Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.9% → 90.6%

Time: 10.8s

Alternatives: 18

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{-A}\\ t_1 := \sqrt{-V}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+221}:\\ \;\;\;\;\frac{t\_0}{t\_1} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-253}:\\ \;\;\;\;c0 \cdot \frac{t\_0}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-313}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{t\_1}\\ \mathbf{elif}\;V \cdot \ell \leq \infty:\\ \;\;\;\;c0 \cdot \left({\left(V \cdot \ell\right)}^{-0.5} \cdot \sqrt{A}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{c0}{V}} \cdot \sqrt{A \cdot \frac{c0}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (- A))) (t_1 (sqrt (- V))))
   (if (<= (* V l) -5e+221)
     (* (/ t_0 t_1) (/ c0 (sqrt l)))
     (if (<= (* V l) -5e-253)
       (* c0 (/ t_0 (sqrt (* V (- l)))))
       (if (<= (* V l) 2e-313)
         (* c0 (/ (sqrt (/ A (- l))) t_1))
         (if (<= (* V l) INFINITY)
           (* c0 (* (pow (* V l) -0.5) (sqrt A)))
           (* (sqrt (/ c0 V)) (sqrt (* A (/ c0 l))))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt(-A);
	double t_1 = sqrt(-V);
	double tmp;
	if ((V * l) <= -5e+221) {
		tmp = (t_0 / t_1) * (c0 / sqrt(l));
	} else if ((V * l) <= -5e-253) {
		tmp = c0 * (t_0 / sqrt((V * -l)));
	} else if ((V * l) <= 2e-313) {
		tmp = c0 * (sqrt((A / -l)) / t_1);
	} else if ((V * l) <= ((double) INFINITY)) {
		tmp = c0 * (pow((V * l), -0.5) * sqrt(A));
	} else {
		tmp = sqrt((c0 / V)) * sqrt((A * (c0 / l)));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt(-A);
	double t_1 = Math.sqrt(-V);
	double tmp;
	if ((V * l) <= -5e+221) {
		tmp = (t_0 / t_1) * (c0 / Math.sqrt(l));
	} else if ((V * l) <= -5e-253) {
		tmp = c0 * (t_0 / Math.sqrt((V * -l)));
	} else if ((V * l) <= 2e-313) {
		tmp = c0 * (Math.sqrt((A / -l)) / t_1);
	} else if ((V * l) <= Double.POSITIVE_INFINITY) {
		tmp = c0 * (Math.pow((V * l), -0.5) * Math.sqrt(A));
	} else {
		tmp = Math.sqrt((c0 / V)) * Math.sqrt((A * (c0 / l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt(-A)
	t_1 = math.sqrt(-V)
	tmp = 0
	if (V * l) <= -5e+221:
		tmp = (t_0 / t_1) * (c0 / math.sqrt(l))
	elif (V * l) <= -5e-253:
		tmp = c0 * (t_0 / math.sqrt((V * -l)))
	elif (V * l) <= 2e-313:
		tmp = c0 * (math.sqrt((A / -l)) / t_1)
	elif (V * l) <= math.inf:
		tmp = c0 * (math.pow((V * l), -0.5) * math.sqrt(A))
	else:
		tmp = math.sqrt((c0 / V)) * math.sqrt((A * (c0 / l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(-A))
	t_1 = sqrt(Float64(-V))
	tmp = 0.0
	if (Float64(V * l) <= -5e+221)
		tmp = Float64(Float64(t_0 / t_1) * Float64(c0 / sqrt(l)));
	elseif (Float64(V * l) <= -5e-253)
		tmp = Float64(c0 * Float64(t_0 / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 2e-313)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / Float64(-l))) / t_1));
	elseif (Float64(V * l) <= Inf)
		tmp = Float64(c0 * Float64((Float64(V * l) ^ -0.5) * sqrt(A)));
	else
		tmp = Float64(sqrt(Float64(c0 / V)) * sqrt(Float64(A * Float64(c0 / l))));
	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 = sqrt(-A);
	t_1 = sqrt(-V);
	tmp = 0.0;
	if ((V * l) <= -5e+221)
		tmp = (t_0 / t_1) * (c0 / sqrt(l));
	elseif ((V * l) <= -5e-253)
		tmp = c0 * (t_0 / sqrt((V * -l)));
	elseif ((V * l) <= 2e-313)
		tmp = c0 * (sqrt((A / -l)) / t_1);
	elseif ((V * l) <= Inf)
		tmp = c0 * (((V * l) ^ -0.5) * sqrt(A));
	else
		tmp = sqrt((c0 / V)) * sqrt((A * (c0 / 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_] := Block[{t$95$0 = N[Sqrt[(-A)], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-V)], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e+221], N[(N[(t$95$0 / t$95$1), $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-253], N[(c0 * N[(t$95$0 / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e-313], N[(c0 * N[(N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], Infinity], N[(c0 * N[(N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(c0 / V), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(A * N[(c0 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -5.0000000000000002e221

   1. Initial program 40.8%

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

      1. associate-/r* 63.7%

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

      2. sqrt-div 45.3%

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

      3. associate-*r/ 45.5%

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

   4. Applied egg-rr 45.5%

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

      1. *-commutative 45.5%

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

      2. associate-/l* 45.3%

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

   6. Simplified 45.3%

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

      1. frac-2neg 45.5%

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

      2. sqrt-div 57.0%

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

   8. Applied egg-rr 56.8%

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

   if -5.0000000000000002e221 < (*.f64 V l) < -4.99999999999999971e-253

   1. Initial program 86.1%

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

      1. frac-2neg 86.1%

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

      2. sqrt-div 99.4%

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

      3. *-commutative 99.4%

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

      4. distribute-rgt-neg-in 99.4%

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

   4. Applied egg-rr 99.4%

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

   if -4.99999999999999971e-253 < (*.f64 V l) < 1.99999999998e-313

   1. Initial program 39.1%

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

      1. associate-/r* 60.6%

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

      2. div-inv 60.5%

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

      3. div-inv 60.5%

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

      4. associate-*l* 34.5%

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

   4. Applied egg-rr 34.5%

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

      1. associate-*r* 60.5%

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

      2. div-inv 60.5%

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

      3. associate-*l/ 60.6%

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

      4. div-inv 60.7%

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

      5. frac-2neg 60.7%

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

      6. sqrt-div 50.0%

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

      7. distribute-neg-frac2 50.0%

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

   6. Applied egg-rr 50.0%

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

      1. distribute-frac-neg2 50.0%

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

      2. distribute-frac-neg 50.0%

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

   8. Simplified 50.0%

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

   if 1.99999999998e-313 < (*.f64 V l) < +inf.0

   1. Initial program 78.2%

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

      1. associate-/r* 73.5%

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

      2. div-inv 73.4%

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

      3. div-inv 73.4%

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

      4. associate-*l* 78.3%

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

   4. Applied egg-rr 78.3%

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

      1. *-commutative 78.3%

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

      2. sqrt-prod 89.3%

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

      3. frac-times 88.3%

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

      4. metadata-eval 88.3%

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

      5. sqrt-div 88.2%

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

      6. metadata-eval 88.2%

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

      7. pow1/2 88.2%

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

      8. pow-flip 88.3%

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

      9. metadata-eval 88.3%

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

   6. Applied egg-rr 88.3%

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

   if +inf.0 < (*.f64 V l)

   1. Initial program 74.1%

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

      1. associate-/r* 73.1%

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

      2. div-inv 73.1%

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

      3. div-inv 73.1%

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

      4. associate-*l* 74.0%

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

   4. Applied egg-rr 74.0%

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

      1. sqrt-prod 48.7%

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

      2. associate-*r* 46.2%

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

      3. frac-times 45.7%

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

      4. metadata-eval 45.7%

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

      5. sqrt-div 45.7%

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

      6. metadata-eval 45.7%

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

      7. div-inv 45.7%

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

      8. associate-*r/ 48.5%

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

      9. clear-num 48.4%

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

      10. un-div-inv 48.5%

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

      11. sqrt-undiv 74.1%

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

   6. Applied egg-rr 74.1%

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

   8. Applied egg-rr 18.8%

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

      1. *-commutative 18.8%

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

      2. unpow1/2 18.8%

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

      3. associate-*l/ 19.6%

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

      4. associate-/l* 19.3%

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

   10. Simplified 19.3%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+221}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{-V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-253}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-313}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq \infty:\\ \;\;\;\;c0 \cdot \left({\left(V \cdot \ell\right)}^{-0.5} \cdot \sqrt{A}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{c0}{V}} \cdot \sqrt{A \cdot \frac{c0}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-305}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+276}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

