Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.7% → 89.7%

Time: 11.9s

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: 73.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+281}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{A}{V}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -5e+281)
   (* c0 (/ (sqrt (/ (- A) l)) (sqrt (- V))))
   (if (<= (* V l) -4e-304)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 0.0)
       (* c0 (* (pow l -0.5) (sqrt (/ A V))))
       (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+281) {
		tmp = c0 * (sqrt((-A / l)) / sqrt(-V));
	} else if ((V * l) <= -4e-304) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (pow(l, -0.5) * sqrt((A / V)));
	} else {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-5d+281)) then
        tmp = c0 * (sqrt((-a / l)) / sqrt(-v))
    else if ((v * l) <= (-4d-304)) then
        tmp = c0 * (sqrt(-a) / sqrt((v * -l)))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 * ((l ** (-0.5d0)) * sqrt((a / v)))
    else
        tmp = c0 * (sqrt(a) * sqrt(((1.0d0 / v) / l)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+281) {
		tmp = c0 * (Math.sqrt((-A / l)) / Math.sqrt(-V));
	} else if ((V * l) <= -4e-304) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (Math.pow(l, -0.5) * Math.sqrt((A / V)));
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -5e+281:
		tmp = c0 * (math.sqrt((-A / l)) / math.sqrt(-V))
	elif (V * l) <= -4e-304:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = c0 * (math.pow(l, -0.5) * math.sqrt((A / V)))
	else:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -5e+281)
		tmp = Float64(c0 * Float64(sqrt(Float64(Float64(-A) / l)) / sqrt(Float64(-V))));
	elseif (Float64(V * l) <= -4e-304)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * Float64((l ^ -0.5) * sqrt(Float64(A / V))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -5e+281)
		tmp = c0 * (sqrt((-A / l)) / sqrt(-V));
	elseif ((V * l) <= -4e-304)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = c0 * ((l ^ -0.5) * sqrt((A / V)));
	else
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -5e+281], N[(c0 * N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -4e-304], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[(N[Power[l, -0.5], $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -5.00000000000000016e281

   1. Initial program 52.0%

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

      1. associate-/r* 71.4%

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

      2. div-inv 71.5%

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

   3. Applied egg-rr 71.5%

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

      1. associate-*l/ 71.6%

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

      2. div-inv 71.5%

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

   5. Applied egg-rr 71.5%

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

      1. frac-2neg 71.5%

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

      2. sqrt-div 56.6%

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

   7. Applied egg-rr 56.6%

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

      1. distribute-neg-frac 56.6%

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

   9. Simplified 56.6%

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

   if -5.00000000000000016e281 < (*.f64 V l) < -3.99999999999999988e-304

   1. Initial program 92.0%

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

      1. frac-2neg 92.0%

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

      2. sqrt-div 99.5%

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

      3. *-commutative 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

   3. Applied egg-rr 99.5%

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

   if -3.99999999999999988e-304 < (*.f64 V l) < -0.0

   1. Initial program 20.7%

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

      1. add-sqr-sqrt 18.5%

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

      2. sqrt-unprod 18.6%

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

      3. pow1/2 18.6%

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

      4. *-commutative 18.6%

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

      5. *-commutative 18.6%

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

      6. swap-sqr 17.7%

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

      7. add-sqr-sqrt 17.7%

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

   3. Applied egg-rr 17.7%

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

      1. unpow1/2 17.7%

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

   5. Simplified 17.7%

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

      1. *-commutative 17.7%

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

      2. sqrt-prod 17.7%

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

      3. sqrt-prod 18.5%

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

      4. add-sqr-sqrt 20.7%

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

      5. associate-/r* 36.8%

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

      6. un-div-inv 36.8%

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

      7. sqrt-prod 44.1%

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

      8. associate-*r* 43.9%

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

      9. inv-pow 43.9%

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

      10. sqrt-pow1 44.0%

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

      11. metadata-eval 44.0%

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

   7. Applied egg-rr 44.0%

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

      1. expm1-log1p-u 26.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(c0 \cdot \sqrt{\frac{A}{V}}\right) \cdot {\ell}^{-0.5}\right)\right)} \]

      2. expm1-udef 16.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(c0 \cdot \sqrt{\frac{A}{V}}\right) \cdot {\ell}^{-0.5}\right)} - 1} \]

      3. *-commutative 16.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{\frac{A}{V}} \cdot c0\right)} \cdot {\ell}^{-0.5}\right)} - 1 \]

      4. associate-*l* 16.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{A}{V}} \cdot \left(c0 \cdot {\ell}^{-0.5}\right)}\right)} - 1 \]

   9. Applied egg-rr 16.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{A}{V}} \cdot \left(c0 \cdot {\ell}^{-0.5}\right)\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 26.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{A}{V}} \cdot \left(c0 \cdot {\ell}^{-0.5}\right)\right)\right)} \]

       2. expm1-log1p 44.0%

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

       3. *-commutative 44.0%

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

       4. associate-*l* 44.1%

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

   11. Simplified 44.1%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 84.0%

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

      1. pow1/2 84.0%

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

      2. clear-num 84.0%

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

      3. inv-pow 84.0%

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

      4. pow-pow 84.5%

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

      5. associate-/l* 74.9%

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

      6. metadata-eval 74.9%

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

   3. Applied egg-rr 74.9%

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

      1. associate-/l* 84.5%

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

   5. Simplified 84.5%

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

      1. add-sqr-sqrt 84.1%

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

      2. sqrt-unprod 84.0%

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

      3. pow-prod-up 84.0%

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

      4. metadata-eval 84.0%

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

      5. inv-pow 84.0%

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

      6. clear-num 84.0%

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

      7. div-inv 83.9%

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

      8. sqrt-prod 92.9%

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

   7. Applied egg-rr 92.9%

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

      1. associate-/r* 94.0%

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

   9. Simplified 94.0%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+281}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{A}{V}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \]

Alternative 2: 89.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(c0 \cdot \frac{\sqrt{-A}}{\sqrt{-V}}\right) \cdot {\ell}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -5e-310)
   (* (* c0 (/ (sqrt (- A)) (sqrt (- V)))) (pow l -0.5))
   (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = (c0 * (sqrt(-A) / sqrt(-V))) * pow(l, -0.5);
	} else {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (a <= (-5d-310)) then
        tmp = (c0 * (sqrt(-a) / sqrt(-v))) * (l ** (-0.5d0))
    else
        tmp = c0 * (sqrt(a) * sqrt(((1.0d0 / v) / l)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = (c0 * (Math.sqrt(-A) / Math.sqrt(-V))) * Math.pow(l, -0.5);
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -5e-310:
		tmp = (c0 * (math.sqrt(-A) / math.sqrt(-V))) * math.pow(l, -0.5)
	else:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -5e-310)
		tmp = Float64(Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(-V)))) * (l ^ -0.5));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (A <= -5e-310)
		tmp = (c0 * (sqrt(-A) / sqrt(-V))) * (l ^ -0.5);
	else
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -5e-310], N[(N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -4.999999999999985e-310

   1. Initial program 77.3%

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

      1. add-sqr-sqrt 44.1%

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

      2. sqrt-unprod 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. pow1/2 34.9%

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

      4. *-commutative 34.9%

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

      5. *-commutative 34.9%

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

      6. swap-sqr 31.8%

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

      7. add-sqr-sqrt 31.8%

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

   3. Applied egg-rr 31.8%

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

      1. unpow1/2 31.8%

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

   5. Simplified 31.8%

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

      1. *-commutative 31.8%

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

      2. sqrt-prod 34.6%

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

      3. sqrt-prod 40.8%

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

      4. add-sqr-sqrt 77.3%

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

      5. associate-/r* 81.9%

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

      6. un-div-inv 82.0%

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

      7. sqrt-prod 51.7%

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

      8. associate-*r* 48.2%

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

      9. inv-pow 48.2%

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

      10. sqrt-pow1 48.2%

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

      11. metadata-eval 48.2%

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

   7. Applied egg-rr 48.2%

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

      1. frac-2neg 48.2%

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

      2. sqrt-div 51.1%

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

   9. Applied egg-rr 51.1%

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

   if -4.999999999999985e-310 < A

   1. Initial program 75.6%

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

      1. pow1/2 75.6%

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

      2. clear-num 75.6%

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

      3. inv-pow 75.6%

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

      4. pow-pow 76.1%

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

      5. associate-/l* 69.4%

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

      6. metadata-eval 69.4%

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

   3. Applied egg-rr 69.4%

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

      1. associate-/l* 76.1%

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

   5. Simplified 76.1%

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

      1. add-sqr-sqrt 75.7%

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

      2. sqrt-unprod 75.6%

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

      3. pow-prod-up 75.6%

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

      4. metadata-eval 75.6%

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

      5. inv-pow 75.6%

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

      6. clear-num 75.6%

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

      7. div-inv 75.5%

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

      8. sqrt-prod 83.5%

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

   7. Applied egg-rr 83.5%

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

      1. associate-/r* 84.5%

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

   9. Simplified 84.5%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(c0 \cdot \frac{\sqrt{-A}}{\sqrt{-V}}\right) \cdot {\ell}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \]

