Henrywood and Agarwal, Equation (3)

Percentage Accurate: 74.4% → 90.9%

Time: 8.3s

Alternatives: 9

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: 74.4% 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: 90.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{V \cdot \frac{\ell}{A}}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-323}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{t_0}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+232}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_0}{c0}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (* V (/ l A)))))
   (if (<= (* V l) (- INFINITY))
     (* c0 (/ (sqrt (/ A V)) (sqrt l)))
     (if (<= (* V l) -4e-323)
       (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
       (if (<= (* V l) 0.0)
         (/ c0 t_0)
         (if (<= (* V l) 5e+232)
           (* c0 (* (sqrt A) (pow (* V l) -0.5)))
           (/ 1.0 (/ t_0 c0))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((V * (l / A)));
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= -4e-323) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / t_0;
	} else if ((V * l) <= 5e+232) {
		tmp = c0 * (sqrt(A) * pow((V * l), -0.5));
	} else {
		tmp = 1.0 / (t_0 / c0);
	}
	return tmp;
}

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((V * (l / A)));
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= -4e-323) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / t_0;
	} else if ((V * l) <= 5e+232) {
		tmp = c0 * (Math.sqrt(A) * Math.pow((V * l), -0.5));
	} else {
		tmp = 1.0 / (t_0 / c0);
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((V * (l / A)))
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= -4e-323:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = c0 / t_0
	elif (V * l) <= 5e+232:
		tmp = c0 * (math.sqrt(A) * math.pow((V * l), -0.5))
	else:
		tmp = 1.0 / (t_0 / c0)
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(V * Float64(l / A)))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= -4e-323)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 / t_0);
	elseif (Float64(V * l) <= 5e+232)
		tmp = Float64(c0 * Float64(sqrt(A) * (Float64(V * l) ^ -0.5)));
	else
		tmp = Float64(1.0 / Float64(t_0 / c0));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((V * (l / A)));
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= -4e-323)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = c0 / t_0;
	elseif ((V * l) <= 5e+232)
		tmp = c0 * (sqrt(A) * ((V * l) ^ -0.5));
	else
		tmp = 1.0 / (t_0 / c0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -4e-323], 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 / t$95$0), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+232], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 / c0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 29.2%

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

      1. associate-/r* 46.0%

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

      2. sqrt-div 26.5%

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

   3. Applied egg-rr 26.5%

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

   if -inf.0 < (*.f64 V l) < -3.95253e-323

   1. Initial program 89.7%

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

      1. frac-2neg 89.7%

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

      2. sqrt-div 98.5%

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

      3. *-commutative 98.5%

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

      4. distribute-rgt-neg-in 98.5%

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

   3. Applied egg-rr 98.5%

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

   if -3.95253e-323 < (*.f64 V l) < 0.0

   1. Initial program 42.7%

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

      1. *-un-lft-identity 42.7%

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

      2. times-frac 76.1%

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

   3. Applied egg-rr 76.1%

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

      1. frac-times 42.7%

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

      2. *-un-lft-identity 42.7%

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

      3. sqrt-div 12.5%

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

      4. clear-num 12.5%

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

      5. div-inv 12.5%

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

      6. sqrt-undiv 42.7%

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

      7. *-un-lft-identity 42.7%

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

      8. times-frac 76.2%

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

      9. /-rgt-identity 76.2%

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

   5. Applied egg-rr 76.2%

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

   if 0.0 < (*.f64 V l) < 4.99999999999999987e232

   1. Initial program 83.4%

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

      1. div-inv 83.4%

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

      2. sqrt-prod 99.4%

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

      3. pow1/2 99.4%

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

      4. inv-pow 99.4%

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

      5. pow-pow 99.4%

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

      6. metadata-eval 99.4%

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

   3. Applied egg-rr 99.4%

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

   if 4.99999999999999987e232 < (*.f64 V l)

   1. Initial program 34.4%

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

      1. *-un-lft-identity 34.4%

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

      2. times-frac 79.8%

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

   3. Applied egg-rr 79.8%

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

      1. *-commutative 79.8%

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

      2. frac-times 34.4%

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

      3. *-un-lft-identity 34.4%

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

      4. sqrt-div 34.4%

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

      5. clear-num 34.5%

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

      6. associate-/r/ 34.6%

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

      7. sqrt-undiv 34.6%

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

      8. *-un-lft-identity 34.6%

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

      9. times-frac 79.9%

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

      10. /-rgt-identity 79.9%

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

   5. Applied egg-rr 79.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{V \cdot \frac{\ell}{A}}}{c0}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-323}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+232}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{V \cdot \frac{\ell}{A}}}{c0}}\\ \end{array} \]