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 4e-305)
     (/ c0 (sqrt (* V (/ l A))))
     (if (<= t_0 2e+276) t_0 (/ c0 (sqrt (* l (/ V 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 <= 4e-305) {
		tmp = c0 / sqrt((V * (l / A)));
	} else if (t_0 <= 2e+276) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt((l * (V / 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 <= 4d-305) then
        tmp = c0 / sqrt((v * (l / a)))
    else if (t_0 <= 2d+276) then
        tmp = t_0
    else
        tmp = c0 / sqrt((l * (v / 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 <= 4e-305) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else if (t_0 <= 2e+276) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt((l * (V / 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 <= 4e-305:
		tmp = c0 / math.sqrt((V * (l / A)))
	elif t_0 <= 2e+276:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt((l * (V / 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 <= 4e-305)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	elseif (t_0 <= 2e+276)
		tmp = t_0;
	else
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / 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 <= 4e-305)
		tmp = c0 / sqrt((V * (l / A)));
	elseif (t_0 <= 2e+276)
		tmp = t_0;
	else
		tmp = c0 / sqrt((l * (V / 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, 4e-305], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+276], t$95$0, N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 3.99999999999999999e-305

   1. Initial program 68.0%

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

      1. associate-/r* 69.7%

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

      2. div-inv 69.6%

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

      3. div-inv 69.6%

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

      4. associate-*l* 68.5%

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

   4. Applied egg-rr 68.5%

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

      1. sqrt-prod 50.4%

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

      2. associate-*r* 48.2%

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

      3. frac-times 47.4%

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

      4. metadata-eval 47.4%

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

      5. sqrt-div 47.4%

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

      6. metadata-eval 47.4%

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

      7. div-inv 47.4%

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

      8. associate-*r/ 49.6%

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

      9. clear-num 49.6%

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

      10. un-div-inv 49.6%

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

      11. sqrt-undiv 67.3%

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

   6. Applied egg-rr 67.3%

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

      1. associate-*r/ 64.8%

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

   8. Simplified 64.8%

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

   if 3.99999999999999999e-305 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.0000000000000001e276

   1. Initial program 98.1%

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

   if 2.0000000000000001e276 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 53.8%

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

      1. associate-/r* 60.7%

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

      2. div-inv 60.7%

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

      3. div-inv 60.7%

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

      4. associate-*l* 53.8%

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

   4. Applied egg-rr 53.8%

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

      1. sqrt-prod 45.2%

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

      2. associate-*r* 45.0%

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

      3. frac-times 44.9%

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

      4. metadata-eval 44.9%

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

      5. sqrt-div 45.0%

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

      6. metadata-eval 45.0%

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

      7. div-inv 44.9%

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

      8. associate-*r/ 45.2%

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

      9. clear-num 45.2%

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

      10. un-div-inv 45.2%

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

      11. sqrt-undiv 57.4%

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

   6. Applied egg-rr 57.4%

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

      1. associate-*l/ 62.7%

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

      2. *-commutative 62.7%

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

   8. Simplified 62.7%

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

Alternative 3: 88.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{-V}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{-193}:\\ \;\;\;\;\frac{\frac{\sqrt{-A}}{t\_0}}{\frac{\sqrt{\ell}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-313}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{t\_0}\\ \mathbf{elif}\;V \cdot \ell \leq \infty:\\ \;\;\;\;c0 \cdot \left({\left(V \cdot \ell\right)}^{-0.5} \cdot \sqrt{A}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{c0}{V}} \cdot \sqrt{A \cdot \frac{c0}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (- V))))
   (if (<= (* V l) -5e-193)
     (/ (/ (sqrt (- A)) t_0) (/ (sqrt l) c0))
     (if (<= (* V l) 2e-313)
       (* c0 (/ (sqrt (/ A (- l))) t_0))
       (if (<= (* V l) INFINITY)
         (* c0 (* (pow (* V l) -0.5) (sqrt A)))
         (* (sqrt (/ c0 V)) (sqrt (* A (/ c0 l)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt(-V);
	double tmp;
	if ((V * l) <= -5e-193) {
		tmp = (sqrt(-A) / t_0) / (sqrt(l) / c0);
	} else if ((V * l) <= 2e-313) {
		tmp = c0 * (sqrt((A / -l)) / t_0);
	} else if ((V * l) <= ((double) INFINITY)) {
		tmp = c0 * (pow((V * l), -0.5) * sqrt(A));
	} else {
		tmp = sqrt((c0 / V)) * sqrt((A * (c0 / l)));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt(-V);
	double tmp;
	if ((V * l) <= -5e-193) {
		tmp = (Math.sqrt(-A) / t_0) / (Math.sqrt(l) / c0);
	} else if ((V * l) <= 2e-313) {
		tmp = c0 * (Math.sqrt((A / -l)) / t_0);
	} else if ((V * l) <= Double.POSITIVE_INFINITY) {
		tmp = c0 * (Math.pow((V * l), -0.5) * Math.sqrt(A));
	} else {
		tmp = Math.sqrt((c0 / V)) * Math.sqrt((A * (c0 / l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt(-V)
	tmp = 0
	if (V * l) <= -5e-193:
		tmp = (math.sqrt(-A) / t_0) / (math.sqrt(l) / c0)
	elif (V * l) <= 2e-313:
		tmp = c0 * (math.sqrt((A / -l)) / t_0)
	elif (V * l) <= math.inf:
		tmp = c0 * (math.pow((V * l), -0.5) * math.sqrt(A))
	else:
		tmp = math.sqrt((c0 / V)) * math.sqrt((A * (c0 / l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(-V))
	tmp = 0.0
	if (Float64(V * l) <= -5e-193)
		tmp = Float64(Float64(sqrt(Float64(-A)) / t_0) / Float64(sqrt(l) / c0));
	elseif (Float64(V * l) <= 2e-313)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / Float64(-l))) / t_0));
	elseif (Float64(V * l) <= Inf)
		tmp = Float64(c0 * Float64((Float64(V * l) ^ -0.5) * sqrt(A)));
	else
		tmp = Float64(sqrt(Float64(c0 / V)) * sqrt(Float64(A * Float64(c0 / l))));
	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 = sqrt(-V);
	tmp = 0.0;
	if ((V * l) <= -5e-193)
		tmp = (sqrt(-A) / t_0) / (sqrt(l) / c0);
	elseif ((V * l) <= 2e-313)
		tmp = c0 * (sqrt((A / -l)) / t_0);
	elseif ((V * l) <= Inf)
		tmp = c0 * (((V * l) ^ -0.5) * sqrt(A));
	else
		tmp = sqrt((c0 / V)) * sqrt((A * (c0 / 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_] := Block[{t$95$0 = N[Sqrt[(-V)], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e-193], N[(N[(N[Sqrt[(-A)], $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e-313], N[(c0 * N[(N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], Infinity], N[(c0 * N[(N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(c0 / V), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(A * N[(c0 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -5.0000000000000005e-193

   1. Initial program 76.7%

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

      1. associate-/r* 75.6%

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

      2. div-inv 75.6%

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

      3. div-inv 75.7%

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

      4. associate-*l* 77.4%

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

   4. Applied egg-rr 77.4%

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

      1. sqrt-prod 0.0%

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

      2. associate-*r* 0.0%

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

      3. frac-times 0.0%

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

      4. metadata-eval 0.0%

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

      5. sqrt-div 0.0%

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

      6. metadata-eval 0.0%

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

      7. div-inv 0.0%

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

      8. *-commutative 0.0%

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

      9. sqrt-prod 0.0%

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

      10. frac-times 0.0%

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

      11. sqrt-div 45.0%

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

      12. *-commutative 45.0%

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

      13. clear-num 45.0%

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

      14. un-div-inv 45.9%

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

   6. Applied egg-rr 45.9%

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

      1. frac-2neg 45.9%

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

      2. sqrt-div 54.6%

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

   8. Applied egg-rr 54.6%

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

   if -5.0000000000000005e-193 < (*.f64 V l) < 1.99999999998e-313

   1. Initial program 47.1%

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

      1. associate-/r* 63.4%

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

      2. div-inv 63.4%

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

      3. div-inv 63.4%

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

      4. associate-*l* 43.6%

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

   4. Applied egg-rr 43.6%

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

      1. associate-*r* 63.4%

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

      2. div-inv 63.4%

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

      3. associate-*l/ 63.4%

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

      4. div-inv 63.4%

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

      5. frac-2neg 63.4%

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

      6. sqrt-div 48.1%

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

      7. distribute-neg-frac2 48.1%

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

   6. Applied egg-rr 48.1%

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

      1. distribute-frac-neg2 48.1%

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

      2. distribute-frac-neg 48.1%

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

   8. Simplified 48.1%

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

   if 1.99999999998e-313 < (*.f64 V l) < +inf.0

   1. Initial program 78.2%

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

      1. associate-/r* 73.5%

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

      2. div-inv 73.4%

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

      3. div-inv 73.4%

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

      4. associate-*l* 78.3%

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

   4. Applied egg-rr 78.3%

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

      1. *-commutative 78.3%

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

      2. sqrt-prod 89.3%

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

      3. frac-times 88.3%

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

      4. metadata-eval 88.3%

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

      5. sqrt-div 88.2%

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

      6. metadata-eval 88.2%

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

      7. pow1/2 88.2%

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

      8. pow-flip 88.3%

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

      9. metadata-eval 88.3%

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

   6. Applied egg-rr 88.3%

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

   if +inf.0 < (*.f64 V l)