Alternative 3: 86.7% accurate, 0.3× speedup

Math

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

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= V -1e-310)
   (* c0 (/ (sqrt (/ (- A) l)) (sqrt (- V))))
   (/ c0 (/ (* (sqrt l) (sqrt V)) (sqrt A)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (V <= -1e-310) {
		tmp = c0 * (sqrt((-A / l)) / sqrt(-V));
	} else {
		tmp = c0 / ((sqrt(l) * sqrt(V)) / sqrt(A));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (v <= (-1d-310)) then
        tmp = c0 * (sqrt((-a / l)) / sqrt(-v))
    else
        tmp = c0 / ((sqrt(l) * sqrt(v)) / sqrt(a))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (V <= -1e-310) {
		tmp = c0 * (Math.sqrt((-A / l)) / Math.sqrt(-V));
	} else {
		tmp = c0 / ((Math.sqrt(l) * Math.sqrt(V)) / Math.sqrt(A));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if V <= -1e-310:
		tmp = c0 * (math.sqrt((-A / l)) / math.sqrt(-V))
	else:
		tmp = c0 / ((math.sqrt(l) * math.sqrt(V)) / math.sqrt(A))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (V <= -1e-310)
		tmp = Float64(c0 * Float64(sqrt(Float64(Float64(-A) / l)) / sqrt(Float64(-V))));
	else
		tmp = Float64(c0 / Float64(Float64(sqrt(l) * sqrt(V)) / sqrt(A)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (V <= -1e-310)
		tmp = c0 * (sqrt((-A / l)) / sqrt(-V));
	else
		tmp = c0 / ((sqrt(l) * sqrt(V)) / sqrt(A));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[V, -1e-310], N[(c0 * N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[V], $MachinePrecision]), $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if V < -9.999999999999969e-311

   1. Initial program 78.8%

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

      1. associate-/r* 76.9%

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

      2. div-inv 76.9%

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

   3. Applied egg-rr 76.9%

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

      1. associate-*l/ 76.4%

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

      2. div-inv 76.3%

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

   5. Applied egg-rr 76.3%

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

      1. frac-2neg 76.3%

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

      2. sqrt-div 82.0%

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

   7. Applied egg-rr 82.0%

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

      1. distribute-neg-frac 82.0%

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

   9. Simplified 82.0%

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

   if -9.999999999999969e-311 < V

   1. Initial program 74.0%

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

      1. sqrt-div 45.4%

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

      2. associate-*r/ 43.3%

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

   3. Applied egg-rr 43.3%

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

      1. associate-/l* 45.4%

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

   5. Simplified 45.4%

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

      1. *-commutative 45.4%

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

      2. sqrt-prod 54.4%

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

   7. Applied egg-rr 54.4%

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

4. Final simplification 68.4%

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

Alternative 4: 86.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+157}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{t_0}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-207}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \left({\ell}^{-0.5} \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (/ A V))))
   (if (<= (* V l) -5e+157)
     (/ c0 (/ (sqrt l) t_0))
     (if (<= (* V l) -1e-207)
       (* c0 (pow (/ (* V l) A) -0.5))
       (if (<= (* V l) 0.0)
         (* c0 (* (pow l -0.5) t_0))
         (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l)))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / V));
	double tmp;
	if ((V * l) <= -5e+157) {
		tmp = c0 / (sqrt(l) / t_0);
	} else if ((V * l) <= -1e-207) {
		tmp = c0 * pow(((V * l) / A), -0.5);
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (pow(l, -0.5) * t_0);
	} else {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a / v))
    if ((v * l) <= (-5d+157)) then
        tmp = c0 / (sqrt(l) / t_0)
    else if ((v * l) <= (-1d-207)) then
        tmp = c0 * (((v * l) / a) ** (-0.5d0))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 * ((l ** (-0.5d0)) * t_0)
    else
        tmp = c0 * (sqrt(a) * sqrt(((1.0d0 / v) / l)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / V));
	double tmp;
	if ((V * l) <= -5e+157) {
		tmp = c0 / (Math.sqrt(l) / t_0);
	} else if ((V * l) <= -1e-207) {
		tmp = c0 * Math.pow(((V * l) / A), -0.5);
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (Math.pow(l, -0.5) * t_0);
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / V))
	tmp = 0
	if (V * l) <= -5e+157:
		tmp = c0 / (math.sqrt(l) / t_0)
	elif (V * l) <= -1e-207:
		tmp = c0 * math.pow(((V * l) / A), -0.5)
	elif (V * l) <= 0.0:
		tmp = c0 * (math.pow(l, -0.5) * t_0)
	else:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(A / V))
	tmp = 0.0
	if (Float64(V * l) <= -5e+157)
		tmp = Float64(c0 / Float64(sqrt(l) / t_0));
	elseif (Float64(V * l) <= -1e-207)
		tmp = Float64(c0 * (Float64(Float64(V * l) / A) ^ -0.5));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * Float64((l ^ -0.5) * t_0));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((A / V));
	tmp = 0.0;
	if ((V * l) <= -5e+157)
		tmp = c0 / (sqrt(l) / t_0);
	elseif ((V * l) <= -1e-207)
		tmp = c0 * (((V * l) / A) ^ -0.5);
	elseif ((V * l) <= 0.0)
		tmp = c0 * ((l ^ -0.5) * t_0);
	else
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e+157], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-207], N[(c0 * N[Power[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[(N[Power[l, -0.5], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -4.99999999999999976e157

   1. Initial program 70.6%

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

      1. pow1/2 70.6%

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

      2. clear-num 70.5%

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

      3. inv-pow 70.5%

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

      4. pow-pow 70.5%

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

      5. associate-/l* 80.0%

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

      6. metadata-eval 80.0%

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

   3. Applied egg-rr 80.0%

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

      1. associate-/l* 70.5%

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

   5. Simplified 70.5%

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

      1. add-sqr-sqrt 70.4%

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

      2. sqrt-unprod 70.5%

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

      3. pow-prod-up 70.5%

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

      4. metadata-eval 70.5%

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

      5. inv-pow 70.5%

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

      6. sqrt-div 70.4%

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

      7. metadata-eval 70.4%

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

      8. sqrt-undiv 0.0%

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

      9. div-inv 0.0%

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

      10. sqrt-undiv 70.6%

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

      11. associate-*r/ 80.2%

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

   7. Applied egg-rr 80.2%

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

      1. associate-*r/ 70.6%

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

      2. sqrt-div 0.0%

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

      3. sqrt-unprod 0.0%

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

      4. *-commutative 0.0%

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

      5. associate-/l* 0.0%

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

      6. sqrt-div 48.8%

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

      7. div-inv 48.8%

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

   9. Applied egg-rr 48.8%

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

       1. associate-*r/ 48.8%

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

       2. *-rgt-identity 48.8%

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

   11. Simplified 48.8%

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

   if -4.99999999999999976e157 < (*.f64 V l) < -9.99999999999999925e-208

   1. Initial program 97.9%

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

      1. pow1/2 97.9%

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

      2. clear-num 97.9%

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

      3. inv-pow 97.9%

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

      4. pow-pow 98.7%

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

      5. associate-/l* 78.6%

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

      6. metadata-eval 78.6%

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

   3. Applied egg-rr 78.6%

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

      1. associate-/l* 98.7%

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

   5. Simplified 98.7%

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

   if -9.99999999999999925e-208 < (*.f64 V l) < -0.0

   1. Initial program 39.1%

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

      1. add-sqr-sqrt 18.3%

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

      2. sqrt-unprod 18.8%

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

      3. pow1/2 18.8%

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

      4. *-commutative 18.8%

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

      5. *-commutative 18.8%

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

      6. swap-sqr 18.2%

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

      7. add-sqr-sqrt 18.3%

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

   3. Applied egg-rr 18.3%

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

      1. unpow1/2 18.3%

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

   5. Simplified 18.3%

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

      1. *-commutative 18.3%

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

      2. sqrt-prod 18.2%

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

      3. sqrt-prod 18.3%

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

      4. add-sqr-sqrt 39.1%

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

      5. associate-/r* 49.0%

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

      6. un-div-inv 49.0%

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

      7. sqrt-prod 48.9%

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

      8. associate-*r* 45.1%

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

      9. inv-pow 45.1%

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

      10. sqrt-pow1 45.2%

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

      11. metadata-eval 45.2%

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

   7. Applied egg-rr 45.2%

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

      1. expm1-log1p-u 23.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(c0 \cdot \sqrt{\frac{A}{V}}\right) \cdot {\ell}^{-0.5}\right)\right)} \]