Alternative 2: 86.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+254}:\\ \;\;\;\;c0 \cdot \frac{t_0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-105}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0 \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+232}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{V \cdot \frac{\ell}{A}}}{c0}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (/ A V))))
   (if (<= (* V l) -2e+254)
     (* c0 (/ t_0 (sqrt l)))
     (if (<= (* V l) -1e-105)
       (* c0 (sqrt (/ A (* V l))))
       (if (<= (* V l) 0.0)
         (* t_0 (/ c0 (sqrt l)))
         (if (<= (* V l) 5e+232)
           (* c0 (* (sqrt A) (pow (* V l) -0.5)))
           (/ 1.0 (/ (sqrt (* V (/ l A))) c0))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / V));
	double tmp;
	if ((V * l) <= -2e+254) {
		tmp = c0 * (t_0 / sqrt(l));
	} else if ((V * l) <= -1e-105) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0 * (c0 / sqrt(l));
	} else if ((V * l) <= 5e+232) {
		tmp = c0 * (sqrt(A) * pow((V * l), -0.5));
	} else {
		tmp = 1.0 / (sqrt((V * (l / A))) / c0);
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a / v))
    if ((v * l) <= (-2d+254)) then
        tmp = c0 * (t_0 / sqrt(l))
    else if ((v * l) <= (-1d-105)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0 * (c0 / sqrt(l))
    else if ((v * l) <= 5d+232) then
        tmp = c0 * (sqrt(a) * ((v * l) ** (-0.5d0)))
    else
        tmp = 1.0d0 / (sqrt((v * (l / a))) / c0)
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / V));
	double tmp;
	if ((V * l) <= -2e+254) {
		tmp = c0 * (t_0 / Math.sqrt(l));
	} else if ((V * l) <= -1e-105) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0 * (c0 / Math.sqrt(l));
	} else if ((V * l) <= 5e+232) {
		tmp = c0 * (Math.sqrt(A) * Math.pow((V * l), -0.5));
	} else {
		tmp = 1.0 / (Math.sqrt((V * (l / A))) / c0);
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / V))
	tmp = 0
	if (V * l) <= -2e+254:
		tmp = c0 * (t_0 / math.sqrt(l))
	elif (V * l) <= -1e-105:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 0.0:
		tmp = t_0 * (c0 / math.sqrt(l))
	elif (V * l) <= 5e+232:
		tmp = c0 * (math.sqrt(A) * math.pow((V * l), -0.5))
	else:
		tmp = 1.0 / (math.sqrt((V * (l / A))) / c0)
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(A / V))
	tmp = 0.0
	if (Float64(V * l) <= -2e+254)
		tmp = Float64(c0 * Float64(t_0 / sqrt(l)));
	elseif (Float64(V * l) <= -1e-105)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(t_0 * Float64(c0 / sqrt(l)));
	elseif (Float64(V * l) <= 5e+232)
		tmp = Float64(c0 * Float64(sqrt(A) * (Float64(V * l) ^ -0.5)));
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(V * Float64(l / A))) / c0));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((A / V));
	tmp = 0.0;
	if ((V * l) <= -2e+254)
		tmp = c0 * (t_0 / sqrt(l));
	elseif ((V * l) <= -1e-105)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 0.0)
		tmp = t_0 * (c0 / sqrt(l));
	elseif ((V * l) <= 5e+232)
		tmp = c0 * (sqrt(A) * ((V * l) ^ -0.5));
	else
		tmp = 1.0 / (sqrt((V * (l / A))) / c0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -2e+254], N[(c0 * N[(t$95$0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-105], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(t$95$0 * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+232], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -1.9999999999999999e254