   1. Initial program 74.1%

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

      1. associate-/r* 73.1%

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

      2. div-inv 73.1%

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

      3. div-inv 73.1%

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

      4. associate-*l* 74.0%

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

   4. Applied egg-rr 74.0%

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

      1. sqrt-prod 48.7%

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

      2. associate-*r* 46.2%

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

      3. frac-times 45.7%

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

      4. metadata-eval 45.7%

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

      5. sqrt-div 45.7%

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

      6. metadata-eval 45.7%

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

      7. div-inv 45.7%

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

      8. associate-*r/ 48.5%

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

      9. clear-num 48.4%

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

      10. un-div-inv 48.5%

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

      11. sqrt-undiv 74.1%

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

   6. Applied egg-rr 74.1%

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

   8. Applied egg-rr 18.8%

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

      1. *-commutative 18.8%

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

      2. unpow1/2 18.8%

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

      3. associate-*l/ 19.6%

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

      4. associate-/l* 19.3%

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

   10. Simplified 19.3%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{-193}:\\ \;\;\;\;\frac{\frac{\sqrt{-A}}{\sqrt{-V}}}{\frac{\sqrt{\ell}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-313}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq \infty:\\ \;\;\;\;c0 \cdot \left({\left(V \cdot \ell\right)}^{-0.5} \cdot \sqrt{A}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{c0}{V}} \cdot \sqrt{A \cdot \frac{c0}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-253}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-313}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq \infty:\\ \;\;\;\;c0 \cdot \left({\left(V \cdot \ell\right)}^{-0.5} \cdot \sqrt{A}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{c0}{V}} \cdot \sqrt{A \cdot \frac{c0}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (/ (sqrt (/ A (- l))) (sqrt (- V))))))
   (if (<= (* V l) (- INFINITY))
     t_0
     (if (<= (* V l) -5e-253)
       (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
       (if (<= (* V l) 2e-313)
         t_0
         (if (<= (* V l) INFINITY)
           (* c0 (* (pow (* V l) -0.5) (sqrt A)))
           (* (sqrt (/ c0 V)) (sqrt (* A (/ c0 l))))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (sqrt((A / -l)) / sqrt(-V));
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((V * l) <= -5e-253) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 2e-313) {
		tmp = t_0;
	} else if ((V * l) <= ((double) INFINITY)) {
		tmp = c0 * (pow((V * l), -0.5) * sqrt(A));
	} else {
		tmp = sqrt((c0 / V)) * sqrt((A * (c0 / l)));
	}
	return tmp;
}

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 / -l)) / Math.sqrt(-V));
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if ((V * l) <= -5e-253) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 2e-313) {
		tmp = t_0;
	} else if ((V * l) <= Double.POSITIVE_INFINITY) {
		tmp = c0 * (Math.pow((V * l), -0.5) * Math.sqrt(A));
	} else {
		tmp = Math.sqrt((c0 / V)) * Math.sqrt((A * (c0 / l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * (math.sqrt((A / -l)) / math.sqrt(-V))
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = t_0
	elif (V * l) <= -5e-253:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 2e-313:
		tmp = t_0
	elif (V * l) <= math.inf:
		tmp = c0 * (math.pow((V * l), -0.5) * math.sqrt(A))
	else:
		tmp = math.sqrt((c0 / V)) * math.sqrt((A * (c0 / l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * Float64(sqrt(Float64(A / Float64(-l))) / sqrt(Float64(-V))))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(V * l) <= -5e-253)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 2e-313)
		tmp = t_0;
	elseif (Float64(V * l) <= Inf)
		tmp = Float64(c0 * Float64((Float64(V * l) ^ -0.5) * sqrt(A)));
	else
		tmp = Float64(sqrt(Float64(c0 / V)) * sqrt(Float64(A * Float64(c0 / l))));
	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 / -l)) / sqrt(-V));
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = t_0;
	elseif ((V * l) <= -5e-253)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 2e-313)
		tmp = t_0;
	elseif ((V * l) <= Inf)
		tmp = c0 * (((V * l) ^ -0.5) * sqrt(A));
	else
		tmp = sqrt((c0 / V)) * sqrt((A * (c0 / 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_] := Block[{t$95$0 = N[(c0 * N[(N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -5e-253], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e-313], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], Infinity], N[(c0 * N[(N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(c0 / V), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(A * N[(c0 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -inf.0 or -4.99999999999999971e-253 < (*.f64 V l) < 1.99999999998e-313

   1. Initial program 37.0%

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

      1. associate-/r* 60.3%

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

      2. div-inv 60.3%

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

      3. div-inv 60.3%

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

      4. associate-*l* 36.3%

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

   4. Applied egg-rr 36.3%

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

      1. associate-*r* 60.3%

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

      2. div-inv 60.3%

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

      3. associate-*l/ 60.3%

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

      4. div-inv 60.3%

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

      5. frac-2neg 60.3%

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

      6. sqrt-div 51.2%

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

      7. distribute-neg-frac2 51.2%

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

   6. Applied egg-rr 51.2%

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

      1. distribute-frac-neg2 51.2%

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

      2. distribute-frac-neg 51.2%

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

   8. Simplified 51.2%

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

   if -inf.0 < (*.f64 V l) < -4.99999999999999971e-253

   1. Initial program 86.4%

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

      1. frac-2neg 86.4%

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

      2. sqrt-div 99.4%

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

      3. *-commutative 99.4%

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

      4. distribute-rgt-neg-in 99.4%

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

   4. Applied egg-rr 99.4%

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

   if 1.99999999998e-313 < (*.f64 V l) < +inf.0

   1. Initial program 78.2%

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

      1. associate-/r* 73.5%

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

      2. div-inv 73.4%

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

      3. div-inv 73.4%

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

      4. associate-*l* 78.3%

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

   4. Applied egg-rr 78.3%

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

      1. *-commutative 78.3%

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

      2. sqrt-prod 89.3%

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

      3. frac-times 88.3%

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

      4. metadata-eval 88.3%

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

      5. sqrt-div 88.2%

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

      6. metadata-eval 88.2%

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

      7. pow1/2 88.2%

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

      8. pow-flip 88.3%

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

      9. metadata-eval 88.3%

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

   6. Applied egg-rr 88.3%

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

   if +inf.0 < (*.f64 V l)

   1. Initial program 74.1%

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

      1. associate-/r* 73.1%

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

      2. div-inv 73.1%

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

      3. div-inv 73.1%

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

      4. associate-*l* 74.0%

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

   4. Applied egg-rr 74.0%

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

      1. sqrt-prod 48.7%

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

      2. associate-*r* 46.2%

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

      3. frac-times 45.7%

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

      4. metadata-eval 45.7%

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

      5. sqrt-div 45.7%

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

      6. metadata-eval 45.7%

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

      7. div-inv 45.7%

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

      8. associate-*r/ 48.5%

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

      9. clear-num 48.4%

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

      10. un-div-inv 48.5%

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

      11. sqrt-undiv 74.1%

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

   6. Applied egg-rr 74.1%

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

   8. Applied egg-rr 18.8%

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

      1. *-commutative 18.8%

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

      2. unpow1/2 18.8%

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

      3. associate-*l/ 19.6%

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

      4. associate-/l* 19.3%

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

   10. Simplified 19.3%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-253}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-313}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq \infty:\\ \;\;\;\;c0 \cdot \left({\left(V \cdot \ell\right)}^{-0.5} \cdot \sqrt{A}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{c0}{V}} \cdot \sqrt{A \cdot \frac{c0}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+221}:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\frac{\sqrt{\ell}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-322}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq \infty\right):\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left({\left(V \cdot \ell\right)}^{-0.5} \cdot \sqrt{A}\right)\\ \end{array} \end{array} \]

