Henrywood and Agarwal, Equation (3)

Percentage Accurate: 74.2% → 89.1%

Time: 10.3s

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: 74.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;\ell \cdot V \leq -2 \cdot 10^{+296}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;\ell \cdot V \leq -1 \cdot 10^{-188}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \left(-V\right)} \cdot \sqrt{\frac{-1}{A}}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0 \cdot t\_0}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{+292}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{c0}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (/ A V))))
   (if (<= (* l V) -2e+296)
     (* c0 (/ (sqrt (/ A (- l))) (sqrt (- V))))
     (if (<= (* l V) -1e-188)
       (/ c0 (* (sqrt (* l (- V))) (sqrt (/ -1.0 A))))
       (if (<= (* l V) 0.0)
         (/ (* c0 t_0) (sqrt l))
         (if (<= (* l V) 2e+292)
           (* c0 (/ (sqrt A) (sqrt (* l V))))
           (* t_0 (/ c0 (sqrt l)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / V));
	double tmp;
	if ((l * V) <= -2e+296) {
		tmp = c0 * (sqrt((A / -l)) / sqrt(-V));
	} else if ((l * V) <= -1e-188) {
		tmp = c0 / (sqrt((l * -V)) * sqrt((-1.0 / A)));
	} else if ((l * V) <= 0.0) {
		tmp = (c0 * t_0) / sqrt(l);
	} else if ((l * V) <= 2e+292) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = t_0 * (c0 / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a / v))
    if ((l * v) <= (-2d+296)) then
        tmp = c0 * (sqrt((a / -l)) / sqrt(-v))
    else if ((l * v) <= (-1d-188)) then
        tmp = c0 / (sqrt((l * -v)) * sqrt(((-1.0d0) / a)))
    else if ((l * v) <= 0.0d0) then
        tmp = (c0 * t_0) / sqrt(l)
    else if ((l * v) <= 2d+292) then
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    else
        tmp = t_0 * (c0 / sqrt(l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / V));
	double tmp;
	if ((l * V) <= -2e+296) {
		tmp = c0 * (Math.sqrt((A / -l)) / Math.sqrt(-V));
	} else if ((l * V) <= -1e-188) {
		tmp = c0 / (Math.sqrt((l * -V)) * Math.sqrt((-1.0 / A)));
	} else if ((l * V) <= 0.0) {
		tmp = (c0 * t_0) / Math.sqrt(l);
	} else if ((l * V) <= 2e+292) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = t_0 * (c0 / Math.sqrt(l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / V))
	tmp = 0
	if (l * V) <= -2e+296:
		tmp = c0 * (math.sqrt((A / -l)) / math.sqrt(-V))
	elif (l * V) <= -1e-188:
		tmp = c0 / (math.sqrt((l * -V)) * math.sqrt((-1.0 / A)))
	elif (l * V) <= 0.0:
		tmp = (c0 * t_0) / math.sqrt(l)
	elif (l * V) <= 2e+292:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = t_0 * (c0 / math.sqrt(l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(A / V))
	tmp = 0.0
	if (Float64(l * V) <= -2e+296)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / Float64(-l))) / sqrt(Float64(-V))));
	elseif (Float64(l * V) <= -1e-188)
		tmp = Float64(c0 / Float64(sqrt(Float64(l * Float64(-V))) * sqrt(Float64(-1.0 / A))));
	elseif (Float64(l * V) <= 0.0)
		tmp = Float64(Float64(c0 * t_0) / sqrt(l));
	elseif (Float64(l * V) <= 2e+292)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = Float64(t_0 * Float64(c0 / sqrt(l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((A / V));
	tmp = 0.0;
	if ((l * V) <= -2e+296)
		tmp = c0 * (sqrt((A / -l)) / sqrt(-V));
	elseif ((l * V) <= -1e-188)
		tmp = c0 / (sqrt((l * -V)) * sqrt((-1.0 / A)));
	elseif ((l * V) <= 0.0)
		tmp = (c0 * t_0) / sqrt(l);
	elseif ((l * V) <= 2e+292)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = t_0 * (c0 / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], -2e+296], N[(c0 * N[(N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -1e-188], N[(c0 / N[(N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(-1.0 / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], N[(N[(c0 * t$95$0), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 2e+292], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -1.99999999999999996e296

   1. Initial program 47.8%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 47.8

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

   4. Applied egg-rr 47.8%

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

      1. associate-/l/ N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. associate-*l/ N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-neg.f64 45.2