      2. expm1-udef 12.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(c0 \cdot \sqrt{\frac{A}{V}}\right) \cdot {\ell}^{-0.5}\right)} - 1} \]

      3. *-commutative 12.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{\frac{A}{V}} \cdot c0\right)} \cdot {\ell}^{-0.5}\right)} - 1 \]

      4. associate-*l* 12.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{A}{V}} \cdot \left(c0 \cdot {\ell}^{-0.5}\right)}\right)} - 1 \]

   9. Applied egg-rr 12.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{A}{V}} \cdot \left(c0 \cdot {\ell}^{-0.5}\right)\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 26.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{A}{V}} \cdot \left(c0 \cdot {\ell}^{-0.5}\right)\right)\right)} \]

       2. expm1-log1p 48.8%

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

       3. *-commutative 48.8%

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

       4. associate-*l* 48.9%

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

   11. Simplified 48.9%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 84.0%

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

      1. pow1/2 84.0%

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

      2. clear-num 84.0%

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

      3. inv-pow 84.0%

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

      4. pow-pow 84.5%

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

      5. associate-/l* 74.9%

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

      6. metadata-eval 74.9%

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

   3. Applied egg-rr 74.9%

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

      1. associate-/l* 84.5%

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

   5. Simplified 84.5%

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

      1. add-sqr-sqrt 84.1%

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

      2. sqrt-unprod 84.0%

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

      3. pow-prod-up 84.0%

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

      4. metadata-eval 84.0%

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

      5. inv-pow 84.0%

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

      6. clear-num 84.0%

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

      7. div-inv 83.9%

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

      8. sqrt-prod 92.9%

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

   7. Applied egg-rr 92.9%

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

      1. associate-/r* 94.0%

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

   9. Simplified 94.0%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+157}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-207}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{A}{V}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \]

Alternative 5: 89.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+281}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{t_0}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \left({\ell}^{-0.5} \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (/ A V))))
   (if (<= (* V l) -5e+281)
     (/ c0 (/ (sqrt l) t_0))
     (if (<= (* V l) -4e-304)
       (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
       (if (<= (* V l) 0.0)
         (* c0 (* (pow l -0.5) t_0))
         (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l)))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / V));
	double tmp;
	if ((V * l) <= -5e+281) {
		tmp = c0 / (sqrt(l) / t_0);
	} else if ((V * l) <= -4e-304) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (pow(l, -0.5) * t_0);
	} else {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a / v))
    if ((v * l) <= (-5d+281)) then
        tmp = c0 / (sqrt(l) / t_0)
    else if ((v * l) <= (-4d-304)) then
        tmp = c0 * (sqrt(-a) / sqrt((v * -l)))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 * ((l ** (-0.5d0)) * t_0)
    else
        tmp = c0 * (sqrt(a) * sqrt(((1.0d0 / v) / l)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / V));
	double tmp;
	if ((V * l) <= -5e+281) {
		tmp = c0 / (Math.sqrt(l) / t_0);
	} else if ((V * l) <= -4e-304) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (Math.pow(l, -0.5) * t_0);
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / V))
	tmp = 0
	if (V * l) <= -5e+281:
		tmp = c0 / (math.sqrt(l) / t_0)
	elif (V * l) <= -4e-304:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = c0 * (math.pow(l, -0.5) * t_0)
	else:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(A / V))
	tmp = 0.0
	if (Float64(V * l) <= -5e+281)
		tmp = Float64(c0 / Float64(sqrt(l) / t_0));
	elseif (Float64(V * l) <= -4e-304)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * Float64((l ^ -0.5) * t_0));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((A / V));
	tmp = 0.0;
	if ((V * l) <= -5e+281)
		tmp = c0 / (sqrt(l) / t_0);
	elseif ((V * l) <= -4e-304)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = c0 * ((l ^ -0.5) * t_0);
	else
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e+281], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -4e-304], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[(N[Power[l, -0.5], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -5.00000000000000016e281

   1. Initial program 52.0%

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

      1. pow1/2 52.0%

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

      2. clear-num 52.0%

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

      3. inv-pow 52.0%

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

      4. pow-pow 52.0%

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

      5. associate-/l* 68.9%

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

      6. metadata-eval 68.9%

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

   3. Applied egg-rr 68.9%

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

      1. associate-/l* 52.0%

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

   5. Simplified 52.0%

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

      1. add-sqr-sqrt 52.0%

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

      2. sqrt-unprod 52.0%

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

      3. pow-prod-up 52.0%

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

      4. metadata-eval 52.0%

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

      5. inv-pow 52.0%

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

      6. sqrt-div 51.9%

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

      7. metadata-eval 51.9%

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

      8. sqrt-undiv 0.0%

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

      9. div-inv 0.0%

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

      10. sqrt-undiv 52.0%

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

      11. associate-*r/ 68.9%

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

   7. Applied egg-rr 68.9%

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

      1. associate-*r/ 52.0%

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

      2. sqrt-div 0.0%

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

      3. sqrt-unprod 0.0%

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

      4. *-commutative 0.0%

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

      5. associate-/l* 0.0%

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

      6. sqrt-div 56.7%

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

      7. div-inv 56.7%

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

   9. Applied egg-rr 56.7%

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

       1. associate-*r/ 56.7%

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

       2. *-rgt-identity 56.7%

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

   11. Simplified 56.7%

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

   if -5.00000000000000016e281 < (*.f64 V l) < -3.99999999999999988e-304

   1. Initial program 92.0%

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

      1. frac-2neg 92.0%

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

      2. sqrt-div 99.5%

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

      3. *-commutative 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

   3. Applied egg-rr 99.5%

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

   if -3.99999999999999988e-304 < (*.f64 V l) < -0.0

   1. Initial program 20.7%

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

      1. add-sqr-sqrt 18.5%

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

      2. sqrt-unprod 18.6%

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

      3. pow1/2 18.6%

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

      4. *-commutative 18.6%

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

      5. *-commutative 18.6%

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

      6. swap-sqr 17.7%

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

      7. add-sqr-sqrt 17.7%

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

   3. Applied egg-rr 17.7%

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

      1. unpow1/2 17.7%

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

   5. Simplified 17.7%

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

      1. *-commutative 17.7%

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

      2. sqrt-prod 17.7%

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

      3. sqrt-prod 18.5%

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

      4. add-sqr-sqrt 20.7%

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

      5. associate-/r* 36.8%

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

      6. un-div-inv 36.8%

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

      7. sqrt-prod 44.1%

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

      8. associate-*r* 43.9%

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

      9. inv-pow 43.9%

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

      10. sqrt-pow1 44.0%

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

      11. metadata-eval 44.0%

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

   7. Applied egg-rr 44.0%

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

      1. expm1-log1p-u 26.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(c0 \cdot \sqrt{\frac{A}{V}}\right) \cdot {\ell}^{-0.5}\right)\right)} \]

      2. expm1-udef 16.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(c0 \cdot \sqrt{\frac{A}{V}}\right) \cdot {\ell}^{-0.5}\right)} - 1} \]

      3. *-commutative 16.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{\frac{A}{V}} \cdot c0\right)} \cdot {\ell}^{-0.5}\right)} - 1 \]

      4. associate-*l* 16.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{A}{V}} \cdot \left(c0 \cdot {\ell}^{-0.5}\right)}\right)} - 1 \]

   9. Applied egg-rr 16.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{A}{V}} \cdot \left(c0 \cdot {\ell}^{-0.5}\right)\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 26.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{A}{V}} \cdot \left(c0 \cdot {\ell}^{-0.5}\right)\right)\right)} \]