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) -5e+221)
   (/ (sqrt (/ A V)) (/ (sqrt l) c0))
   (if (<= (* V l) -1e-322)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (or (<= (* V l) 0.0) (not (<= (* V l) INFINITY)))
       (sqrt (* (/ A l) (* c0 (/ c0 V))))
       (* c0 (* (pow (* V l) -0.5) (sqrt A)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+221) {
		tmp = sqrt((A / V)) / (sqrt(l) / c0);
	} else if ((V * l) <= -1e-322) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= ((double) INFINITY))) {
		tmp = sqrt(((A / l) * (c0 * (c0 / V))));
	} else {
		tmp = c0 * (pow((V * l), -0.5) * sqrt(A));
	}
	return tmp;
}

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) <= -5e+221) {
		tmp = Math.sqrt((A / V)) / (Math.sqrt(l) / c0);
	} else if ((V * l) <= -1e-322) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= Double.POSITIVE_INFINITY)) {
		tmp = Math.sqrt(((A / l) * (c0 * (c0 / V))));
	} else {
		tmp = c0 * (Math.pow((V * l), -0.5) * Math.sqrt(A));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -5e+221:
		tmp = math.sqrt((A / V)) / (math.sqrt(l) / c0)
	elif (V * l) <= -1e-322:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif ((V * l) <= 0.0) or not ((V * l) <= math.inf):
		tmp = math.sqrt(((A / l) * (c0 * (c0 / V))))
	else:
		tmp = c0 * (math.pow((V * l), -0.5) * math.sqrt(A))
	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) <= -5e+221)
		tmp = Float64(sqrt(Float64(A / V)) / Float64(sqrt(l) / c0));
	elseif (Float64(V * l) <= -1e-322)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif ((Float64(V * l) <= 0.0) || !(Float64(V * l) <= Inf))
		tmp = sqrt(Float64(Float64(A / l) * Float64(c0 * Float64(c0 / V))));
	else
		tmp = Float64(c0 * Float64((Float64(V * l) ^ -0.5) * sqrt(A)));
	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) <= -5e+221)
		tmp = sqrt((A / V)) / (sqrt(l) / c0);
	elseif ((V * l) <= -1e-322)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif (((V * l) <= 0.0) || ~(((V * l) <= Inf)))
		tmp = sqrt(((A / l) * (c0 * (c0 / V))));
	else
		tmp = c0 * (((V * l) ^ -0.5) * sqrt(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_] := If[LessEqual[N[(V * l), $MachinePrecision], -5e+221], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-322], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(V * l), $MachinePrecision], Infinity]], $MachinePrecision]], N[Sqrt[N[(N[(A / l), $MachinePrecision] * N[(c0 * N[(c0 / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(c0 * N[(N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -5.0000000000000002e221

   1. Initial program 40.8%

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

      1. associate-/r* 63.7%

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

      2. div-inv 63.7%

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

      3. div-inv 63.7%

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

      4. associate-*l* 44.1%

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

   4. Applied egg-rr 44.1%

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

      1. sqrt-prod 0.0%

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

      2. associate-*r* 0.0%

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

      3. frac-times 0.0%

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

      4. metadata-eval 0.0%

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

      5. sqrt-div 0.0%

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

      6. metadata-eval 0.0%

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

      7. div-inv 0.0%

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

      8. *-commutative 0.0%

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

      9. sqrt-prod 0.0%

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

      10. frac-times 0.0%

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

      11. sqrt-div 45.3%

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

      12. *-commutative 45.3%

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

      13. clear-num 45.4%

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

      14. un-div-inv 45.5%

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

   6. Applied egg-rr 45.5%

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

   if -5.0000000000000002e221 < (*.f64 V l) < -9.88131e-323

   1. Initial program 85.5%

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

      1. frac-2neg 85.5%

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

      2. sqrt-div 98.6%

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

      3. *-commutative 98.6%

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

      4. distribute-rgt-neg-in 98.6%

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

   4. Applied egg-rr 98.6%

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

   if -9.88131e-323 < (*.f64 V l) < 0.0 or +inf.0 < (*.f64 V l)

   1. Initial program 34.4%

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

      1. add-sqr-sqrt 1.5%

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

      2. sqrt-unprod 1.7%

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

      3. *-commutative 1.7%

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

      4. *-commutative 1.7%

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

      5. swap-sqr 0.9%

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

      6. add-sqr-sqrt 0.9%

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

      7. pow2 0.9%

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

   4. Applied egg-rr 0.9%

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

      1. associate-*l/ 0.7%

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

      2. *-commutative 0.7%

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

      3. times-frac 6.8%

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

   6. Simplified 6.8%

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

      1. div-inv 6.9%

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

      2. unpow2 6.9%

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

      3. associate-*l* 12.3%

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

   8. Applied egg-rr 12.3%

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

   9. Taylor expanded in c0 around 0 12.3%

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

   if 0.0 < (*.f64 V l) < +inf.0

   1. Initial program 78.2%

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

      1. associate-/r* 73.7%

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

      2. div-inv 73.6%

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

      3. div-inv 73.6%

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

      4. associate-*l* 77.7%

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

   4. Applied egg-rr 77.7%

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

      1. *-commutative 77.7%

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

      2. sqrt-prod 88.6%

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

      3. frac-times 87.7%

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

      4. metadata-eval 87.7%

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

      5. sqrt-div 88.1%

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

      6. metadata-eval 88.1%

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

      7. pow1/2 88.1%

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

      8. pow-flip 88.2%

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

      9. metadata-eval 88.2%

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

   6. Applied egg-rr 88.2%

      \[\leadsto c0 \cdot \color{blue}{\left({\left(V \cdot \ell\right)}^{-0.5} \cdot \sqrt{A}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+221}:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\frac{\sqrt{\ell}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-322}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq \infty\right):\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left({\left(V \cdot \ell\right)}^{-0.5} \cdot \sqrt{A}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+160}:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\frac{\sqrt{\ell}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-322}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq \infty\right):\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left({\left(V \cdot \ell\right)}^{-0.5} \cdot \sqrt{A}\right)\\ \end{array} \end{array} \]

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+160)
   (/ (sqrt (/ A V)) (/ (sqrt l) c0))
   (if (<= (* V l) -1e-322)
     (* c0 (sqrt (/ A (* V l))))
     (if (or (<= (* V l) 0.0) (not (<= (* V l) INFINITY)))
       (sqrt (* (/ A l) (* c0 (/ c0 V))))
       (* c0 (* (pow (* V l) -0.5) (sqrt A)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+160) {
		tmp = sqrt((A / V)) / (sqrt(l) / c0);
	} else if ((V * l) <= -1e-322) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= ((double) INFINITY))) {
		tmp = sqrt(((A / l) * (c0 * (c0 / V))));
	} else {
		tmp = c0 * (pow((V * l), -0.5) * sqrt(A));
	}
	return tmp;
}

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+160) {
		tmp = Math.sqrt((A / V)) / (Math.sqrt(l) / c0);
	} else if ((V * l) <= -1e-322) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= Double.POSITIVE_INFINITY)) {
		tmp = Math.sqrt(((A / l) * (c0 * (c0 / V))));
	} else {
		tmp = c0 * (Math.pow((V * l), -0.5) * Math.sqrt(A));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -2e+160:
		tmp = math.sqrt((A / V)) / (math.sqrt(l) / c0)
	elif (V * l) <= -1e-322:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif ((V * l) <= 0.0) or not ((V * l) <= math.inf):
		tmp = math.sqrt(((A / l) * (c0 * (c0 / V))))
	else:
		tmp = c0 * (math.pow((V * l), -0.5) * math.sqrt(A))
	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+160)
		tmp = Float64(sqrt(Float64(A / V)) / Float64(sqrt(l) / c0));
	elseif (Float64(V * l) <= -1e-322)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif ((Float64(V * l) <= 0.0) || !(Float64(V * l) <= Inf))
		tmp = sqrt(Float64(Float64(A / l) * Float64(c0 * Float64(c0 / V))));
	else
		tmp = Float64(c0 * Float64((Float64(V * l) ^ -0.5) * sqrt(A)));
	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+160)
		tmp = sqrt((A / V)) / (sqrt(l) / c0);
	elseif ((V * l) <= -1e-322)
		tmp = c0 * sqrt((A / (V * l)));
	elseif (((V * l) <= 0.0) || ~(((V * l) <= Inf)))
		tmp = sqrt(((A / l) * (c0 * (c0 / V))));
	else
		tmp = c0 * (((V * l) ^ -0.5) * sqrt(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_] := If[LessEqual[N[(V * l), $MachinePrecision], -2e+160], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-322], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(V * l), $MachinePrecision], Infinity]], $MachinePrecision]], N[Sqrt[N[(N[(A / l), $MachinePrecision] * N[(c0 * N[(c0 / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(c0 * N[(N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -2.00000000000000001e160