   1. Initial program 34.1%

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

      1. associate-/r* 47.4%

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

      2. sqrt-div 33.2%

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

   3. Applied egg-rr 33.2%

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

   if -1.9999999999999999e254 < (*.f64 V l) < -9.99999999999999965e-106

   1. Initial program 95.3%

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

   if -9.99999999999999965e-106 < (*.f64 V l) < 0.0

   1. Initial program 60.5%

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

      1. *-un-lft-identity 60.5%

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

      2. times-frac 75.3%

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

   3. Applied egg-rr 75.3%

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

      1. *-commutative 75.3%

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

      2. frac-times 60.5%

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

      3. *-un-lft-identity 60.5%

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

      4. sqrt-div 6.8%

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

      5. clear-num 6.8%

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

      6. associate-/r/ 6.8%

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

      7. sqrt-undiv 60.5%

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

      8. *-un-lft-identity 60.5%

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

      9. times-frac 75.3%

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

      10. /-rgt-identity 75.3%

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

   5. Applied egg-rr 75.3%

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

      1. associate-/r/ 75.2%

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

      2. associate-*l/ 75.4%

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

      3. *-lft-identity 75.4%

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

      4. *-commutative 75.4%

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

      5. associate-*l/ 60.5%

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

      6. associate-/l* 77.2%

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

   7. Simplified 77.2%

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

      1. *-un-lft-identity 77.2%

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

      2. associate-*l/ 77.0%

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

      3. metadata-eval 77.0%

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

      4. sqrt-div 77.2%

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

      5. clear-num 77.2%

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

      6. associate-/r* 60.5%

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

      7. sqrt-undiv 6.8%

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

      8. associate-*l/ 6.8%

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

      9. sqrt-prod 13.6%

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

      10. times-frac 11.8%

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

      11. sqrt-div 47.3%

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

   9. Applied egg-rr 47.3%

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

   if 0.0 < (*.f64 V l) < 4.99999999999999987e232

   1. Initial program 83.4%

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

      1. div-inv 83.4%

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

      2. sqrt-prod 99.4%

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

      3. pow1/2 99.4%

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

      4. inv-pow 99.4%

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

      5. pow-pow 99.4%

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

      6. metadata-eval 99.4%

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

   3. Applied egg-rr 99.4%

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

   if 4.99999999999999987e232 < (*.f64 V l)

   1. Initial program 34.4%

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

      1. *-un-lft-identity 34.4%

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

      2. times-frac 79.8%

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

   3. Applied egg-rr 79.8%

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

      1. *-commutative 79.8%

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

      2. frac-times 34.4%

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

      3. *-un-lft-identity 34.4%

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

      4. sqrt-div 34.4%

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

      5. clear-num 34.5%

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

      6. associate-/r/ 34.6%

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

      7. sqrt-undiv 34.6%

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

      8. *-un-lft-identity 34.6%

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

      9. times-frac 79.9%

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

      10. /-rgt-identity 79.9%

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

   5. Applied egg-rr 79.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{V \cdot \frac{\ell}{A}}}{c0}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+254}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-105}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+232}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{V \cdot \frac{\ell}{A}}}{c0}}\\ \end{array} \]