       2. expm1-log1p 44.0%

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

       3. *-commutative 44.0%

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

       4. associate-*l* 44.1%

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

   11. Simplified 44.1%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 84.0%

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

      1. pow1/2 84.0%

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

      2. clear-num 84.0%

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

      3. inv-pow 84.0%

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

      4. pow-pow 84.5%

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

      5. associate-/l* 74.9%

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

      6. metadata-eval 74.9%

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

   3. Applied egg-rr 74.9%

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

      1. associate-/l* 84.5%

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

   5. Simplified 84.5%

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

      1. add-sqr-sqrt 84.1%

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

      2. sqrt-unprod 84.0%

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

      3. pow-prod-up 84.0%

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

      4. metadata-eval 84.0%

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

      5. inv-pow 84.0%

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

      6. clear-num 84.0%

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

      7. div-inv 83.9%

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

      8. sqrt-prod 92.9%

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

   7. Applied egg-rr 92.9%

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

      1. associate-/r* 94.0%

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

   9. Simplified 94.0%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+281}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{A}{V}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \]

Alternative 6: 86.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-235}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ c0 (/ (sqrt l) (sqrt (/ A V))))))
   (if (<= (* V l) -5e+157)
     t_0
     (if (<= (* V l) -1e-235)
       (* c0 (pow (/ (* V l) A) -0.5))
       (if (<= (* V l) 0.0) t_0 (* c0 (* (sqrt A) (pow (* V l) -0.5))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 / (sqrt(l) / sqrt((A / V)));
	double tmp;
	if ((V * l) <= -5e+157) {
		tmp = t_0;
	} else if ((V * l) <= -1e-235) {
		tmp = c0 * pow(((V * l) / A), -0.5);
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = c0 * (sqrt(A) * pow((V * l), -0.5));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 / (sqrt(l) / sqrt((a / v)))
    if ((v * l) <= (-5d+157)) then
        tmp = t_0
    else if ((v * l) <= (-1d-235)) then
        tmp = c0 * (((v * l) / a) ** (-0.5d0))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else
        tmp = c0 * (sqrt(a) * ((v * l) ** (-0.5d0)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 / (Math.sqrt(l) / Math.sqrt((A / V)));
	double tmp;
	if ((V * l) <= -5e+157) {
		tmp = t_0;
	} else if ((V * l) <= -1e-235) {
		tmp = c0 * Math.pow(((V * l) / A), -0.5);
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.pow((V * l), -0.5));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = c0 / (math.sqrt(l) / math.sqrt((A / V)))
	tmp = 0
	if (V * l) <= -5e+157:
		tmp = t_0
	elif (V * l) <= -1e-235:
		tmp = c0 * math.pow(((V * l) / A), -0.5)
	elif (V * l) <= 0.0:
		tmp = t_0
	else:
		tmp = c0 * (math.sqrt(A) * math.pow((V * l), -0.5))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 / Float64(sqrt(l) / sqrt(Float64(A / V))))
	tmp = 0.0
	if (Float64(V * l) <= -5e+157)
		tmp = t_0;
	elseif (Float64(V * l) <= -1e-235)
		tmp = Float64(c0 * (Float64(Float64(V * l) / A) ^ -0.5));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = Float64(c0 * Float64(sqrt(A) * (Float64(V * l) ^ -0.5)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 / (sqrt(l) / sqrt((A / V)));
	tmp = 0.0;
	if ((V * l) <= -5e+157)
		tmp = t_0;
	elseif ((V * l) <= -1e-235)
		tmp = c0 * (((V * l) / A) ^ -0.5);
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = c0 * (sqrt(A) * ((V * l) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 / N[(N[Sqrt[l], $MachinePrecision] / N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e+157], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -1e-235], N[(c0 * N[Power[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -4.99999999999999976e157 or -9.9999999999999996e-236 < (*.f64 V l) < -0.0

   1. Initial program 51.6%

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

      1. pow1/2 51.6%

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

      2. clear-num 51.6%

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

      3. inv-pow 51.6%

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

      4. pow-pow 51.6%

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

      5. associate-/l* 62.5%

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

      6. metadata-eval 62.5%

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

   3. Applied egg-rr 62.5%

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

      1. associate-/l* 51.6%

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

   5. Simplified 51.6%

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

      1. add-sqr-sqrt 51.5%

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

      2. sqrt-unprod 51.6%

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

      3. pow-prod-up 51.6%

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

      4. metadata-eval 51.6%

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

      5. inv-pow 51.6%

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

      6. sqrt-div 51.5%

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

      7. metadata-eval 51.5%

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

      8. sqrt-undiv 1.9%

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

      9. div-inv 1.9%

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

      10. sqrt-undiv 51.6%

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

      11. associate-*r/ 62.6%

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

   7. Applied egg-rr 62.6%

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

      1. associate-*r/ 51.6%

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

      2. sqrt-div 1.9%

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

      3. sqrt-unprod 9.5%

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

      4. *-commutative 9.5%

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

      5. associate-/l* 9.5%

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

      6. sqrt-div 48.8%

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

      7. div-inv 48.7%

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

   9. Applied egg-rr 48.7%

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

       1. associate-*r/ 48.8%

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

       2. *-rgt-identity 48.8%

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

   11. Simplified 48.8%

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

   if -4.99999999999999976e157 < (*.f64 V l) < -9.9999999999999996e-236

   1. Initial program 98.0%

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

      1. pow1/2 98.0%

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

      2. clear-num 98.0%

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

      3. inv-pow 98.0%

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

      4. pow-pow 98.8%

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

      5. associate-/l* 80.1%

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

      6. metadata-eval 80.1%

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

   3. Applied egg-rr 80.1%

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

      1. associate-/l* 98.8%

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

   5. Simplified 98.8%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 84.0%

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

      1. div-inv 83.9%

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

      2. sqrt-prod 92.9%

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

      3. pow1/2 92.9%

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

      4. inv-pow 92.9%

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

      5. pow-pow 92.8%

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

      6. metadata-eval 92.8%

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

   3. Applied egg-rr 92.8%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+157}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-235}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \end{array} \]

Alternative 7: 86.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+157}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{t_0}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-207}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \left({\ell}^{-0.5} \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (/ A V))))
   (if (<= (* V l) -5e+157)
     (/ c0 (/ (sqrt l) t_0))
     (if (<= (* V l) -1e-207)
       (* c0 (pow (/ (* V l) A) -0.5))
       (if (<= (* V l) 0.0)
         (* c0 (* (pow l -0.5) t_0))
         (* c0 (* (sqrt A) (pow (* V l) -0.5))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / V));
	double tmp;
	if ((V * l) <= -5e+157) {
		tmp = c0 / (sqrt(l) / t_0);
	} else if ((V * l) <= -1e-207) {
		tmp = c0 * pow(((V * l) / A), -0.5);
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (pow(l, -0.5) * t_0);
	} else {
		tmp = c0 * (sqrt(A) * pow((V * l), -0.5));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a / v))
    if ((v * l) <= (-5d+157)) then
        tmp = c0 / (sqrt(l) / t_0)
    else if ((v * l) <= (-1d-207)) then
        tmp = c0 * (((v * l) / a) ** (-0.5d0))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 * ((l ** (-0.5d0)) * t_0)
    else
        tmp = c0 * (sqrt(a) * ((v * l) ** (-0.5d0)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / V));
	double tmp;
	if ((V * l) <= -5e+157) {
		tmp = c0 / (Math.sqrt(l) / t_0);
	} else if ((V * l) <= -1e-207) {
		tmp = c0 * Math.pow(((V * l) / A), -0.5);
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (Math.pow(l, -0.5) * t_0);
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.pow((V * l), -0.5));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / V))
	tmp = 0
	if (V * l) <= -5e+157:
		tmp = c0 / (math.sqrt(l) / t_0)
	elif (V * l) <= -1e-207:
		tmp = c0 * math.pow(((V * l) / A), -0.5)
	elif (V * l) <= 0.0:
		tmp = c0 * (math.pow(l, -0.5) * t_0)
	else:
		tmp = c0 * (math.sqrt(A) * math.pow((V * l), -0.5))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(A / V))
	tmp = 0.0
	if (Float64(V * l) <= -5e+157)
		tmp = Float64(c0 / Float64(sqrt(l) / t_0));
	elseif (Float64(V * l) <= -1e-207)
		tmp = Float64(c0 * (Float64(Float64(V * l) / A) ^ -0.5));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * Float64((l ^ -0.5) * t_0));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * (Float64(V * l) ^ -0.5)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((A / V));
	tmp = 0.0;
	if ((V * l) <= -5e+157)
		tmp = c0 / (sqrt(l) / t_0);
	elseif ((V * l) <= -1e-207)
		tmp = c0 * (((V * l) / A) ^ -0.5);
	elseif ((V * l) <= 0.0)
		tmp = c0 * ((l ^ -0.5) * t_0);
	else
		tmp = c0 * (sqrt(A) * ((V * l) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e+157], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-207], N[(c0 * N[Power[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[(N[Power[l, -0.5], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -4.99999999999999976e157