   1. Initial program 50.5%

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

      1. associate-/r* 66.1%

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

      2. div-inv 66.1%

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

      3. div-inv 66.1%

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

      4. associate-*l* 52.8%

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

   4. Applied egg-rr 52.8%

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

      1. sqrt-prod 0.0%

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

      2. associate-*r* 0.0%

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

      3. frac-times 0.0%

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

      4. metadata-eval 0.0%

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

      5. sqrt-div 0.0%

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

      6. metadata-eval 0.0%

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

      7. div-inv 0.0%

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

      8. *-commutative 0.0%

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

      9. sqrt-prod 0.0%

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

      10. frac-times 0.0%

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

      11. sqrt-div 44.2%

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

      12. *-commutative 44.2%

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

      13. clear-num 44.2%

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

      14. un-div-inv 44.3%

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

   6. Applied egg-rr 44.3%

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

   if -2.00000000000000001e160 < (*.f64 V l) < -9.88131e-323

   1. Initial program 87.7%

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

   if -9.88131e-323 < (*.f64 V l) < 0.0 or +inf.0 < (*.f64 V l)

   1. Initial program 34.4%

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

      1. add-sqr-sqrt 1.5%

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

      2. sqrt-unprod 1.7%

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

      3. *-commutative 1.7%

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

      4. *-commutative 1.7%

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

      5. swap-sqr 0.9%

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

      6. add-sqr-sqrt 0.9%

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

      7. pow2 0.9%

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

   4. Applied egg-rr 0.9%

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

      1. associate-*l/ 0.7%

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

      2. *-commutative 0.7%

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

      3. times-frac 6.8%

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

   6. Simplified 6.8%

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

      1. div-inv 6.9%

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

      2. unpow2 6.9%

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

      3. associate-*l* 12.3%

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

   8. Applied egg-rr 12.3%

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

   9. Taylor expanded in c0 around 0 12.3%

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

   if 0.0 < (*.f64 V l) < +inf.0

   1. Initial program 78.2%

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

      1. associate-/r* 73.7%

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

      2. div-inv 73.6%

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

      3. div-inv 73.6%

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

      4. associate-*l* 77.7%

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

   4. Applied egg-rr 77.7%

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

      1. *-commutative 77.7%

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

      2. sqrt-prod 88.6%

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

      3. frac-times 87.7%

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

      4. metadata-eval 87.7%

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

      5. sqrt-div 88.1%

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

      6. metadata-eval 88.1%

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

      7. pow1/2 88.1%

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

      8. pow-flip 88.2%

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

      9. metadata-eval 88.2%

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

   6. Applied egg-rr 88.2%

      \[\leadsto c0 \cdot \color{blue}{\left({\left(V \cdot \ell\right)}^{-0.5} \cdot \sqrt{A}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+160}:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\frac{\sqrt{\ell}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-322}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq \infty\right):\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left({\left(V \cdot \ell\right)}^{-0.5} \cdot \sqrt{A}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-253}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-313}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq \infty:\\ \;\;\;\;c0 \cdot \left({\left(V \cdot \ell\right)}^{-0.5} \cdot \sqrt{A}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \end{array} \end{array} \]

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 (- l))) (sqrt (- V))))))
   (if (<= (* V l) (- INFINITY))
     t_0
     (if (<= (* V l) -5e-253)
       (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
       (if (<= (* V l) 2e-313)
         t_0
         (if (<= (* V l) INFINITY)
           (* c0 (* (pow (* V l) -0.5) (sqrt A)))
           (sqrt (* (/ A l) (* c0 (/ c0 V))))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (sqrt((A / -l)) / sqrt(-V));
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((V * l) <= -5e-253) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 2e-313) {
		tmp = t_0;
	} else if ((V * l) <= ((double) INFINITY)) {
		tmp = c0 * (pow((V * l), -0.5) * sqrt(A));
	} else {
		tmp = sqrt(((A / l) * (c0 * (c0 / V))));
	}
	return tmp;
}

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 / -l)) / Math.sqrt(-V));
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if ((V * l) <= -5e-253) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 2e-313) {
		tmp = t_0;
	} else if ((V * l) <= Double.POSITIVE_INFINITY) {
		tmp = c0 * (Math.pow((V * l), -0.5) * Math.sqrt(A));
	} else {
		tmp = Math.sqrt(((A / l) * (c0 * (c0 / V))));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * (math.sqrt((A / -l)) / math.sqrt(-V))
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = t_0
	elif (V * l) <= -5e-253:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 2e-313:
		tmp = t_0
	elif (V * l) <= math.inf:
		tmp = c0 * (math.pow((V * l), -0.5) * math.sqrt(A))
	else:
		tmp = math.sqrt(((A / l) * (c0 * (c0 / V))))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * Float64(sqrt(Float64(A / Float64(-l))) / sqrt(Float64(-V))))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(V * l) <= -5e-253)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 2e-313)
		tmp = t_0;
	elseif (Float64(V * l) <= Inf)
		tmp = Float64(c0 * Float64((Float64(V * l) ^ -0.5) * sqrt(A)));
	else
		tmp = sqrt(Float64(Float64(A / l) * Float64(c0 * Float64(c0 / V))));
	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 / -l)) / sqrt(-V));
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = t_0;
	elseif ((V * l) <= -5e-253)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 2e-313)
		tmp = t_0;
	elseif ((V * l) <= Inf)
		tmp = c0 * (((V * l) ^ -0.5) * sqrt(A));
	else
		tmp = sqrt(((A / l) * (c0 * (c0 / V))));
	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[(N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -5e-253], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e-313], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], Infinity], N[(c0 * N[(N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(A / l), $MachinePrecision] * N[(c0 * N[(c0 / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -inf.0 or -4.99999999999999971e-253 < (*.f64 V l) < 1.99999999998e-313

   1. Initial program 37.0%

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

      1. associate-/r* 60.3%

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

      2. div-inv 60.3%

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

      3. div-inv 60.3%

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

      4. associate-*l* 36.3%

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

   4. Applied egg-rr 36.3%

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

      1. associate-*r* 60.3%

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

      2. div-inv 60.3%

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

      3. associate-*l/ 60.3%

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

      4. div-inv 60.3%

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

      5. frac-2neg 60.3%

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

      6. sqrt-div 51.2%

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

      7. distribute-neg-frac2 51.2%

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

   6. Applied egg-rr 51.2%

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

      1. distribute-frac-neg2 51.2%

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

      2. distribute-frac-neg 51.2%

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

   8. Simplified 51.2%

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

   if -inf.0 < (*.f64 V l) < -4.99999999999999971e-253

   1. Initial program 86.4%

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

      1. frac-2neg 86.4%

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

      2. sqrt-div 99.4%

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

      3. *-commutative 99.4%

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

      4. distribute-rgt-neg-in 99.4%

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

   4. Applied egg-rr 99.4%

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

   if 1.99999999998e-313 < (*.f64 V l) < +inf.0

   1. Initial program 78.2%

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

      1. associate-/r* 73.5%

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

      2. div-inv 73.4%

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

      3. div-inv 73.4%

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

      4. associate-*l* 78.3%

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

   4. Applied egg-rr 78.3%

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

      1. *-commutative 78.3%

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

      2. sqrt-prod 89.3%

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

      3. frac-times 88.3%

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

      4. metadata-eval 88.3%

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

      5. sqrt-div 88.2%

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

      6. metadata-eval 88.2%

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

      7. pow1/2 88.2%

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

      8. pow-flip 88.3%

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

      9. metadata-eval 88.3%

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

   6. Applied egg-rr 88.3%

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

   if +inf.0 < (*.f64 V l)

   1. Initial program 74.1%

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

      1. add-sqr-sqrt 39.8%

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

      2. sqrt-unprod 30.1%

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

      3. *-commutative 30.1%

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

      4. *-commutative 30.1%

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

      5. swap-sqr 26.2%

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

      6. add-sqr-sqrt 26.3%

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

      7. pow2 26.3%

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

   4. Applied egg-rr 26.3%

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

      1. associate-*l/ 26.1%

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

      2. *-commutative 26.1%

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

      3. times-frac 25.8%

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

   6. Simplified 25.8%

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

      1. div-inv 25.8%

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

      2. unpow2 25.8%

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

      3. associate-*l* 30.8%

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

   8. Applied egg-rr 30.8%

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

   9. Taylor expanded in c0 around 0 30.8%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-253}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-313}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq \infty:\\ \;\;\;\;c0 \cdot \left({\left(V \cdot \ell\right)}^{-0.5} \cdot \sqrt{A}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+160}:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\frac{\sqrt{\ell}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-322}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq \infty\right):\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