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

   6. Applied egg-rr 45.2%

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

   if -1.99999999999999996e296 < (*.f64 V l) < -9.9999999999999995e-189

   1. Initial program 87.6%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 87.7

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

   4. Applied egg-rr 87.7%

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

      1. lift-*.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. sqrt-prod N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 99.5

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

   6. Applied egg-rr 99.5%

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

   if -9.9999999999999995e-189 < (*.f64 V l) < 0.0

   1. Initial program 49.7%

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

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 29.5

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

   4. Applied egg-rr 29.5%

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

   if 0.0 < (*.f64 V l) < 2e292

   1. Initial program 83.0%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 99.3

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

   4. Applied egg-rr 99.3%

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

   if 2e292 < (*.f64 V l)

   1. Initial program 22.5%

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

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 48.2

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

   4. Applied egg-rr 48.2%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -2 \cdot 10^{+296}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;\ell \cdot V \leq -1 \cdot 10^{-188}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \left(-V\right)} \cdot \sqrt{\frac{-1}{A}}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{+292}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;\ell \cdot V \leq -2 \cdot 10^{+296}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;\ell \cdot V \leq -1 \cdot 10^{-188}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0 \cdot t\_0}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{+292}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{c0}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (/ A V))))
   (if (<= (* l V) -2e+296)
     (* c0 (/ (sqrt (/ A (- l))) (sqrt (- V))))
     (if (<= (* l V) -1e-188)
       (* c0 (/ (sqrt (- A)) (sqrt (* l (- V)))))
       (if (<= (* l V) 0.0)
         (/ (* c0 t_0) (sqrt l))
         (if (<= (* l V) 2e+292)
           (* c0 (/ (sqrt A) (sqrt (* l V))))
           (* t_0 (/ c0 (sqrt l)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / V));
	double tmp;
	if ((l * V) <= -2e+296) {
		tmp = c0 * (sqrt((A / -l)) / sqrt(-V));
	} else if ((l * V) <= -1e-188) {
		tmp = c0 * (sqrt(-A) / sqrt((l * -V)));
	} else if ((l * V) <= 0.0) {
		tmp = (c0 * t_0) / sqrt(l);
	} else if ((l * V) <= 2e+292) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = t_0 * (c0 / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a / v))
    if ((l * v) <= (-2d+296)) then
        tmp = c0 * (sqrt((a / -l)) / sqrt(-v))
    else if ((l * v) <= (-1d-188)) then
        tmp = c0 * (sqrt(-a) / sqrt((l * -v)))
    else if ((l * v) <= 0.0d0) then
        tmp = (c0 * t_0) / sqrt(l)
    else if ((l * v) <= 2d+292) then
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    else
        tmp = t_0 * (c0 / sqrt(l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / V));
	double tmp;
	if ((l * V) <= -2e+296) {
		tmp = c0 * (Math.sqrt((A / -l)) / Math.sqrt(-V));
	} else if ((l * V) <= -1e-188) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((l * -V)));
	} else if ((l * V) <= 0.0) {
		tmp = (c0 * t_0) / Math.sqrt(l);
	} else if ((l * V) <= 2e+292) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = t_0 * (c0 / Math.sqrt(l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / V))
	tmp = 0
	if (l * V) <= -2e+296:
		tmp = c0 * (math.sqrt((A / -l)) / math.sqrt(-V))
	elif (l * V) <= -1e-188:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((l * -V)))
	elif (l * V) <= 0.0:
		tmp = (c0 * t_0) / math.sqrt(l)
	elif (l * V) <= 2e+292:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = t_0 * (c0 / math.sqrt(l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(A / V))
	tmp = 0.0
	if (Float64(l * V) <= -2e+296)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / Float64(-l))) / sqrt(Float64(-V))));
	elseif (Float64(l * V) <= -1e-188)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(l * Float64(-V)))));
	elseif (Float64(l * V) <= 0.0)
		tmp = Float64(Float64(c0 * t_0) / sqrt(l));
	elseif (Float64(l * V) <= 2e+292)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = Float64(t_0 * Float64(c0 / sqrt(l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((A / V));
	tmp = 0.0;
	if ((l * V) <= -2e+296)
		tmp = c0 * (sqrt((A / -l)) / sqrt(-V));
	elseif ((l * V) <= -1e-188)
		tmp = c0 * (sqrt(-A) / sqrt((l * -V)));
	elseif ((l * V) <= 0.0)
		tmp = (c0 * t_0) / sqrt(l);
	elseif ((l * V) <= 2e+292)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = t_0 * (c0 / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], -2e+296], N[(c0 * N[(N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -1e-188], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], N[(N[(c0 * t$95$0), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 2e+292], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -1.99999999999999996e296