   1. Initial program 70.6%

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

      1. pow1/2 70.6%

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

      2. clear-num 70.5%

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

      3. inv-pow 70.5%

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

      4. pow-pow 70.5%

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

      5. associate-/l* 80.0%

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

      6. metadata-eval 80.0%

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

   3. Applied egg-rr 80.0%

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

      1. associate-/l* 70.5%

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

   5. Simplified 70.5%

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

      1. add-sqr-sqrt 70.4%

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

      2. sqrt-unprod 70.5%

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

      3. pow-prod-up 70.5%

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

      4. metadata-eval 70.5%

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

      5. inv-pow 70.5%

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

      6. sqrt-div 70.4%

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

      7. metadata-eval 70.4%

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

      8. sqrt-undiv 0.0%

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

      9. div-inv 0.0%

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

      10. sqrt-undiv 70.6%

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

      11. associate-*r/ 80.2%

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

   7. Applied egg-rr 80.2%

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

      1. associate-*r/ 70.6%

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

      2. sqrt-div 0.0%

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

      3. sqrt-unprod 0.0%

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

      4. *-commutative 0.0%

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

      5. associate-/l* 0.0%

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

      6. sqrt-div 48.8%

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

      7. div-inv 48.8%

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

   9. Applied egg-rr 48.8%

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

       1. associate-*r/ 48.8%

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

       2. *-rgt-identity 48.8%

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

   11. Simplified 48.8%

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

   if -4.99999999999999976e157 < (*.f64 V l) < -9.99999999999999925e-208

   1. Initial program 97.9%

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

      1. pow1/2 97.9%

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

      2. clear-num 97.9%

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

      3. inv-pow 97.9%

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

      4. pow-pow 98.7%

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

      5. associate-/l* 78.6%

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

      6. metadata-eval 78.6%

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

   3. Applied egg-rr 78.6%

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

      1. associate-/l* 98.7%

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

   5. Simplified 98.7%

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

   if -9.99999999999999925e-208 < (*.f64 V l) < -0.0

   1. Initial program 39.1%

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

      1. add-sqr-sqrt 18.3%

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

      2. sqrt-unprod 18.8%

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

      3. pow1/2 18.8%

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

      4. *-commutative 18.8%

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

      5. *-commutative 18.8%

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

      6. swap-sqr 18.2%

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

      7. add-sqr-sqrt 18.3%

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

   3. Applied egg-rr 18.3%

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

      1. unpow1/2 18.3%

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

   5. Simplified 18.3%

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

      1. *-commutative 18.3%

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

      2. sqrt-prod 18.2%

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

      3. sqrt-prod 18.3%

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

      4. add-sqr-sqrt 39.1%

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

      5. associate-/r* 49.0%

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

      6. un-div-inv 49.0%

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

      7. sqrt-prod 48.9%

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

      8. associate-*r* 45.1%

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

      9. inv-pow 45.1%

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

      10. sqrt-pow1 45.2%

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

      11. metadata-eval 45.2%

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

   7. Applied egg-rr 45.2%

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

      1. expm1-log1p-u 23.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(c0 \cdot \sqrt{\frac{A}{V}}\right) \cdot {\ell}^{-0.5}\right)\right)} \]

      2. expm1-udef 12.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(c0 \cdot \sqrt{\frac{A}{V}}\right) \cdot {\ell}^{-0.5}\right)} - 1} \]

      3. *-commutative 12.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{\frac{A}{V}} \cdot c0\right)} \cdot {\ell}^{-0.5}\right)} - 1 \]

      4. associate-*l* 12.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{A}{V}} \cdot \left(c0 \cdot {\ell}^{-0.5}\right)}\right)} - 1 \]

   9. Applied egg-rr 12.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{A}{V}} \cdot \left(c0 \cdot {\ell}^{-0.5}\right)\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 26.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{A}{V}} \cdot \left(c0 \cdot {\ell}^{-0.5}\right)\right)\right)} \]

       2. expm1-log1p 48.8%

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

       3. *-commutative 48.8%

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

       4. associate-*l* 48.9%

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

   11. Simplified 48.9%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 84.0%

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

      1. div-inv 83.9%

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

      2. sqrt-prod 92.9%

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

      3. pow1/2 92.9%

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

      4. inv-pow 92.9%

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

      5. pow-pow 92.8%

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

      6. metadata-eval 92.8%

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

   3. Applied egg-rr 92.8%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+157}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-207}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{A}{V}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \end{array} \]

Alternative 8: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-218}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (/ (sqrt (/ A V)) (sqrt l)))))
   (if (<= (* V l) -5e+157)
     t_0
     (if (<= (* V l) -2e-218)
       (* c0 (pow (/ (* V l) A) -0.5))
       (if (<= (* V l) 0.0) t_0 (* (sqrt A) (/ c0 (sqrt (* V l)))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (sqrt((A / V)) / sqrt(l));
	double tmp;
	if ((V * l) <= -5e+157) {
		tmp = t_0;
	} else if ((V * l) <= -2e-218) {
		tmp = c0 * pow(((V * l) / A), -0.5);
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * (sqrt((a / v)) / sqrt(l))
    if ((v * l) <= (-5d+157)) then
        tmp = t_0
    else if ((v * l) <= (-2d-218)) then
        tmp = c0 * (((v * l) / a) ** (-0.5d0))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else
        tmp = sqrt(a) * (c0 / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	double tmp;
	if ((V * l) <= -5e+157) {
		tmp = t_0;
	} else if ((V * l) <= -2e-218) {
		tmp = c0 * Math.pow(((V * l) / A), -0.5);
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(A) * (c0 / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	tmp = 0
	if (V * l) <= -5e+157:
		tmp = t_0
	elif (V * l) <= -2e-218:
		tmp = c0 * math.pow(((V * l) / A), -0.5)
	elif (V * l) <= 0.0:
		tmp = t_0
	else:
		tmp = math.sqrt(A) * (c0 / math.sqrt((V * l)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)))
	tmp = 0.0
	if (Float64(V * l) <= -5e+157)
		tmp = t_0;
	elseif (Float64(V * l) <= -2e-218)
		tmp = Float64(c0 * (Float64(Float64(V * l) / A) ^ -0.5));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = Float64(sqrt(A) * Float64(c0 / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * (sqrt((A / V)) / sqrt(l));
	tmp = 0.0;
	if ((V * l) <= -5e+157)
		tmp = t_0;
	elseif ((V * l) <= -2e-218)
		tmp = c0 * (((V * l) / A) ^ -0.5);
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e+157], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -2e-218], N[(c0 * N[Power[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, N[(N[Sqrt[A], $MachinePrecision] * N[(c0 / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -4.99999999999999976e157 or -2.0000000000000001e-218 < (*.f64 V l) < -0.0

   1. Initial program 53.2%

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

      1. associate-/r* 63.7%

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

      2. sqrt-div 49.4%

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

   3. Applied egg-rr 49.4%

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

   if -4.99999999999999976e157 < (*.f64 V l) < -2.0000000000000001e-218

   1. Initial program 98.0%

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

      1. pow1/2 98.0%

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

      2. clear-num 97.9%

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

      3. inv-pow 97.9%

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

      4. pow-pow 98.7%

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

      5. associate-/l* 79.0%

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

      6. metadata-eval 79.0%

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

   3. Applied egg-rr 79.0%

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

      1. associate-/l* 98.7%

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

   5. Simplified 98.7%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 84.0%

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

      1. add-sqr-sqrt 48.4%

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

      2. sqrt-unprod 31.9%

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

      3. pow1/2 31.9%

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

      4. *-commutative 31.9%

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

      5. *-commutative 31.9%

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

      6. swap-sqr 26.7%

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

      7. add-sqr-sqrt 26.6%

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

   3. Applied egg-rr 26.6%

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

      1. unpow1/2 26.6%

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

   5. Simplified 26.6%

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

      1. sqrt-prod 30.7%

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

      2. sqrt-div 31.5%

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

      3. clear-num 31.5%

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

      4. sqrt-prod 49.5%

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

      5. add-sqr-sqrt 92.8%

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

      6. associate-/r/ 92.1%

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

      7. clear-num 92.7%

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

      8. associate-/r/ 91.2%

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

   7. Applied egg-rr 91.2%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+157}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-218}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \end{array} \]