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+160)
   (/ (sqrt (/ A V)) (/ (sqrt l) c0))
   (if (<= (* V l) -1e-322)
     (* c0 (sqrt (/ A (* V l))))
     (if (or (<= (* V l) 0.0) (not (<= (* V l) INFINITY)))
       (sqrt (* (/ A l) (* c0 (/ c0 V))))
       (* c0 (/ (sqrt A) (sqrt (* 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+160) {
		tmp = sqrt((A / V)) / (sqrt(l) / c0);
	} else if ((V * l) <= -1e-322) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= ((double) INFINITY))) {
		tmp = sqrt(((A / l) * (c0 * (c0 / V))));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

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+160) {
		tmp = Math.sqrt((A / V)) / (Math.sqrt(l) / c0);
	} else if ((V * l) <= -1e-322) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= Double.POSITIVE_INFINITY)) {
		tmp = Math.sqrt(((A / l) * (c0 * (c0 / V))));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((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+160:
		tmp = math.sqrt((A / V)) / (math.sqrt(l) / c0)
	elif (V * l) <= -1e-322:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif ((V * l) <= 0.0) or not ((V * l) <= math.inf):
		tmp = math.sqrt(((A / l) * (c0 * (c0 / V))))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((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+160)
		tmp = Float64(sqrt(Float64(A / V)) / Float64(sqrt(l) / c0));
	elseif (Float64(V * l) <= -1e-322)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif ((Float64(V * l) <= 0.0) || !(Float64(V * l) <= Inf))
		tmp = sqrt(Float64(Float64(A / l) * Float64(c0 * Float64(c0 / V))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(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+160)
		tmp = sqrt((A / V)) / (sqrt(l) / c0);
	elseif ((V * l) <= -1e-322)
		tmp = c0 * sqrt((A / (V * l)));
	elseif (((V * l) <= 0.0) || ~(((V * l) <= Inf)))
		tmp = sqrt(((A / l) * (c0 * (c0 / V))));
	else
		tmp = c0 * (sqrt(A) / sqrt((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+160], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-322], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(V * l), $MachinePrecision], Infinity]], $MachinePrecision]], N[Sqrt[N[(N[(A / l), $MachinePrecision] * N[(c0 * N[(c0 / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -2.00000000000000001e160

   1. Initial program 50.5%

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

      1. associate-/r* 66.1%

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

      2. div-inv 66.1%

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

      3. div-inv 66.1%

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

      4. associate-*l* 52.8%

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

   4. Applied egg-rr 52.8%

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

      1. sqrt-prod 0.0%

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

      2. associate-*r* 0.0%

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

      3. frac-times 0.0%

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

      4. metadata-eval 0.0%

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

      5. sqrt-div 0.0%

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

      6. metadata-eval 0.0%

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

      7. div-inv 0.0%

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

      8. *-commutative 0.0%

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

      9. sqrt-prod 0.0%

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

      10. frac-times 0.0%

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

      11. sqrt-div 44.2%

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

      12. *-commutative 44.2%

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

      13. clear-num 44.2%

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

      14. un-div-inv 44.3%

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

   6. Applied egg-rr 44.3%

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

   if -2.00000000000000001e160 < (*.f64 V l) < -9.88131e-323

   1. Initial program 87.7%

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

   if -9.88131e-323 < (*.f64 V l) < 0.0 or +inf.0 < (*.f64 V l)

   1. Initial program 34.4%

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

      1. add-sqr-sqrt 1.5%

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

      2. sqrt-unprod 1.7%

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

      3. *-commutative 1.7%

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

      4. *-commutative 1.7%

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

      5. swap-sqr 0.9%

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

      6. add-sqr-sqrt 0.9%

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

      7. pow2 0.9%

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

   4. Applied egg-rr 0.9%

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

      1. associate-*l/ 0.7%

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

      2. *-commutative 0.7%

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

      3. times-frac 6.8%

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

   6. Simplified 6.8%

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

      1. div-inv 6.9%

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

      2. unpow2 6.9%

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

      3. associate-*l* 12.3%

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

   8. Applied egg-rr 12.3%

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

   9. Taylor expanded in c0 around 0 12.3%

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

   if 0.0 < (*.f64 V l) < +inf.0

   1. Initial program 78.2%

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

      1. sqrt-div 88.2%

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

      2. associate-*r/ 82.9%

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

   4. Applied egg-rr 82.9%

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

      1. associate-/l* 88.2%

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

   6. Simplified 88.2%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+160}:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\frac{\sqrt{\ell}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-322}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq \infty\right):\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+160}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-322}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq \infty\right):\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

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+160)
   (* (/ c0 (sqrt l)) (sqrt (/ A V)))
   (if (<= (* V l) -1e-322)
     (* c0 (sqrt (/ A (* V l))))
     (if (or (<= (* V l) 0.0) (not (<= (* V l) INFINITY)))
       (sqrt (* (/ A l) (* c0 (/ c0 V))))
       (* c0 (/ (sqrt A) (sqrt (* 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+160) {
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	} else if ((V * l) <= -1e-322) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= ((double) INFINITY))) {
		tmp = sqrt(((A / l) * (c0 * (c0 / V))));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

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+160) {
		tmp = (c0 / Math.sqrt(l)) * Math.sqrt((A / V));
	} else if ((V * l) <= -1e-322) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= Double.POSITIVE_INFINITY)) {
		tmp = Math.sqrt(((A / l) * (c0 * (c0 / V))));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((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+160:
		tmp = (c0 / math.sqrt(l)) * math.sqrt((A / V))
	elif (V * l) <= -1e-322:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif ((V * l) <= 0.0) or not ((V * l) <= math.inf):
		tmp = math.sqrt(((A / l) * (c0 * (c0 / V))))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((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+160)
		tmp = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(A / V)));
	elseif (Float64(V * l) <= -1e-322)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif ((Float64(V * l) <= 0.0) || !(Float64(V * l) <= Inf))
		tmp = sqrt(Float64(Float64(A / l) * Float64(c0 * Float64(c0 / V))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(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+160)
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	elseif ((V * l) <= -1e-322)
		tmp = c0 * sqrt((A / (V * l)));
	elseif (((V * l) <= 0.0) || ~(((V * l) <= Inf)))
		tmp = sqrt(((A / l) * (c0 * (c0 / V))));
	else
		tmp = c0 * (sqrt(A) / sqrt((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+160], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-322], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(V * l), $MachinePrecision], Infinity]], $MachinePrecision]], N[Sqrt[N[(N[(A / l), $MachinePrecision] * N[(c0 * N[(c0 / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -2.00000000000000001e160

   1. Initial program 50.5%

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

      1. associate-/r* 66.1%

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

      2. sqrt-div 44.2%

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

      3. associate-*r/ 44.3%

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

   4. Applied egg-rr 44.3%

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

      1. *-commutative 44.3%

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

      2. associate-/l* 44.2%

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

   6. Simplified 44.2%

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

   if -2.00000000000000001e160 < (*.f64 V l) < -9.88131e-323

   1. Initial program 87.7%

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

   if -9.88131e-323 < (*.f64 V l) < 0.0 or +inf.0 < (*.f64 V l)

   1. Initial program 34.4%

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

      1. add-sqr-sqrt 1.5%

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

      2. sqrt-unprod 1.7%

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

      3. *-commutative 1.7%

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

      4. *-commutative 1.7%

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

      5. swap-sqr 0.9%

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

      6. add-sqr-sqrt 0.9%

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

      7. pow2 0.9%

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

   4. Applied egg-rr 0.9%

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

      1. associate-*l/ 0.7%

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

      2. *-commutative 0.7%

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

      3. times-frac 6.8%

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

   6. Simplified 6.8%

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

      1. div-inv 6.9%

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

      2. unpow2 6.9%

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

      3. associate-*l* 12.3%

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

   8. Applied egg-rr 12.3%

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

   9. Taylor expanded in c0 around 0 12.3%

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

   if 0.0 < (*.f64 V l) < +inf.0

   1. Initial program 78.2%

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

      1. sqrt-div 88.2%

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

      2. associate-*r/ 82.9%

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

   4. Applied egg-rr 82.9%

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

      1. associate-/l* 88.2%

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

   6. Simplified 88.2%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+160}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-322}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq \infty\right):\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+119}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-322}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq \infty\right):\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