   1. Initial program 47.8%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 47.8

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

   4. Applied egg-rr 47.8%

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

      1. associate-/l/ N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. associate-*l/ N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-neg.f64 45.2

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

   6. Applied egg-rr 45.2%

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

   if -1.99999999999999996e296 < (*.f64 V l) < -9.9999999999999995e-189

   1. Initial program 87.6%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 0.0

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

   4. Applied egg-rr 0.0%

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

      1. lift-*.f64 N/A

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

      2. sqrt-undiv N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 99.4

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

   6. Applied egg-rr 99.4%

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

   if -9.9999999999999995e-189 < (*.f64 V l) < 0.0

   1. Initial program 49.7%

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

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 29.5

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

   4. Applied egg-rr 29.5%

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

   if 0.0 < (*.f64 V l) < 2e292

   1. Initial program 83.0%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 99.3

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

   4. Applied egg-rr 99.3%

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

   if 2e292 < (*.f64 V l)

   1. Initial program 22.5%

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

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 48.2

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

   4. Applied egg-rr 48.2%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -2 \cdot 10^{+296}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;\ell \cdot V \leq -1 \cdot 10^{-188}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{+292}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ t_1 := t\_0 \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \cdot V \leq -1 \cdot 10^{-188}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0 \cdot t\_0}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{+292}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (/ A V))) (t_1 (* t_0 (/ c0 (sqrt l)))))
   (if (<= (* l V) (- INFINITY))
     t_1
     (if (<= (* l V) -1e-188)
       (* c0 (/ (sqrt (- A)) (sqrt (* l (- V)))))
       (if (<= (* l V) 0.0)
         (/ (* c0 t_0) (sqrt l))
         (if (<= (* l V) 2e+292) (* c0 (/ (sqrt A) (sqrt (* l V)))) t_1))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / V));
	double t_1 = t_0 * (c0 / sqrt(l));
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = t_1;
	} else if ((l * V) <= -1e-188) {
		tmp = c0 * (sqrt(-A) / sqrt((l * -V)));
	} else if ((l * V) <= 0.0) {
		tmp = (c0 * t_0) / sqrt(l);
	} else if ((l * V) <= 2e+292) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / V));
	double t_1 = t_0 * (c0 / Math.sqrt(l));
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if ((l * V) <= -1e-188) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((l * -V)));
	} else if ((l * V) <= 0.0) {
		tmp = (c0 * t_0) / Math.sqrt(l);
	} else if ((l * V) <= 2e+292) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / V))
	t_1 = t_0 * (c0 / math.sqrt(l))
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = t_1
	elif (l * V) <= -1e-188:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((l * -V)))
	elif (l * V) <= 0.0:
		tmp = (c0 * t_0) / math.sqrt(l)
	elif (l * V) <= 2e+292:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = t_1
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(A / V))
	t_1 = Float64(t_0 * Float64(c0 / sqrt(l)))
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = t_1;
	elseif (Float64(l * V) <= -1e-188)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(l * Float64(-V)))));
	elseif (Float64(l * V) <= 0.0)
		tmp = Float64(Float64(c0 * t_0) / sqrt(l));
	elseif (Float64(l * V) <= 2e+292)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((A / V));
	t_1 = t_0 * (c0 / sqrt(l));
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = t_1;
	elseif ((l * V) <= -1e-188)
		tmp = c0 * (sqrt(-A) / sqrt((l * -V)));
	elseif ((l * V) <= 0.0)
		tmp = (c0 * t_0) / sqrt(l);
	elseif ((l * V) <= 2e+292)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], t$95$1, If[LessEqual[N[(l * V), $MachinePrecision], -1e-188], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], N[(N[(c0 * t$95$0), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 2e+292], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -inf.0 or 2e292 < (*.f64 V l)

