* 67.8%

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

      3. sqrt-div 35.6%

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

      4. associate-*l/ 29.0%

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

   4. Applied egg-rr 29.0%

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

      1. associate-/l* 35.7%

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

      2. *-commutative 35.7%

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

      3. clear-num 35.7%

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

      4. sqrt-div 35.6%

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

      5. metadata-eval 35.6%

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

      6. times-frac 35.7%

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

      7. sqrt-prod 69.4%

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

      8. associate-*r/ 48.9%

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

      9. *-commutative 48.9%

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

      10. associate-*r/ 69.4%

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

      11. *-rgt-identity 69.4%

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

      12. add-cube-cbrt 69.1%

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

      13. associate-/l* 69.0%

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

      14. pow2 69.0%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{c0}\right)}^{2}} \cdot \frac{\sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}} \]

   6. Applied egg-rr 69.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{c0}\right)}^{2} \cdot \frac{\sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 69.1%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{c0}\right)}^{2} \cdot \sqrt[3]{c0}}{\sqrt{V \cdot \frac{\ell}{A}}}} \]

      2. unpow2 69.1%

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

      3. rem-3cbrt-lft 69.4%

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

   8. Simplified 69.4%

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

   if 5.0000000023e-315 < (*.f64 V l) < 1e206

   1. Initial program 88.6%

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

      1. sqrt-div 99.3%

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

      2. div-inv 99.1%

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

   4. Applied egg-rr 99.1%

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

      1. associate-*r/ 99.3%

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

      2. *-rgt-identity 99.3%

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

   6. Simplified 99.3%

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

   if 1e206 < (*.f64 V l)

   1. Initial program 52.3%

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

      1. associate-/r* 76.0%

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

   3. Simplified 76.0%

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

Alternative 10: 74.2% accurate, 1.0× speedup

Math

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

FPCore

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: c0, A, 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 73.3%

   \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
2. Add Preprocessing
3. Add Preprocessing

Reproduce

herbie shell --seed 2024113 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))