Alternative 3: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+254}:\\ \;\;\;\;c0 \cdot \frac{t_0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-105}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0 \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+232}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{V \cdot \frac{\ell}{A}}}{c0}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (/ A V))))
   (if (<= (* V l) -2e+254)
     (* c0 (/ t_0 (sqrt l)))
     (if (<= (* V l) -1e-105)
       (* c0 (sqrt (/ A (* V l))))
       (if (<= (* V l) 0.0)
         (* t_0 (/ c0 (sqrt l)))
         (if (<= (* V l) 5e+232)
           (* (sqrt A) (/ c0 (sqrt (* V l))))
           (/ 1.0 (/ (sqrt (* V (/ l A))) c0))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / V));
	double tmp;
	if ((V * l) <= -2e+254) {
		tmp = c0 * (t_0 / sqrt(l));
	} else if ((V * l) <= -1e-105) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0 * (c0 / sqrt(l));
	} else if ((V * l) <= 5e+232) {
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	} else {
		tmp = 1.0 / (sqrt((V * (l / A))) / c0);
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a / v))
    if ((v * l) <= (-2d+254)) then
        tmp = c0 * (t_0 / sqrt(l))
    else if ((v * l) <= (-1d-105)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0 * (c0 / sqrt(l))
    else if ((v * l) <= 5d+232) then
        tmp = sqrt(a) * (c0 / sqrt((v * l)))
    else
        tmp = 1.0d0 / (sqrt((v * (l / a))) / c0)
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / V));
	double tmp;
	if ((V * l) <= -2e+254) {
		tmp = c0 * (t_0 / Math.sqrt(l));
	} else if ((V * l) <= -1e-105) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0 * (c0 / Math.sqrt(l));
	} else if ((V * l) <= 5e+232) {
		tmp = Math.sqrt(A) * (c0 / Math.sqrt((V * l)));
	} else {
		tmp = 1.0 / (Math.sqrt((V * (l / A))) / c0);
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / V))
	tmp = 0
	if (V * l) <= -2e+254:
		tmp = c0 * (t_0 / math.sqrt(l))
	elif (V * l) <= -1e-105:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 0.0:
		tmp = t_0 * (c0 / math.sqrt(l))
	elif (V * l) <= 5e+232:
		tmp = math.sqrt(A) * (c0 / math.sqrt((V * l)))
	else:
		tmp = 1.0 / (math.sqrt((V * (l / A))) / c0)
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(A / V))
	tmp = 0.0
	if (Float64(V * l) <= -2e+254)
		tmp = Float64(c0 * Float64(t_0 / sqrt(l)));
	elseif (Float64(V * l) <= -1e-105)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(t_0 * Float64(c0 / sqrt(l)));
	elseif (Float64(V * l) <= 5e+232)
		tmp = Float64(sqrt(A) * Float64(c0 / sqrt(Float64(V * l))));
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(V * Float64(l / A))) / c0));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((A / V));
	tmp = 0.0;
	if ((V * l) <= -2e+254)
		tmp = c0 * (t_0 / sqrt(l));
	elseif ((V * l) <= -1e-105)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 0.0)
		tmp = t_0 * (c0 / sqrt(l));
	elseif ((V * l) <= 5e+232)
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	else
		tmp = 1.0 / (sqrt((V * (l / A))) / c0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -2e+254], N[(c0 * N[(t$95$0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-105], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(t$95$0 * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+232], N[(N[Sqrt[A], $MachinePrecision] * N[(c0 / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -1.9999999999999999e254

   1. Initial program 34.1%

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

      1. associate-/r* 47.4%

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

      2. sqrt-div 33.2%

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

   3. Applied egg-rr 33.2%

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

   if -1.9999999999999999e254 < (*.f64 V l) < -9.99999999999999965e-106

   1. Initial program 95.3%

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

   if -9.99999999999999965e-106 < (*.f64 V l) < 0.0

   1. Initial program 60.5%

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

      1. *-un-lft-identity 60.5%

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

      2. times-frac 75.3%

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

   3. Applied egg-rr 75.3%

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

      1. *-commutative 75.3%

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

      2. frac-times 60.5%

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

      3. *-un-lft-identity 60.5%

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

      4. sqrt-div 6.8%

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

      5. clear-num 6.8%

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

      6. associate-/r/ 6.8%

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

      7. sqrt-undiv 60.5%

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

      8. *-un-lft-identity 60.5%

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

      9. times-frac 75.3%

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

      10. /-rgt-identity 75.3%

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

   5. Applied egg-rr 75.3%

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

      1. associate-/r/ 75.2%

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

      2. associate-*l/ 75.4%

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

      3. *-lft-identity 75.4%

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

      4. *-commutative 75.4%

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

      5. associate-*l/ 60.5%

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

      6. associate-/l* 77.2%

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

   7. Simplified 77.2%

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

      1. *-un-lft-identity 77.2%

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

      2. associate-*l/ 77.0%

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

      3. metadata-eval 77.0%

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

      4. sqrt-div 77.2%

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

      5. clear-num 77.2%

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

      6. associate-/r* 60.5%

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

      7. sqrt-undiv 6.8%

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

      8. associate-*l/ 6.8%

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

      9. sqrt-prod 13.6%

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

      10. times-frac 11.8%

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

      11. sqrt-div 47.3%

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

   9. Applied egg-rr 47.3%

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

   if 0.0 < (*.f64 V l) < 4.99999999999999987e232

   1. Initial program 83.4%

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

      1. sqrt-div 99.3%

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

      2. associate-*r/ 95.2%

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

   3. Applied egg-rr 95.2%

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

      1. associate-*l/ 97.2%

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

   5. Simplified 97.2%

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

   if 4.99999999999999987e232 < (*.f64 V l)