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+119)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* V l) -1e-322)
     (* c0 (sqrt (/ A (* V l))))
     (if (or (<= (* V l) 0.0) (not (<= (* V l) INFINITY)))
       (sqrt (* (/ A l) (* c0 (/ c0 V))))
       (* c0 (/ (sqrt A) (sqrt (* 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+119) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= -1e-322) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= ((double) INFINITY))) {
		tmp = sqrt(((A / l) * (c0 * (c0 / V))));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

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+119) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= -1e-322) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= Double.POSITIVE_INFINITY)) {
		tmp = Math.sqrt(((A / l) * (c0 * (c0 / V))));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((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+119:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= -1e-322:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif ((V * l) <= 0.0) or not ((V * l) <= math.inf):
		tmp = math.sqrt(((A / l) * (c0 * (c0 / V))))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((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+119)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= -1e-322)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif ((Float64(V * l) <= 0.0) || !(Float64(V * l) <= Inf))
		tmp = sqrt(Float64(Float64(A / l) * Float64(c0 * Float64(c0 / V))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(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+119)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= -1e-322)
		tmp = c0 * sqrt((A / (V * l)));
	elseif (((V * l) <= 0.0) || ~(((V * l) <= Inf)))
		tmp = sqrt(((A / l) * (c0 * (c0 / V))));
	else
		tmp = c0 * (sqrt(A) / sqrt((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+119], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-322], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(V * l), $MachinePrecision], Infinity]], $MachinePrecision]], N[Sqrt[N[(N[(A / l), $MachinePrecision] * N[(c0 * N[(c0 / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -9.99999999999999944e118

   1. Initial program 56.2%

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

      1. associate-/r* 67.1%

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

      2. sqrt-div 44.8%

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

      3. associate-*r/ 44.9%

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

   4. Applied egg-rr 44.9%

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

      1. associate-/l* 44.8%

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

   6. Simplified 44.8%

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

   if -9.99999999999999944e118 < (*.f64 V l) < -9.88131e-323

   1. Initial program 86.9%

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

   if -9.88131e-323 < (*.f64 V l) < 0.0 or +inf.0 < (*.f64 V l)

   1. Initial program 34.4%

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

      1. add-sqr-sqrt 1.5%

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

      2. sqrt-unprod 1.7%

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

      3. *-commutative 1.7%

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

      4. *-commutative 1.7%

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

      5. swap-sqr 0.9%

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

      6. add-sqr-sqrt 0.9%

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

      7. pow2 0.9%

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

   4. Applied egg-rr 0.9%

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

      1. associate-*l/ 0.7%

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

      2. *-commutative 0.7%

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

      3. times-frac 6.8%

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

   6. Simplified 6.8%

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

      1. div-inv 6.9%

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

      2. unpow2 6.9%

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

      3. associate-*l* 12.3%

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

   8. Applied egg-rr 12.3%

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

   9. Taylor expanded in c0 around 0 12.3%

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

   if 0.0 < (*.f64 V l) < +inf.0

   1. Initial program 78.2%

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

      1. sqrt-div 88.2%

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

      2. associate-*r/ 82.9%

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

   4. Applied egg-rr 82.9%

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

      1. associate-/l* 88.2%

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

   6. Simplified 88.2%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+119}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-322}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq \infty\right):\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-322}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq \infty\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

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

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt(((A / l) * (c0 * (c0 / V))));
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((V * l) <= -1e-322) {
		tmp = c0 / sqrt(((V * l) / A));
	} else if (((V * l) <= 0.0) || !((V * l) <= ((double) INFINITY))) {
		tmp = t_0;
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Java

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

Python

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

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(Float64(A / l) * Float64(c0 * Float64(c0 / V))))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(V * l) <= -1e-322)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	elseif ((Float64(V * l) <= 0.0) || !(Float64(V * l) <= Inf))
		tmp = t_0;
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(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)
	t_0 = sqrt(((A / l) * (c0 * (c0 / V))));
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = t_0;
	elseif ((V * l) <= -1e-322)
		tmp = c0 / sqrt(((V * l) / A));
	elseif (((V * l) <= 0.0) || ~(((V * l) <= Inf)))
		tmp = t_0;
	else
		tmp = c0 * (sqrt(A) / sqrt((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_] := Block[{t$95$0 = N[Sqrt[N[(N[(A / l), $MachinePrecision] * N[(c0 * N[(c0 / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -1e-322], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(V * l), $MachinePrecision], Infinity]], $MachinePrecision]], t$95$0, N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -inf.0 or -9.88131e-323 < (*.f64 V l) < 0.0 or +inf.0 < (*.f64 V l)

   1. Initial program 34.6%

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

      1. add-sqr-sqrt 18.1%

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

      2. sqrt-unprod 18.2%

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

      3. *-commutative 18.2%

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

      4. *-commutative 18.2%

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

      5. swap-sqr 17.2%

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

      6. add-sqr-sqrt 17.2%

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

      7. pow2 17.2%

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

   4. Applied egg-rr 17.2%

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

      1. associate-*l/ 16.8%

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

      2. *-commutative 16.8%

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

      3. times-frac 25.3%

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

   6. Simplified 25.3%

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

      1. div-inv 25.4%

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

      2. unpow2 25.4%

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

      3. associate-*l* 30.9%

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

   8. Applied egg-rr 30.9%

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

   9. Taylor expanded in c0 around 0 30.9%

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

   if -inf.0 < (*.f64 V l) < -9.88131e-323

   1. Initial program 85.9%

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

      1. associate-/r* 79.6%

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

      2. div-inv 79.6%

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

      3. div-inv 79.6%

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

      4. associate-*l* 85.5%

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

   4. Applied egg-rr 85.5%

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

      1. sqrt-prod 0.0%

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

      2. associate-*r* 0.0%

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

      3. frac-times 0.0%

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

      4. metadata-eval 0.0%

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

      5. sqrt-div 0.0%

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

      6. metadata-eval 0.0%

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

      7. div-inv 0.0%

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

      8. associate-*r/ 0.0%

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

      9. clear-num 0.0%

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

      10. un-div-inv 0.0%

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

      11. sqrt-undiv 85.9%

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

   6. Applied egg-rr 85.9%

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

   if 0.0 < (*.f64 V l) < +inf.0

   1. Initial program 78.2%

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

      1. sqrt-div 88.2%

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

      2. associate-*r/ 82.9%

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

   4. Applied egg-rr 82.9%

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

      1. associate-/l* 88.2%

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

   6. Simplified 88.2%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-322}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq \infty\right):\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-305}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 (<= t_0 4e-305) (/ c0 (sqrt (* V (/ l A)))) 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 <= 4e-305) {
		tmp = c0 / sqrt((V * (l / A)));
	} 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 <= 4d-305) then
        tmp = c0 / sqrt((v * (l / a)))
    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 <= 4e-305) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} 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 <= 4e-305:
		tmp = c0 / math.sqrt((V * (l / A)))
	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 <= 4e-305)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	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 <= 4e-305)
		tmp = c0 / sqrt((V * (l / A)));
	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[LessEqual[t$95$0, 4e-305], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 3.99999999999999999e-305

   1. Initial program 68.0%

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

      1. associate-/r* 69.7%

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

      2. div-inv 69.6%

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

      3. div-inv 69.6%

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

      4. associate-*l* 68.5%

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

   4. Applied egg-rr 68.5%

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

      1. sqrt-prod 50.4%

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

      2. associate-*r* 48.2%

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

      3. frac-times 47.4%

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

      4. metadata-eval 47.4%

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

      5. sqrt-div 47.4%

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

      6. metadata-eval 47.4%

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

      7. div-inv 47.4%

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

      8. associate-*r/ 49.6%

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

      9. clear-num 49.6%

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

      10. un-div-inv 49.6%

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

      11. sqrt-undiv 67.3%

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

   6. Applied egg-rr 67.3%

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

      1. associate-*r/ 64.8%

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

   8. Simplified 64.8%

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

   if 3.99999999999999999e-305 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 85.1%

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

Alternative 13: 76.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-305}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 (<= t_0 4e-305) (* c0 (sqrt (/ (/ A l) V))) 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 <= 4e-305) {
		tmp = c0 * sqrt(((A / l) / V));
	} 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 <= 4d-305) then
        tmp = c0 * sqrt(((a / l) / v))
    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 <= 4e-305) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} 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 <= 4e-305:
		tmp = c0 * math.sqrt(((A / l) / V))
	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 <= 4e-305)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	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 <= 4e-305)
		tmp = c0 * sqrt(((A / l) / V));
	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[LessEqual[t$95$0, 4e-305], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 3.99999999999999999e-305