   1. Initial program 35.4%

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

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 47.8

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

   4. Applied egg-rr 47.8%

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

   if -inf.0 < (*.f64 V l) < -9.9999999999999995e-189

   1. Initial program 87.8%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 0.0

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

   4. Applied egg-rr 0.0%

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

      1. lift-*.f64 N/A

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

      2. sqrt-undiv N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 99.4

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

   6. Applied egg-rr 99.4%

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

   if -9.9999999999999995e-189 < (*.f64 V l) < 0.0

   1. Initial program 49.7%

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

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 29.5

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

   4. Applied egg-rr 29.5%

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

   if 0.0 < (*.f64 V l) < 2e292

   1. Initial program 83.0%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 99.3

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

   4. Applied egg-rr 99.3%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq -1 \cdot 10^{-188}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{+292}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq -4 \cdot 10^{-280}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{+292}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* (sqrt (/ A V)) (/ c0 (sqrt l)))))
   (if (<= (* l V) (- INFINITY))
     t_0
     (if (<= (* l V) -4e-280)
       (* c0 (/ (sqrt (- A)) (sqrt (* l (- V)))))
       (if (<= (* l V) 0.0)
         t_0
         (if (<= (* l V) 2e+292) (* c0 (/ (sqrt A) (sqrt (* l V)))) t_0))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / V)) * (c0 / sqrt(l));
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((l * V) <= -4e-280) {
		tmp = c0 * (sqrt(-A) / sqrt((l * -V)));
	} else if ((l * V) <= 0.0) {
		tmp = t_0;
	} else if ((l * V) <= 2e+292) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / V)) * (c0 / Math.sqrt(l));
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if ((l * V) <= -4e-280) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((l * -V)));
	} else if ((l * V) <= 0.0) {
		tmp = t_0;
	} else if ((l * V) <= 2e+292) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / V)) * (c0 / math.sqrt(l))
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = t_0
	elif (l * V) <= -4e-280:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((l * -V)))
	elif (l * V) <= 0.0:
		tmp = t_0
	elif (l * V) <= 2e+292:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(sqrt(Float64(A / V)) * Float64(c0 / sqrt(l)))
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(l * V) <= -4e-280)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(l * Float64(-V)))));
	elseif (Float64(l * V) <= 0.0)
		tmp = t_0;
	elseif (Float64(l * V) <= 2e+292)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((A / V)) * (c0 / sqrt(l));
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = t_0;
	elseif ((l * V) <= -4e-280)
		tmp = c0 * (sqrt(-A) / sqrt((l * -V)));
	elseif ((l * V) <= 0.0)
		tmp = t_0;
	elseif ((l * V) <= 2e+292)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(l * V), $MachinePrecision], -4e-280], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(l * V), $MachinePrecision], 2e+292], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -inf.0 or -3.9999999999999998e-280 < (*.f64 V l) < 0.0 or 2e292 < (*.f64 V l)

   1. Initial program 38.5%

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

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 39.8

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

   4. Applied egg-rr 39.8%

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

   if -inf.0 < (*.f64 V l) < -3.9999999999999998e-280

   1. Initial program 87.0%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 0.0

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

   4. Applied egg-rr 0.0%

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

      1. lift-*.f64 N/A

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

      2. sqrt-undiv N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 99.5

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

   6. Applied egg-rr 99.5%

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

   if 0.0 < (*.f64 V l) < 2e292

   1. Initial program 83.0%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 99.3

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

   4. Applied egg-rr 99.3%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq -4 \cdot 10^{-280}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{+292}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq -4 \cdot 10^{-280}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{+292}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (/ (sqrt (/ A V)) (sqrt l)))))
   (if (<= (* l V) (- INFINITY))
     t_0
     (if (<= (* l V) -4e-280)
       (* c0 (/ (sqrt (- A)) (sqrt (* l (- V)))))
       (if (<= (* l V) 0.0)
         t_0
         (if (<= (* l V) 2e+292) (* c0 (/ (sqrt A) (sqrt (* l V)))) t_0))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (sqrt((A / V)) / sqrt(l));
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((l * V) <= -4e-280) {
		tmp = c0 * (sqrt(-A) / sqrt((l * -V)));
	} else if ((l * V) <= 0.0) {
		tmp = t_0;
	} else if ((l * V) <= 2e+292) {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if ((l * V) <= -4e-280) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((l * -V)));
	} else if ((l * V) <= 0.0) {
		tmp = t_0;
	} else if ((l * V) <= 2e+292) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = t_0
	elif (l * V) <= -4e-280:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((l * -V)))
	elif (l * V) <= 0.0:
		tmp = t_0
	elif (l * V) <= 2e+292:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	else:
		tmp = t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)))
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(l * V) <= -4e-280)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(l * Float64(-V)))));
	elseif (Float64(l * V) <= 0.0)
		tmp = t_0;
	elseif (Float64(l * V) <= 2e+292)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * (sqrt((A / V)) / sqrt(l));
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = t_0;
	elseif ((l * V) <= -4e-280)
		tmp = c0 * (sqrt(-A) / sqrt((l * -V)));
	elseif ((l * V) <= 0.0)
		tmp = t_0;
	elseif ((l * V) <= 2e+292)
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(l * V), $MachinePrecision], -4e-280], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(l * V), $MachinePrecision], 2e+292], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -inf.0 or -3.9999999999999998e-280 < (*.f64 V l) < 0.0 or 2e292 < (*.f64 V l)