   1. Initial program 34.4%

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

      1. *-un-lft-identity 34.4%

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

      2. times-frac 79.8%

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

   3. Applied egg-rr 79.8%

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

      1. *-commutative 79.8%

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

      2. frac-times 34.4%

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

      3. *-un-lft-identity 34.4%

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

      4. sqrt-div 34.4%

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

      5. clear-num 34.5%

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

      6. associate-/r/ 34.6%

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

      7. sqrt-undiv 34.6%

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

      8. *-un-lft-identity 34.6%

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

      9. times-frac 79.9%

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

      10. /-rgt-identity 79.9%

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

   5. Applied egg-rr 79.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{V \cdot \frac{\ell}{A}}}{c0}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+254}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-105}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+232}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{V \cdot \frac{\ell}{A}}}{c0}}\\ \end{array} \]

Alternative 4: 86.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+254}:\\ \;\;\;\;c0 \cdot \frac{t_0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-105}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0 \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+232}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{V \cdot \frac{\ell}{A}}}{c0}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (/ A V))))
   (if (<= (* V l) -2e+254)
     (* c0 (/ t_0 (sqrt l)))
     (if (<= (* V l) -1e-105)
       (* c0 (sqrt (/ A (* V l))))
       (if (<= (* V l) 0.0)
         (* t_0 (/ c0 (sqrt l)))
         (if (<= (* V l) 5e+232)
           (* c0 (/ (sqrt A) (sqrt (* V l))))
           (/ 1.0 (/ (sqrt (* V (/ l A))) c0))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / V));
	double tmp;
	if ((V * l) <= -2e+254) {
		tmp = c0 * (t_0 / sqrt(l));
	} else if ((V * l) <= -1e-105) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0 * (c0 / sqrt(l));
	} else if ((V * l) <= 5e+232) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = 1.0 / (sqrt((V * (l / A))) / c0);
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a / v))
    if ((v * l) <= (-2d+254)) then
        tmp = c0 * (t_0 / sqrt(l))
    else if ((v * l) <= (-1d-105)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0 * (c0 / sqrt(l))
    else if ((v * l) <= 5d+232) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = 1.0d0 / (sqrt((v * (l / a))) / c0)
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / V));
	double tmp;
	if ((V * l) <= -2e+254) {
		tmp = c0 * (t_0 / Math.sqrt(l));
	} else if ((V * l) <= -1e-105) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0 * (c0 / Math.sqrt(l));
	} else if ((V * l) <= 5e+232) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = 1.0 / (Math.sqrt((V * (l / A))) / c0);
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / V))
	tmp = 0
	if (V * l) <= -2e+254:
		tmp = c0 * (t_0 / math.sqrt(l))
	elif (V * l) <= -1e-105:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 0.0:
		tmp = t_0 * (c0 / math.sqrt(l))
	elif (V * l) <= 5e+232:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = 1.0 / (math.sqrt((V * (l / A))) / c0)
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(A / V))
	tmp = 0.0
	if (Float64(V * l) <= -2e+254)
		tmp = Float64(c0 * Float64(t_0 / sqrt(l)));
	elseif (Float64(V * l) <= -1e-105)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(t_0 * Float64(c0 / sqrt(l)));
	elseif (Float64(V * l) <= 5e+232)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(V * Float64(l / A))) / c0));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((A / V));
	tmp = 0.0;
	if ((V * l) <= -2e+254)
		tmp = c0 * (t_0 / sqrt(l));
	elseif ((V * l) <= -1e-105)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 0.0)
		tmp = t_0 * (c0 / sqrt(l));
	elseif ((V * l) <= 5e+232)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = 1.0 / (sqrt((V * (l / A))) / c0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -2e+254], N[(c0 * N[(t$95$0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-105], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(t$95$0 * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+232], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -1.9999999999999999e254