Alternative 9: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-235}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ c0 (/ (sqrt l) (sqrt (/ A V))))))
   (if (<= (* V l) -5e+157)
     t_0
     (if (<= (* V l) -1e-235)
       (* c0 (pow (/ (* V l) A) -0.5))
       (if (<= (* V l) 0.0) t_0 (* (sqrt A) (/ c0 (sqrt (* V l)))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 / (sqrt(l) / sqrt((A / V)));
	double tmp;
	if ((V * l) <= -5e+157) {
		tmp = t_0;
	} else if ((V * l) <= -1e-235) {
		tmp = c0 * pow(((V * l) / A), -0.5);
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 / (sqrt(l) / sqrt((a / v)))
    if ((v * l) <= (-5d+157)) then
        tmp = t_0
    else if ((v * l) <= (-1d-235)) then
        tmp = c0 * (((v * l) / a) ** (-0.5d0))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else
        tmp = sqrt(a) * (c0 / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 / (Math.sqrt(l) / Math.sqrt((A / V)));
	double tmp;
	if ((V * l) <= -5e+157) {
		tmp = t_0;
	} else if ((V * l) <= -1e-235) {
		tmp = c0 * Math.pow(((V * l) / A), -0.5);
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(A) * (c0 / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = c0 / (math.sqrt(l) / math.sqrt((A / V)))
	tmp = 0
	if (V * l) <= -5e+157:
		tmp = t_0
	elif (V * l) <= -1e-235:
		tmp = c0 * math.pow(((V * l) / A), -0.5)
	elif (V * l) <= 0.0:
		tmp = t_0
	else:
		tmp = math.sqrt(A) * (c0 / math.sqrt((V * l)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 / Float64(sqrt(l) / sqrt(Float64(A / V))))
	tmp = 0.0
	if (Float64(V * l) <= -5e+157)
		tmp = t_0;
	elseif (Float64(V * l) <= -1e-235)
		tmp = Float64(c0 * (Float64(Float64(V * l) / A) ^ -0.5));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = Float64(sqrt(A) * Float64(c0 / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 / (sqrt(l) / sqrt((A / V)));
	tmp = 0.0;
	if ((V * l) <= -5e+157)
		tmp = t_0;
	elseif ((V * l) <= -1e-235)
		tmp = c0 * (((V * l) / A) ^ -0.5);
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 / N[(N[Sqrt[l], $MachinePrecision] / N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e+157], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -1e-235], N[(c0 * N[Power[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, N[(N[Sqrt[A], $MachinePrecision] * N[(c0 / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -4.99999999999999976e157 or -9.9999999999999996e-236 < (*.f64 V l) < -0.0

   1. Initial program 51.6%

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

      1. pow1/2 51.6%

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

      2. clear-num 51.6%

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

      3. inv-pow 51.6%

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

      4. pow-pow 51.6%

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

      5. associate-/l* 62.5%

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

      6. metadata-eval 62.5%

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

   3. Applied egg-rr 62.5%

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

      1. associate-/l* 51.6%

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

   5. Simplified 51.6%

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

      1. add-sqr-sqrt 51.5%

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

      2. sqrt-unprod 51.6%

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

      3. pow-prod-up 51.6%

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

      4. metadata-eval 51.6%

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

      5. inv-pow 51.6%

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

      6. sqrt-div 51.5%

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

      7. metadata-eval 51.5%

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

      8. sqrt-undiv 1.9%

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

      9. div-inv 1.9%

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

      10. sqrt-undiv 51.6%

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

      11. associate-*r/ 62.6%

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

   7. Applied egg-rr 62.6%

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

      1. associate-*r/ 51.6%

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

      2. sqrt-div 1.9%

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

      3. sqrt-unprod 9.5%

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

      4. *-commutative 9.5%

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

      5. associate-/l* 9.5%

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

      6. sqrt-div 48.8%

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

      7. div-inv 48.7%

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

   9. Applied egg-rr 48.7%

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

       1. associate-*r/ 48.8%

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

       2. *-rgt-identity 48.8%

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

   11. Simplified 48.8%

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

   if -4.99999999999999976e157 < (*.f64 V l) < -9.9999999999999996e-236

   1. Initial program 98.0%

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

      1. pow1/2 98.0%

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

      2. clear-num 98.0%

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

      3. inv-pow 98.0%

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

      4. pow-pow 98.8%

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

      5. associate-/l* 80.1%

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

      6. metadata-eval 80.1%

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

   3. Applied egg-rr 80.1%

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

      1. associate-/l* 98.8%

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

   5. Simplified 98.8%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 84.0%

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

      1. add-sqr-sqrt 48.4%

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

      2. sqrt-unprod 31.9%

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

      3. pow1/2 31.9%

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

      4. *-commutative 31.9%

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

      5. *-commutative 31.9%

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

      6. swap-sqr 26.7%

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

      7. add-sqr-sqrt 26.6%

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

   3. Applied egg-rr 26.6%

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

      1. unpow1/2 26.6%

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

   5. Simplified 26.6%

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

      1. sqrt-prod 30.7%

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

      2. sqrt-div 31.5%

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

      3. clear-num 31.5%

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

      4. sqrt-prod 49.5%

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

      5. add-sqr-sqrt 92.8%

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

      6. associate-/r/ 92.1%

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

      7. clear-num 92.7%

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

      8. associate-/r/ 91.2%

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

   7. Applied egg-rr 91.2%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+157}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-235}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \end{array} \]

Alternative 10: 86.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-235}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ c0 (/ (sqrt l) (sqrt (/ A V))))))
   (if (<= (* V l) -5e+157)
     t_0
     (if (<= (* V l) -1e-235)
       (* c0 (pow (/ (* V l) A) -0.5))
       (if (<= (* V l) 0.0) t_0 (/ c0 (/ (sqrt (* V l)) (sqrt A))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 / (sqrt(l) / sqrt((A / V)));
	double tmp;
	if ((V * l) <= -5e+157) {
		tmp = t_0;
	} else if ((V * l) <= -1e-235) {
		tmp = c0 * pow(((V * l) / A), -0.5);
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 / (sqrt(l) / sqrt((a / v)))
    if ((v * l) <= (-5d+157)) then
        tmp = t_0
    else if ((v * l) <= (-1d-235)) then
        tmp = c0 * (((v * l) / a) ** (-0.5d0))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 / (Math.sqrt(l) / Math.sqrt((A / V)));
	double tmp;
	if ((V * l) <= -5e+157) {
		tmp = t_0;
	} else if ((V * l) <= -1e-235) {
		tmp = c0 * Math.pow(((V * l) / A), -0.5);
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = c0 / (math.sqrt(l) / math.sqrt((A / V)))
	tmp = 0
	if (V * l) <= -5e+157:
		tmp = t_0
	elif (V * l) <= -1e-235:
		tmp = c0 * math.pow(((V * l) / A), -0.5)
	elif (V * l) <= 0.0:
		tmp = t_0
	else:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 / Float64(sqrt(l) / sqrt(Float64(A / V))))
	tmp = 0.0
	if (Float64(V * l) <= -5e+157)
		tmp = t_0;
	elseif (Float64(V * l) <= -1e-235)
		tmp = Float64(c0 * (Float64(Float64(V * l) / A) ^ -0.5));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 / (sqrt(l) / sqrt((A / V)));
	tmp = 0.0;
	if ((V * l) <= -5e+157)
		tmp = t_0;
	elseif ((V * l) <= -1e-235)
		tmp = c0 * (((V * l) / A) ^ -0.5);
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 / N[(N[Sqrt[l], $MachinePrecision] / N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e+157], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -1e-235], N[(c0 * N[Power[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -4.99999999999999976e157 or -9.9999999999999996e-236 < (*.f64 V l) < -0.0

   1. Initial program 51.6%

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

      1. pow1/2 51.6%

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

      2. clear-num 51.6%

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

      3. inv-pow 51.6%

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

      4. pow-pow 51.6%

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

      5. associate-/l* 62.5%

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

      6. metadata-eval 62.5%

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

   3. Applied egg-rr 62.5%

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

      1. associate-/l* 51.6%

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

   5. Simplified 51.6%

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

      1. add-sqr-sqrt 51.5%

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

      2. sqrt-unprod 51.6%

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

      3. pow-prod-up 51.6%

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

      4. metadata-eval 51.6%

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

      5. inv-pow 51.6%

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

      6. sqrt-div 51.5%

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

      7. metadata-eval 51.5%

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

      8. sqrt-undiv 1.9%

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

      9. div-inv 1.9%

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

      10. sqrt-undiv 51.6%

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

      11. associate-*r/ 62.6%

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

   7. Applied egg-rr 62.6%

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

      1. associate-*r/ 51.6%

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

      2. sqrt-div 1.9%

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

      3. sqrt-unprod 9.5%

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

      4. *-commutative 9.5%

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

      5. associate-/l* 9.5%

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

      6. sqrt-div 48.8%

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

      7. div-inv 48.7%

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

   9. Applied egg-rr 48.7%

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

       1. associate-*r/ 48.8%

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

       2. *-rgt-identity 48.8%

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

   11. Simplified 48.8%

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

   if -4.99999999999999976e157 < (*.f64 V l) < -9.9999999999999996e-236

   1. Initial program 98.0%

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

      1. pow1/2 98.0%

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

      2. clear-num 98.0%

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

      3. inv-pow 98.0%

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

      4. pow-pow 98.8%

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

      5. associate-/l* 80.1%

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

      6. metadata-eval 80.1%

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

   3. Applied egg-rr 80.1%

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

      1. associate-/l* 98.8%

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

   5. Simplified 98.8%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 84.0%

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

      1. sqrt-div 92.7%

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

      2. associate-*r/ 87.5%

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

   3. Applied egg-rr 87.5%

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

      1. associate-/l* 92.7%

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

   5. Simplified 92.7%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+157}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-235}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \]