   1. Initial program 38.5%

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

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 39.8

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

   4. Applied egg-rr 39.8%

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

   if -inf.0 < (*.f64 V l) < -3.9999999999999998e-280

   1. Initial program 87.0%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 0.0

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

   4. Applied egg-rr 0.0%

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

      1. lift-*.f64 N/A

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

      2. sqrt-undiv N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 99.5

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

   6. Applied egg-rr 99.5%

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

   if 0.0 < (*.f64 V l) < 2e292

   1. Initial program 83.0%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 99.3

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

   4. Applied egg-rr 99.3%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq -4 \cdot 10^{-280}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{+292}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ t_1 := \frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{if}\;t\_0 \leq 10^{-303}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+298}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* l V))) (t_1 (/ c0 (sqrt (* V (/ l A))))))
   (if (<= t_0 1e-303) t_1 (if (<= t_0 5e+298) (* c0 (sqrt t_0)) t_1))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double t_1 = c0 / sqrt((V * (l / A)));
	double tmp;
	if (t_0 <= 1e-303) {
		tmp = t_1;
	} else if (t_0 <= 5e+298) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a / (l * v)
    t_1 = c0 / sqrt((v * (l / a)))
    if (t_0 <= 1d-303) then
        tmp = t_1
    else if (t_0 <= 5d+298) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double t_1 = c0 / Math.sqrt((V * (l / A)));
	double tmp;
	if (t_0 <= 1e-303) {
		tmp = t_1;
	} else if (t_0 <= 5e+298) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (l * V)
	t_1 = c0 / math.sqrt((V * (l / A)))
	tmp = 0
	if t_0 <= 1e-303:
		tmp = t_1
	elif t_0 <= 5e+298:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = t_1
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(l * V))
	t_1 = Float64(c0 / sqrt(Float64(V * Float64(l / A))))
	tmp = 0.0
	if (t_0 <= 1e-303)
		tmp = t_1;
	elseif (t_0 <= 5e+298)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (l * V);
	t_1 = c0 / sqrt((V * (l / A)));
	tmp = 0.0;
	if (t_0 <= 1e-303)
		tmp = t_1;
	elseif (t_0 <= 5e+298)
		tmp = c0 * sqrt(t_0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-303], t$95$1, If[LessEqual[t$95$0, 5e+298], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 9.99999999999999931e-304 or 5.0000000000000003e298 < (/.f64 A (*.f64 V l))

   1. Initial program 36.0%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 36.4

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

   4. Applied egg-rr 36.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 47.2

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

   6. Applied egg-rr 47.2%

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

   if 9.99999999999999931e-304 < (/.f64 A (*.f64 V l)) < 5.0000000000000003e298

   1. Initial program 99.3%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 10^{-303}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;\frac{A}{\ell \cdot V} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ t_1 := c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{if}\;t\_0 \leq 10^{-303}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+282}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* l V))) (t_1 (* c0 (sqrt (/ (/ A l) V)))))
   (if (<= t_0 1e-303) t_1 (if (<= t_0 1e+282) (* c0 (sqrt t_0)) t_1))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double t_1 = c0 * sqrt(((A / l) / V));
	double tmp;
	if (t_0 <= 1e-303) {
		tmp = t_1;
	} else if (t_0 <= 1e+282) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a / (l * v)
    t_1 = c0 * sqrt(((a / l) / v))
    if (t_0 <= 1d-303) then
        tmp = t_1
    else if (t_0 <= 1d+282) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double t_1 = c0 * Math.sqrt(((A / l) / V));
	double tmp;
	if (t_0 <= 1e-303) {
		tmp = t_1;
	} else if (t_0 <= 1e+282) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (l * V)
	t_1 = c0 * math.sqrt(((A / l) / V))
	tmp = 0
	if t_0 <= 1e-303:
		tmp = t_1
	elif t_0 <= 1e+282:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = t_1
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(l * V))
	t_1 = Float64(c0 * sqrt(Float64(Float64(A / l) / V)))
	tmp = 0.0
	if (t_0 <= 1e-303)
		tmp = t_1;
	elseif (t_0 <= 1e+282)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (l * V);
	t_1 = c0 * sqrt(((A / l) / V));
	tmp = 0.0;
	if (t_0 <= 1e-303)
		tmp = t_1;
	elseif (t_0 <= 1e+282)
		tmp = c0 * sqrt(t_0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-303], t$95$1, If[LessEqual[t$95$0, 1e+282], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 9.99999999999999931e-304 or 1.00000000000000003e282 < (/.f64 A (*.f64 V l))