   1. Initial program 34.1%

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

      1. associate-/r* 47.4%

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

      2. sqrt-div 33.2%

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

   3. Applied egg-rr 33.2%

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

   if -1.9999999999999999e254 < (*.f64 V l) < -9.99999999999999965e-106

   1. Initial program 95.3%

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

   if -9.99999999999999965e-106 < (*.f64 V l) < 0.0

   1. Initial program 60.5%

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

      1. *-un-lft-identity 60.5%

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

      2. times-frac 75.3%

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

   3. Applied egg-rr 75.3%

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

      1. *-commutative 75.3%

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

      2. frac-times 60.5%

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

      3. *-un-lft-identity 60.5%

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

      4. sqrt-div 6.8%

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

      5. clear-num 6.8%

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

      6. associate-/r/ 6.8%

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

      7. sqrt-undiv 60.5%

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

      8. *-un-lft-identity 60.5%

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

      9. times-frac 75.3%

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

      10. /-rgt-identity 75.3%

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

   5. Applied egg-rr 75.3%

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

      1. associate-/r/ 75.2%

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

      2. associate-*l/ 75.4%

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

      3. *-lft-identity 75.4%

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

      4. *-commutative 75.4%

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

      5. associate-*l/ 60.5%

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

      6. associate-/l* 77.2%

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

   7. Simplified 77.2%

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

      1. *-un-lft-identity 77.2%

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

      2. associate-*l/ 77.0%

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

      3. metadata-eval 77.0%

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

      4. sqrt-div 77.2%

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

      5. clear-num 77.2%

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

      6. associate-/r* 60.5%

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

      7. sqrt-undiv 6.8%

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

      8. associate-*l/ 6.8%

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

      9. sqrt-prod 13.6%

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

      10. times-frac 11.8%

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

      11. sqrt-div 47.3%

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

   9. Applied egg-rr 47.3%

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

   if 0.0 < (*.f64 V l) < 4.99999999999999987e232

   1. Initial program 83.4%

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

      1. sqrt-div 99.3%

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

      2. associate-*r/ 95.2%

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

   3. Applied egg-rr 95.2%

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

      1. *-commutative 95.2%

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

      2. associate-*l/ 99.3%

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

   5. Simplified 99.3%

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

   if 4.99999999999999987e232 < (*.f64 V l)

   1. Initial program 34.4%

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

      1. *-un-lft-identity 34.4%

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

      2. times-frac 79.8%

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

   3. Applied egg-rr 79.8%

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

      1. *-commutative 79.8%

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

      2. frac-times 34.4%

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

      3. *-un-lft-identity 34.4%

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

      4. sqrt-div 34.4%

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

      5. clear-num 34.5%

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

      6. associate-/r/ 34.6%

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

      7. sqrt-undiv 34.6%

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

      8. *-un-lft-identity 34.6%

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

      9. times-frac 79.9%

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

      10. /-rgt-identity 79.9%

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

   5. Applied egg-rr 79.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{V \cdot \frac{\ell}{A}}}{c0}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+254}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-105}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+232}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{V \cdot \frac{\ell}{A}}}{c0}}\\ \end{array} \]

Alternative 5: 81.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 8.5 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{V \cdot \frac{\ell}{A}}}{c0}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= l 8.5e-305)
   (/ 1.0 (/ (sqrt (* V (/ l A))) c0))
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= 8.5e-305) {
		tmp = 1.0 / (sqrt((V * (l / A))) / c0);
	} else {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= 8.5d-305) then
        tmp = 1.0d0 / (sqrt((v * (l / a))) / c0)
    else
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= 8.5e-305) {
		tmp = 1.0 / (Math.sqrt((V * (l / A))) / c0);
	} else {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if l <= 8.5e-305:
		tmp = 1.0 / (math.sqrt((V * (l / A))) / c0)
	else:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (l <= 8.5e-305)
		tmp = Float64(1.0 / Float64(sqrt(Float64(V * Float64(l / A))) / c0));
	else
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (l <= 8.5e-305)
		tmp = 1.0 / (sqrt((V * (l / A))) / c0);
	else
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[l, 8.5e-305], N[(1.0 / N[(N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 8.4999999999999997e-305

   1. Initial program 72.0%

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

      1. *-un-lft-identity 72.0%

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

      2. times-frac 70.5%

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

   3. Applied egg-rr 70.5%

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

      1. *-commutative 70.5%

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

      2. frac-times 72.0%

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

      3. *-un-lft-identity 72.0%

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

      4. sqrt-div 44.4%

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

      5. clear-num 44.4%

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

      6. associate-/r/ 44.4%

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

      7. sqrt-undiv 72.5%

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

      8. *-un-lft-identity 72.5%

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

      9. times-frac 71.0%

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

      10. /-rgt-identity 71.0%

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

   5. Applied egg-rr 71.0%

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

   if 8.4999999999999997e-305 < l

   1. Initial program 73.9%

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

      1. associate-/r* 80.0%

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

      2. sqrt-div 87.9%

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

   3. Applied egg-rr 87.9%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8.5 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{V \cdot \frac{\ell}{A}}}{c0}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 6: 80.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 10^{+275}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (or (<= t_0 0.0) (not (<= t_0 1e+275)))
     (* c0 (sqrt (/ (/ A V) l)))
     (* c0 (sqrt t_0)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 1e+275)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = c0 * sqrt(t_0);
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 1d+275))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = c0 * sqrt(t_0)
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 1e+275)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 1e+275):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 1e+275))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(c0 * sqrt(t_0));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 1e+275)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = c0 * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 1e+275]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 9.9999999999999996e274 < (/.f64 A (*.f64 V l))