   1. Initial program 68.0%

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

   3. Taylor expanded in c0 around 0 68.0%

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

      1. *-commutative 68.0%

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

      2. associate-/r* 65.3%

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

   5. Simplified 65.3%

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

   if 3.99999999999999999e-305 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 85.1%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 4 \cdot 10^{-305}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 76.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-305}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 (<= t_0 4e-305) (* 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 <= 4e-305) {
		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 <= 4d-305) 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 <= 4e-305) {
		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 <= 4e-305:
		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 <= 4e-305)
		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 <= 4e-305)
		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[LessEqual[t$95$0, 4e-305], 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)))) < 3.99999999999999999e-305

   1. Initial program 68.0%

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

      1. *-commutative 68.0%

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

      2. associate-/l/ 69.7%

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

   3. Simplified 69.7%

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

   if 3.99999999999999999e-305 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 85.1%

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

Alternative 15: 74.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+295}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \left(c0 \cdot \frac{1}{V}\right)\right)}\\ \end{array} \end{array} \]

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 (/ A (* V l))))
   (if (<= t_0 0.0)
     (sqrt (* (/ A l) (* c0 (/ c0 V))))
     (if (<= t_0 5e+295)
       (* c0 (sqrt t_0))
       (sqrt (* (/ A l) (* c0 (* c0 (/ 1.0 V)))))))))

C

assert(c0 < A && A < V && 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 = sqrt(((A / l) * (c0 * (c0 / V))));
	} else if (t_0 <= 5e+295) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = sqrt(((A / l) * (c0 * (c0 * (1.0 / V)))));
	}
	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 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = sqrt(((a / l) * (c0 * (c0 / v))))
    else if (t_0 <= 5d+295) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = sqrt(((a / l) * (c0 * (c0 * (1.0d0 / v)))))
    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 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.sqrt(((A / l) * (c0 * (c0 / V))));
	} else if (t_0 <= 5e+295) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = Math.sqrt(((A / l) * (c0 * (c0 * (1.0 / V)))));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.sqrt(((A / l) * (c0 * (c0 / V))))
	elif t_0 <= 5e+295:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = math.sqrt(((A / l) * (c0 * (c0 * (1.0 / V)))))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = sqrt(Float64(Float64(A / l) * Float64(c0 * Float64(c0 / V))));
	elseif (t_0 <= 5e+295)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = sqrt(Float64(Float64(A / l) * Float64(c0 * Float64(c0 * Float64(1.0 / V)))));
	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 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = sqrt(((A / l) * (c0 * (c0 / V))));
	elseif (t_0 <= 5e+295)
		tmp = c0 * sqrt(t_0);
	else
		tmp = sqrt(((A / l) * (c0 * (c0 * (1.0 / V)))));
	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[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[Sqrt[N[(N[(A / l), $MachinePrecision] * N[(c0 * N[(c0 / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 5e+295], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(A / l), $MachinePrecision] * N[(c0 * N[(c0 * N[(1.0 / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 34.9%

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

      1. add-sqr-sqrt 34.9%

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

      2. sqrt-unprod 34.9%

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

      3. *-commutative 34.9%

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

      4. *-commutative 34.9%

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

      5. swap-sqr 33.8%

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

      6. add-sqr-sqrt 33.8%

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

      7. pow2 33.8%

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

   4. Applied egg-rr 33.8%

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

      1. associate-*l/ 35.2%

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

      2. *-commutative 35.2%

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

      3. times-frac 39.0%

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

   6. Simplified 39.0%

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

      1. div-inv 39.0%

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

      2. unpow2 39.0%

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

      3. associate-*l* 44.6%

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

   8. Applied egg-rr 44.6%

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

   9. Taylor expanded in c0 around 0 44.6%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.99999999999999991e295

   1. Initial program 98.6%

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

   if 4.99999999999999991e295 < (/.f64 A (*.f64 V l))

   1. Initial program 37.5%

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

      1. add-sqr-sqrt 17.5%

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

      2. sqrt-unprod 17.7%

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

      3. *-commutative 17.7%

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

      4. *-commutative 17.7%

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

      5. swap-sqr 17.1%

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

      6. add-sqr-sqrt 17.1%

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

      7. pow2 17.1%

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

   4. Applied egg-rr 17.1%

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

      1. associate-*l/ 20.9%

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

      2. *-commutative 20.9%

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

      3. times-frac 22.2%

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

   6. Simplified 22.2%

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

      1. div-inv 22.2%

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

      2. unpow2 22.2%

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

      3. associate-*l* 31.3%

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

   8. Applied egg-rr 31.3%

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

Alternative 16: 74.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 5 \cdot 10^{+295}\right):\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \end{array} \end{array} \]

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 (/ A (* V l))))
   (if (or (<= t_0 0.0) (not (<= t_0 5e+295)))
     (sqrt (* (/ A l) (* c0 (/ c0 V))))
     (* c0 (sqrt t_0)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+295)) {
		tmp = sqrt(((A / l) * (c0 * (c0 / V))));
	} else {
		tmp = c0 * sqrt(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 = a / (v * l)
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 5d+295))) then
        tmp = sqrt(((a / l) * (c0 * (c0 / v))))
    else
        tmp = c0 * sqrt(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 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+295)) {
		tmp = Math.sqrt(((A / l) * (c0 * (c0 / V))));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 5e+295):
		tmp = math.sqrt(((A / l) * (c0 * (c0 / V))))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 5e+295))
		tmp = sqrt(Float64(Float64(A / l) * Float64(c0 * Float64(c0 / V))));
	else
		tmp = Float64(c0 * sqrt(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 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 5e+295)))
		tmp = sqrt(((A / l) * (c0 * (c0 / V))));
	else
		tmp = c0 * sqrt(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[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 5e+295]], $MachinePrecision]], N[Sqrt[N[(N[(A / l), $MachinePrecision] * N[(c0 * N[(c0 / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 4.99999999999999991e295 < (/.f64 A (*.f64 V l))

   1. Initial program 36.0%

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

      1. add-sqr-sqrt 27.4%

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

      2. sqrt-unprod 27.5%

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

      3. *-commutative 27.5%

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

      4. *-commutative 27.5%

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

      5. swap-sqr 26.6%

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

      6. add-sqr-sqrt 26.6%

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

      7. pow2 26.6%

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

   4. Applied egg-rr 26.6%

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

      1. associate-*l/ 29.1%

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

      2. *-commutative 29.1%

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

      3. times-frac 31.8%

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

   6. Simplified 31.8%

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

      1. div-inv 31.8%

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

      2. unpow2 31.8%

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

      3. associate-*l* 38.9%

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

   8. Applied egg-rr 38.9%

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

   9. Taylor expanded in c0 around 0 38.9%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.99999999999999991e295

   1. Initial program 98.6%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+295}\right):\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 77.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{-284}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (/ A (* V l)) 5e-284)
   (* c0 (sqrt (/ (/ A 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 tmp;
	if ((A / (V * l)) <= 5e-284) {
		tmp = c0 * sqrt(((A / 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) :: tmp
    if ((a / (v * l)) <= 5d-284) then
        tmp = c0 * sqrt(((a / 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 tmp;
	if ((A / (V * l)) <= 5e-284) {
		tmp = c0 * Math.sqrt(((A / 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):
	tmp = 0
	if (A / (V * l)) <= 5e-284:
		tmp = c0 * math.sqrt(((A / 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)
	tmp = 0.0
	if (Float64(A / Float64(V * l)) <= 5e-284)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(c0 / sqrt(Float64(Float64(V * 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)
	tmp = 0.0;
	if ((A / (V * l)) <= 5e-284)
		tmp = c0 * sqrt(((A / 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_] := If[LessEqual[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision], 5e-284], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 4.99999999999999973e-284

   1. Initial program 39.1%

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

      1. *-commutative 39.1%

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

      2. associate-/l/ 53.0%

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

   3. Simplified 53.0%

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

   if 4.99999999999999973e-284 < (/.f64 A (*.f64 V l))

   1. Initial program 85.3%

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

      1. associate-/r* 79.5%

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

      2. div-inv 79.5%

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

      3. div-inv 79.5%

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

      4. associate-*l* 84.7%

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

   4. Applied egg-rr 84.7%

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

      1. sqrt-prod 52.4%

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

      2. associate-*r* 49.1%

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

      3. frac-times 49.1%

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

      4. metadata-eval 49.1%

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

      5. sqrt-div 49.2%

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

      6. metadata-eval 49.2%

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

      7. div-inv 49.1%

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

      8. associate-*r/ 52.8%

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

      9. clear-num 52.8%

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

      10. un-div-inv 52.8%

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

      11. sqrt-undiv 85.8%

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

   6. Applied egg-rr 85.8%

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

Alternative 18: 73.9% 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 74.1%

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

Reproduce

herbie shell --seed 2024163 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))