   1. Initial program 36.6%

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 46.7

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

   4. Applied egg-rr 46.7%

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

   if 9.99999999999999931e-304 < (/.f64 A (*.f64 V l)) < 1.00000000000000003e282

   1. Initial program 99.3%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 10^{-303}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\frac{A}{\ell \cdot V} \leq 10^{+282}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\left(c0 \cdot \sqrt{A}\right) \cdot \frac{\sqrt{\frac{-1}{V}}}{\sqrt{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= l -2e-311)
   (* (* c0 (sqrt A)) (/ (sqrt (/ -1.0 V)) (sqrt (- l))))
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -2e-311) {
		tmp = (c0 * sqrt(A)) * (sqrt((-1.0 / V)) / sqrt(-l));
	} else {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-2d-311)) then
        tmp = (c0 * sqrt(a)) * (sqrt(((-1.0d0) / v)) / sqrt(-l))
    else
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -2e-311) {
		tmp = (c0 * Math.sqrt(A)) * (Math.sqrt((-1.0 / V)) / Math.sqrt(-l));
	} else {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if l <= -2e-311:
		tmp = (c0 * math.sqrt(A)) * (math.sqrt((-1.0 / V)) / math.sqrt(-l))
	else:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (l <= -2e-311)
		tmp = Float64(Float64(c0 * sqrt(A)) * Float64(sqrt(Float64(-1.0 / V)) / sqrt(Float64(-l))));
	else
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (l <= -2e-311)
		tmp = (c0 * sqrt(A)) * (sqrt((-1.0 / V)) / sqrt(-l));
	else
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[l, -2e-311], N[(N[(c0 * N[Sqrt[A], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(-1.0 / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.9999999999999e-311

   1. Initial program 70.8%

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

      1. lift-*.f64 N/A

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

      2. div-inv N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 33.8

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

   4. Applied egg-rr 33.8%

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

      1. associate-/r* N/A

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

      2. frac-2neg N/A

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

      3. sqrt-div N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. distribute-frac-neg2 N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 42.1

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

   6. Applied egg-rr 42.1%

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

   if -1.9999999999999e-311 < l

   1. Initial program 72.0%

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

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 81.1

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

   4. Applied egg-rr 81.1%

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

Alternative 9: 86.6% accurate, 0.5× speedup

Math

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

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= V -1e-310)
   (* c0 (/ (sqrt (/ A (- l))) (sqrt (- V))))
   (* (* c0 (sqrt A)) (/ (/ 1.0 (sqrt V)) (sqrt l)))))

C

assert(c0 < A && A < V && 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(A)) * ((1.0 / sqrt(V)) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (v <= (-1d-310)) then
        tmp = c0 * (sqrt((a / -l)) / sqrt(-v))
    else
        tmp = (c0 * sqrt(a)) * ((1.0d0 / sqrt(v)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (V <= -1e-310) {
		tmp = c0 * (Math.sqrt((A / -l)) / Math.sqrt(-V));
	} else {
		tmp = (c0 * Math.sqrt(A)) * ((1.0 / Math.sqrt(V)) / Math.sqrt(l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, 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(A)) * ((1.0 / math.sqrt(V)) / math.sqrt(l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (V <= -1e-310)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / Float64(-l))) / sqrt(Float64(-V))));
	else
		tmp = Float64(Float64(c0 * sqrt(A)) * Float64(Float64(1.0 / sqrt(V)) / sqrt(l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (V <= -1e-310)
		tmp = c0 * (sqrt((A / -l)) / sqrt(-V));
	else
		tmp = (c0 * sqrt(A)) * ((1.0 / sqrt(V)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[V, -1e-310], N[(c0 * N[(N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * N[Sqrt[A], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[Sqrt[V], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if V < -9.999999999999969e-311