   1. Initial program 38.0%

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

      1. *-un-lft-identity 38.0%

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

      2. times-frac 58.0%

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

   3. Applied egg-rr 58.0%

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

      1. frac-times 38.0%

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

      2. *-un-lft-identity 38.0%

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

      3. associate-/r* 57.1%

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

   5. Applied egg-rr 57.1%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 9.9999999999999996e274

   1. Initial program 98.8%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 10^{+275}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]

Alternative 7: 80.9% accurate, 0.9× speedup

Math

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

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 1e+281) (* c0 (sqrt t_0)) (/ c0 (sqrt (* V (/ l A))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 1e+281) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 1d+281) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 1e+281) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 1e+281:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 1e+281)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 1e+281)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+281], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 27.8%

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

      1. *-un-lft-identity 27.8%

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

      2. times-frac 51.3%

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

   3. Applied egg-rr 51.3%

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

      1. frac-times 27.8%

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

      2. *-un-lft-identity 27.8%

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

      3. associate-/r* 51.3%

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

   5. Applied egg-rr 51.3%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 1e281

   1. Initial program 98.8%

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

   if 1e281 < (/.f64 A (*.f64 V l))

   1. Initial program 46.7%

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

      1. *-un-lft-identity 46.7%

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

      2. times-frac 63.6%

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

   3. Applied egg-rr 63.6%

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

      1. frac-times 46.7%

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

      2. *-un-lft-identity 46.7%

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

      3. sqrt-div 33.4%

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

      4. clear-num 33.4%

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

      5. div-inv 33.5%

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

      6. sqrt-undiv 47.8%

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

      7. *-un-lft-identity 47.8%

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

      8. times-frac 64.9%

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

      9. /-rgt-identity 64.9%

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

   5. Applied egg-rr 64.9%

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

4. Final simplification 81.7%

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

Alternative 8: 81.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 10^{+275}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 1e+275) (* c0 (sqrt t_0)) (/ c0 (sqrt (* l (/ V A))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 1e+275) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((l * (V / A)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 1d+275) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((l * (v / a)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 1e+275) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((l * (V / A)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 1e+275:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((l * (V / A)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 1e+275)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 1e+275)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((l * (V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+275], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 27.8%

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

      1. *-un-lft-identity 27.8%

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

      2. times-frac 51.3%

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

   3. Applied egg-rr 51.3%

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

      1. frac-times 27.8%

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

      2. *-un-lft-identity 27.8%

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

      3. associate-/r* 51.3%

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

   5. Applied egg-rr 51.3%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 9.9999999999999996e274

   1. Initial program 98.8%

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

   if 9.9999999999999996e274 < (/.f64 A (*.f64 V l))

   1. Initial program 47.6%

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

      1. *-un-lft-identity 47.6%

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

      2. times-frac 64.3%

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

   3. Applied egg-rr 64.3%

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

      1. *-commutative 64.3%

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

      2. frac-times 47.6%

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

      3. *-un-lft-identity 47.6%

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

      4. sqrt-div 32.9%

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

      5. clear-num 32.8%

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

      6. associate-/r/ 32.8%

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

      7. sqrt-undiv 48.8%

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

      8. *-un-lft-identity 48.8%

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

      9. times-frac 65.5%

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

      10. /-rgt-identity 65.5%

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

   5. Applied egg-rr 65.5%

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

      1. associate-/r/ 65.4%

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

      2. associate-*l/ 65.5%

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

      3. *-lft-identity 65.5%

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

      4. *-commutative 65.5%

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

      5. associate-*l/ 48.8%

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

      6. associate-/l* 62.6%

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

   7. Simplified 62.6%

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

   8. Taylor expanded in l around 0 48.8%

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

      1. associate-*l/ 64.2%

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

      2. *-commutative 64.2%

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

   10. Simplified 64.2%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 10^{+275}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]

Alternative 9: 74.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

2. Final simplification 72.9%

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

Reproduce

herbie shell --seed 2023228 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))