Alternative 11: 81.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 5 \cdot 10^{+300}\right):\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (or (<= t_0 0.0) (not (<= t_0 5e+300)))
     (* c0 (/ (sqrt (/ A V)) (sqrt l)))
     (* c0 (pow (/ (* V l) A) -0.5)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+300)) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else {
		tmp = c0 * pow(((V * l) / A), -0.5);
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 5d+300))) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else
        tmp = c0 * (((v * l) / a) ** (-0.5d0))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+300)) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else {
		tmp = c0 * Math.pow(((V * l) / A), -0.5);
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 5e+300):
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	else:
		tmp = c0 * math.pow(((V * l) / A), -0.5)
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 5e+300))
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	else
		tmp = Float64(c0 * (Float64(Float64(V * l) / A) ^ -0.5));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 5e+300)))
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	else
		tmp = c0 * (((V * l) / A) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := 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+300]], $MachinePrecision]], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Power[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 5.00000000000000026e300 < (/.f64 A (*.f64 V l))

   1. Initial program 25.3%

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

      1. associate-/r* 40.5%

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

      2. sqrt-div 42.9%

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

   3. Applied egg-rr 42.9%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 5.00000000000000026e300

   1. Initial program 99.6%

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

      1. pow1/2 99.6%

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

      2. clear-num 99.6%

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

      3. inv-pow 99.6%

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

      4. pow-pow 99.6%

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

      5. associate-/l* 86.3%

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

      6. metadata-eval 86.3%

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

   3. Applied egg-rr 86.3%

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

      1. associate-/l* 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+300}\right):\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \end{array} \]

Alternative 12: 79.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+292}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{V \cdot \frac{\ell}{A}}}{c0}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_0 5e+292)
       (* c0 (pow (/ (* V l) A) -0.5))
       (/ 1.0 (/ (sqrt (* V (/ l A))) c0))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_0 <= 5e+292) {
		tmp = c0 * pow(((V * l) / A), -0.5);
	} else {
		tmp = 1.0 / (sqrt((V * (l / A))) / c0);
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_0 <= 5d+292) then
        tmp = c0 * (((v * l) / a) ** (-0.5d0))
    else
        tmp = 1.0d0 / (sqrt((v * (l / a))) / c0)
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 5e+292) {
		tmp = c0 * Math.pow(((V * l) / A), -0.5);
	} else {
		tmp = 1.0 / (Math.sqrt((V * (l / A))) / c0);
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_0 <= 5e+292:
		tmp = c0 * math.pow(((V * l) / A), -0.5)
	else:
		tmp = 1.0 / (math.sqrt((V * (l / A))) / c0)
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 5e+292)
		tmp = Float64(c0 * (Float64(Float64(V * l) / A) ^ -0.5));
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(V * Float64(l / A))) / c0));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_0 <= 5e+292)
		tmp = c0 * (((V * l) / A) ^ -0.5);
	else
		tmp = 1.0 / (sqrt((V * (l / A))) / c0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+292], N[(c0 * N[Power[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 32.3%

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

      1. associate-/r* 51.1%

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

      2. div-inv 51.1%

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

   3. Applied egg-rr 51.1%

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

      1. associate-*l/ 51.1%

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

      2. div-inv 51.1%

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

   5. Applied egg-rr 51.1%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.9999999999999996e292

   1. Initial program 99.6%

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

      1. pow1/2 99.6%

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

      2. clear-num 99.6%

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

      3. inv-pow 99.6%

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

      4. pow-pow 99.6%

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

      5. associate-/l* 86.6%

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

      6. metadata-eval 86.6%

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

   3. Applied egg-rr 86.6%

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

      1. associate-/l* 99.6%

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

   5. Simplified 99.6%

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

   if 4.9999999999999996e292 < (/.f64 A (*.f64 V l))

   1. Initial program 26.0%

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

      1. pow1/2 26.0%

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

      2. clear-num 25.9%

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

      3. inv-pow 25.9%

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

      4. pow-pow 28.2%

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

      5. associate-/l* 35.6%

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

      6. metadata-eval 35.6%

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

   3. Applied egg-rr 35.6%

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

      1. associate-/l* 28.2%

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

   5. Simplified 28.2%

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

      1. add-sqr-sqrt 28.1%

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

      2. sqrt-unprod 25.9%

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

      3. pow-prod-up 25.9%

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

      4. metadata-eval 25.9%

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

      5. inv-pow 25.9%

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

      6. sqrt-div 28.2%

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

      7. metadata-eval 28.2%

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

      8. sqrt-undiv 23.5%

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

      9. div-inv 23.4%

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

      10. clear-num 23.5%

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

      11. sqrt-undiv 28.2%

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

      12. associate-*r/ 35.7%

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

   7. Applied egg-rr 35.7%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+292}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{V \cdot \frac{\ell}{A}}}{c0}}\\ \end{array} \]

Alternative 13: 79.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+292}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_0 5e+292)
       (* c0 (pow (/ (* V l) A) -0.5))
       (/ c0 (sqrt (/ V (/ A l))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_0 <= 5e+292) {
		tmp = c0 * pow(((V * l) / A), -0.5);
	} else {
		tmp = c0 / sqrt((V / (A / l)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_0 <= 5d+292) then
        tmp = c0 * (((v * l) / a) ** (-0.5d0))
    else
        tmp = c0 / sqrt((v / (a / l)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 5e+292) {
		tmp = c0 * Math.pow(((V * l) / A), -0.5);
	} else {
		tmp = c0 / Math.sqrt((V / (A / l)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_0 <= 5e+292:
		tmp = c0 * math.pow(((V * l) / A), -0.5)
	else:
		tmp = c0 / math.sqrt((V / (A / l)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 5e+292)
		tmp = Float64(c0 * (Float64(Float64(V * l) / A) ^ -0.5));
	else
		tmp = Float64(c0 / sqrt(Float64(V / Float64(A / l))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_0 <= 5e+292)
		tmp = c0 * (((V * l) / A) ^ -0.5);
	else
		tmp = c0 / sqrt((V / (A / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+292], N[(c0 * N[Power[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 32.3%

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

      1. associate-/r* 51.1%

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

      2. div-inv 51.1%

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

   3. Applied egg-rr 51.1%

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

      1. associate-*l/ 51.1%

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

      2. div-inv 51.1%

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

   5. Applied egg-rr 51.1%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.9999999999999996e292

   1. Initial program 99.6%

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

      1. pow1/2 99.6%

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

      2. clear-num 99.6%

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

      3. inv-pow 99.6%

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

      4. pow-pow 99.6%

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

      5. associate-/l* 86.6%

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

      6. metadata-eval 86.6%

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

   3. Applied egg-rr 86.6%

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

      1. associate-/l* 99.6%

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

   5. Simplified 99.6%

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

   if 4.9999999999999996e292 < (/.f64 A (*.f64 V l))

   1. Initial program 26.0%

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

      1. pow1/2 26.0%

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

      2. clear-num 25.9%

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

      3. inv-pow 25.9%

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

      4. pow-pow 28.2%

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

      5. associate-/l* 35.6%

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

      6. metadata-eval 35.6%

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

   3. Applied egg-rr 35.6%

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

      1. associate-/l* 28.2%

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

   5. Simplified 28.2%

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

      1. add-sqr-sqrt 28.1%

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

      2. sqrt-unprod 25.9%

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

      3. pow-prod-up 25.9%

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

      4. metadata-eval 25.9%

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

      5. inv-pow 25.9%

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

      6. sqrt-div 28.2%

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

      7. metadata-eval 28.2%

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

      8. sqrt-undiv 23.5%

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

      9. div-inv 23.4%

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

      10. sqrt-undiv 28.2%

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

      11. associate-*r/ 35.6%

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

   7. Applied egg-rr 35.6%

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

      1. expm1-log1p-u 35.6%

         \[\leadsto \frac{c0}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{V \cdot \frac{\ell}{A}}\right)\right)}} \]