   1. Initial program 68.3%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 68.4

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

   4. Applied egg-rr 68.4%

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

      1. associate-/l/ N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. associate-*l/ N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-neg.f64 81.2

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

   6. Applied egg-rr 81.2%

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

   if -9.999999999999969e-311 < V

   1. Initial program 74.0%

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

      1. lift-*.f64 N/A

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

      2. div-inv N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 36.8

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

   4. Applied egg-rr 36.8%

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

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

      3. pow1/2 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow1/2 N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-/.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 43.0

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

   6. Applied egg-rr 43.0%

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

4. Final simplification 60.6%

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

Alternative 10: 88.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot \frac{c0}{\sqrt{\ell}}}{\sqrt{-V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -2e-310)
   (/ (* (sqrt (- A)) (/ c0 (sqrt l))) (sqrt (- V)))
   (* c0 (/ (sqrt A) (sqrt (* l V))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -2e-310) {
		tmp = (sqrt(-A) * (c0 / sqrt(l))) / sqrt(-V);
	} else {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (a <= (-2d-310)) then
        tmp = (sqrt(-a) * (c0 / sqrt(l))) / sqrt(-v)
    else
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -2e-310) {
		tmp = (Math.sqrt(-A) * (c0 / Math.sqrt(l))) / Math.sqrt(-V);
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -2e-310:
		tmp = (math.sqrt(-A) * (c0 / math.sqrt(l))) / math.sqrt(-V)
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -2e-310)
		tmp = Float64(Float64(sqrt(Float64(-A)) * Float64(c0 / sqrt(l))) / sqrt(Float64(-V)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (A <= -2e-310)
		tmp = (sqrt(-A) * (c0 / sqrt(l))) / sqrt(-V);
	else
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -2e-310], N[(N[(N[Sqrt[(-A)], $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -1.999999999999994e-310

   1. Initial program 74.3%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 74.3

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

   4. Applied egg-rr 74.3%

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

      1. *-lft-identity N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. lift-/.f64 N/A

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

      6. un-div-inv N/A

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

      7. sqrt-undiv N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lift-sqrt.f64 N/A

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

      14. sqrt-div N/A

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

      15. lift-/.f64 N/A

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

      16. clear-num N/A

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

      17. frac-times N/A

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

   6. Applied egg-rr 41.1%

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

   if -1.999999999999994e-310 < A

   1. Initial program 68.0%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 81.2

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

   4. Applied egg-rr 81.2%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot \frac{c0}{\sqrt{\ell}}}{\sqrt{-V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -4 \cdot 10^{-280}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) -4e-280)
   (* c0 (/ (sqrt (- A)) (sqrt (* l (- V)))))
   (if (<= (* l V) 0.0)
     (/ c0 (sqrt (* V (/ l A))))
     (* c0 (/ (sqrt A) (sqrt (* l V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -4e-280) {
		tmp = c0 * (sqrt(-A) / sqrt((l * -V)));
	} else if ((l * V) <= 0.0) {
		tmp = c0 / sqrt((V * (l / A)));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((l * v) <= (-4d-280)) then
        tmp = c0 * (sqrt(-a) / sqrt((l * -v)))
    else if ((l * v) <= 0.0d0) then
        tmp = c0 / sqrt((v * (l / a)))
    else
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -4e-280) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((l * -V)));
	} else if ((l * V) <= 0.0) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -4e-280:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((l * -V)))
	elif (l * V) <= 0.0:
		tmp = c0 / math.sqrt((V * (l / A)))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= -4e-280)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(l * Float64(-V)))));
	elseif (Float64(l * V) <= 0.0)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -4e-280)
		tmp = c0 * (sqrt(-A) / sqrt((l * -V)));
	elseif ((l * V) <= 0.0)
		tmp = c0 / sqrt((V * (l / A)));
	else
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], -4e-280], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -3.9999999999999998e-280

   1. Initial program 79.1%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 0.0

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

   4. Applied egg-rr 0.0%

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

      1. lift-*.f64 N/A

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

      2. sqrt-undiv N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 89.2

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

   6. Applied egg-rr 89.2%

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

   if -3.9999999999999998e-280 < (*.f64 V l) < 0.0

   1. Initial program 41.5%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 41.5

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

   4. Applied egg-rr 41.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 59.7

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

   6. Applied egg-rr 59.7%

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

   if 0.0 < (*.f64 V l)