      2. expm1-udef 19.6%

         \[\leadsto \frac{c0}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{V \cdot \frac{\ell}{A}}\right)} - 1}} \]

   9. Applied egg-rr 19.6%

      \[\leadsto \frac{c0}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{V \cdot \frac{\ell}{A}}\right)} - 1}} \]
   10. Step-by-step derivation

       1. expm1-def 35.6%

          \[\leadsto \frac{c0}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{V \cdot \frac{\ell}{A}}\right)\right)}} \]

       2. expm1-log1p 35.6%

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

       3. associate-*r/ 28.2%

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

       4. associate-/l* 35.6%

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

   11. Simplified 35.6%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+292}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \end{array} \]

Alternative 14: 79.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 2 \cdot 10^{+299}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (or (<= t_0 0.0) (not (<= t_0 2e+299)))
     (* c0 (sqrt (/ (/ A l) V)))
     (* c0 (sqrt t_0)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 2e+299)) {
		tmp = c0 * sqrt(((A / l) / V));
	} else {
		tmp = c0 * sqrt(t_0);
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 2d+299))) then
        tmp = c0 * sqrt(((a / l) / v))
    else
        tmp = c0 * sqrt(t_0)
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 2e+299)) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 2e+299):
		tmp = c0 * math.sqrt(((A / l) / V))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 2e+299))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	else
		tmp = Float64(c0 * sqrt(t_0));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 2e+299)))
		tmp = c0 * sqrt(((A / l) / V));
	else
		tmp = c0 * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 2e+299]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $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 2.0000000000000001e299 < (/.f64 A (*.f64 V l))

   1. Initial program 26.3%

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

      1. associate-/r* 41.3%

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

      2. div-inv 41.3%

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

   3. Applied egg-rr 41.3%

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

      1. associate-*l/ 40.0%

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

      2. div-inv 40.0%

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

   5. Applied egg-rr 40.0%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 2.0000000000000001e299

   1. Initial program 99.6%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+299}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]

Alternative 15: 79.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_0 5e+300) (* c0 (sqrt t_0)) (* c0 (sqrt (/ (/ A V) l)))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_0 <= 5e+300) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_0 <= 5d+300) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 * sqrt(((a / v) / l))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 5e+300) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_0 <= 5e+300:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 5e+300)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_0 <= 5e+300)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+300], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 32.3%

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

      1. associate-/r* 51.1%

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

      2. div-inv 51.1%

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

   3. Applied egg-rr 51.1%

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

      1. associate-*l/ 51.1%

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

      2. div-inv 51.1%

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

   5. Applied egg-rr 51.1%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 5.00000000000000026e300

   1. Initial program 99.6%

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

   if 5.00000000000000026e300 < (/.f64 A (*.f64 V l))

   1. Initial program 18.7%

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

      1. add-sqr-sqrt 11.7%

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

      2. sqrt-unprod 11.9%

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

      3. pow1/2 11.9%

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

      4. *-commutative 11.9%

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

      5. *-commutative 11.9%

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

      6. swap-sqr 11.1%

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

      7. add-sqr-sqrt 11.1%

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

   3. Applied egg-rr 11.1%

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

      1. unpow1/2 11.1%

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

   5. Simplified 11.1%

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

      1. sqrt-prod 11.1%

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

      2. associate-/r* 18.3%

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

      3. sqrt-prod 21.0%

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

      4. add-sqr-sqrt 30.5%

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

   7. Applied egg-rr 30.5%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+300}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]

Alternative 16: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_0 2e+299) (* c0 (sqrt t_0)) (/ c0 (sqrt (* V (/ l A))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_0 <= 2e+299) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_0 <= 2d+299) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 2e+299) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_0 <= 2e+299:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 2e+299)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_0 <= 2e+299)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+299], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 32.3%

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

      1. associate-/r* 51.1%

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

      2. div-inv 51.1%

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

   3. Applied egg-rr 51.1%

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

      1. associate-*l/ 51.1%

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

      2. div-inv 51.1%

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

   5. Applied egg-rr 51.1%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 2.0000000000000001e299

   1. Initial program 99.6%

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

   if 2.0000000000000001e299 < (/.f64 A (*.f64 V l))

   1. Initial program 20.7%

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

      1. pow1/2 20.7%

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

      2. clear-num 20.7%

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

      3. inv-pow 20.7%

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

      4. pow-pow 23.1%

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

      5. associate-/l* 31.0%

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

      6. metadata-eval 31.0%

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

   3. Applied egg-rr 31.0%

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

      1. associate-/l* 23.1%

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

   5. Simplified 23.1%

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

      1. add-sqr-sqrt 23.1%

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

      2. sqrt-unprod 20.7%

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

      3. pow-prod-up 20.7%

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

      4. metadata-eval 20.7%

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

      5. inv-pow 20.7%

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

      6. sqrt-div 23.1%

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

      7. metadata-eval 23.1%

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

      8. sqrt-undiv 20.4%

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

      9. div-inv 20.4%

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

      10. sqrt-undiv 23.1%

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

      11. associate-*r/ 31.1%

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

   7. Applied egg-rr 31.1%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+299}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]

Alternative 17: 80.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_0 2e+299) (* c0 (sqrt t_0)) (/ c0 (sqrt (/ V (/ A l))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_0 <= 2e+299) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((V / (A / l)));
	}
	return tmp;
}

Fortran

NOTE: V and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_0 <= 2d+299) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((v / (a / l)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 2e+299) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((V / (A / l)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_0 <= 2e+299:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((V / (A / l)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 2e+299)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(V / Float64(A / l))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_0 <= 2e+299)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((V / (A / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+299], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 32.3%

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

      1. associate-/r* 51.1%

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

      2. div-inv 51.1%

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

   3. Applied egg-rr 51.1%

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

      1. associate-*l/ 51.1%

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

      2. div-inv 51.1%

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

   5. Applied egg-rr 51.1%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 2.0000000000000001e299

   1. Initial program 99.6%

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

   if 2.0000000000000001e299 < (/.f64 A (*.f64 V l))

   1. Initial program 20.7%

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

      1. pow1/2 20.7%

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

      2. clear-num 20.7%

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

      3. inv-pow 20.7%

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

      4. pow-pow 23.1%

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

      5. associate-/l* 31.0%

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

      6. metadata-eval 31.0%

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

   3. Applied egg-rr 31.0%

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

      1. associate-/l* 23.1%

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

   5. Simplified 23.1%

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

      1. add-sqr-sqrt 23.1%

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

      2. sqrt-unprod 20.7%

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

      3. pow-prod-up 20.7%

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

      4. metadata-eval 20.7%

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

      5. inv-pow 20.7%

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

      6. sqrt-div 23.1%

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

      7. metadata-eval 23.1%

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

      8. sqrt-undiv 20.4%

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

      9. div-inv 20.4%

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

      10. sqrt-undiv 23.1%

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

      11. associate-*r/ 31.1%

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

   7. Applied egg-rr 31.1%

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

      1. expm1-log1p-u 31.1%

         \[\leadsto \frac{c0}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{V \cdot \frac{\ell}{A}}\right)\right)}} \]

      2. expm1-udef 18.4%

         \[\leadsto \frac{c0}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{V \cdot \frac{\ell}{A}}\right)} - 1}} \]

   9. Applied egg-rr 18.4%

      \[\leadsto \frac{c0}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{V \cdot \frac{\ell}{A}}\right)} - 1}} \]
   10. Step-by-step derivation

       1. expm1-def 31.1%

          \[\leadsto \frac{c0}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{V \cdot \frac{\ell}{A}}\right)\right)}} \]

       2. expm1-log1p 31.1%

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

       3. associate-*r/ 23.1%

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

       4. associate-/l* 31.0%

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

   11. Simplified 31.0%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+299}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \end{array} \]

Alternative 18: 73.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	return c0 * math.sqrt((A / (V * l)))

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	return Float64(c0 * sqrt(Float64(A / Float64(V * l))))
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp = code(c0, A, V, l)
	tmp = c0 * sqrt((A / (V * l)));
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.4%

   \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]

2. Final simplification 76.4%

   \[\leadsto c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]

Reproduce

herbie shell --seed 2023229 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))