   1. Initial program 73.9%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 88.7

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

   4. Applied egg-rr 88.7%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -4 \cdot 10^{-280}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.8% accurate, 0.5× speedup

Math

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

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) -1e-249)
   (* c0 (sqrt (* A (/ 1.0 (* l V)))))
   (if (<= (* l V) 0.0)
     (/ c0 (sqrt (* V (/ l A))))
     (* c0 (/ (sqrt A) (sqrt (* l V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -1e-249) {
		tmp = c0 * sqrt((A * (1.0 / (l * V))));
	} else if ((l * V) <= 0.0) {
		tmp = c0 / sqrt((V * (l / A)));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((l * v) <= (-1d-249)) then
        tmp = c0 * sqrt((a * (1.0d0 / (l * v))))
    else if ((l * v) <= 0.0d0) then
        tmp = c0 / sqrt((v * (l / a)))
    else
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -1e-249) {
		tmp = c0 * Math.sqrt((A * (1.0 / (l * V))));
	} else if ((l * V) <= 0.0) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -1e-249:
		tmp = c0 * math.sqrt((A * (1.0 / (l * V))))
	elif (l * V) <= 0.0:
		tmp = c0 / math.sqrt((V * (l / A)))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= -1e-249)
		tmp = Float64(c0 * sqrt(Float64(A * Float64(1.0 / Float64(l * V)))));
	elseif (Float64(l * V) <= 0.0)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -1e-249)
		tmp = c0 * sqrt((A * (1.0 / (l * V))));
	elseif ((l * V) <= 0.0)
		tmp = c0 / sqrt((V * (l / A)));
	else
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], -1e-249], N[(c0 * N[Sqrt[N[(A * N[(1.0 / N[(l * V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -1.00000000000000005e-249

   1. Initial program 79.4%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 79.4

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

   4. Applied egg-rr 79.4%

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

   if -1.00000000000000005e-249 < (*.f64 V l) < 0.0

   1. Initial program 43.5%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 43.5

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

   4. Applied egg-rr 43.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 60.4

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

   6. Applied egg-rr 60.4%

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

   if 0.0 < (*.f64 V l)

   1. Initial program 73.9%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 88.7

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

   4. Applied egg-rr 88.7%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -1 \cdot 10^{-249}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{1}{\ell \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 17.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6 \cdot 10^{-233}:\\ \;\;\;\;\frac{c0}{\sqrt{-V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{1}{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= l 6e-233) (/ c0 (sqrt (- V))) (* c0 (sqrt (* A (/ 1.0 V))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= 6e-233) {
		tmp = c0 / sqrt(-V);
	} else {
		tmp = c0 * sqrt((A * (1.0 / V)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= 6d-233) then
        tmp = c0 / sqrt(-v)
    else
        tmp = c0 * sqrt((a * (1.0d0 / v)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= 6e-233) {
		tmp = c0 / Math.sqrt(-V);
	} else {
		tmp = c0 * Math.sqrt((A * (1.0 / V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if l <= 6e-233:
		tmp = c0 / math.sqrt(-V)
	else:
		tmp = c0 * math.sqrt((A * (1.0 / V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (l <= 6e-233)
		tmp = Float64(c0 / sqrt(Float64(-V)));
	else
		tmp = Float64(c0 * sqrt(Float64(A * Float64(1.0 / V))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (l <= 6e-233)
		tmp = c0 / sqrt(-V);
	else
		tmp = c0 * sqrt((A * (1.0 / V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[l, 6e-233], N[(c0 / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(A * N[(1.0 / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.99999999999999997e-233

   1. Initial program 71.3%

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

   3. Taylor expanded in c0 around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. rem-square-sqrt N/A

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

      4. unpow2 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

   5. Simplified 7.8%

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

   6. Taylor expanded in l around -inf

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

      1. lower-/.f64 6.2

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

   8. Simplified 6.2%

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

      1. frac-2neg N/A

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

      2. metadata-eval N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\mathsf{neg}\left(V\right)}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\mathsf{neg}\left(V\right)}}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{\color{blue}{\sqrt{\mathsf{neg}\left(V\right)}}} \]

      8. lower-neg.f64 6.2

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

   10. Applied egg-rr 6.2%

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

   if 5.99999999999999997e-233 < l

   1. Initial program 71.5%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 71.6

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

   4. Applied egg-rr 71.6%

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

   5. Taylor expanded in l around inf

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

      1. lower-/.f64 17.8

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

   7. Simplified 17.8%

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

4. Final simplification 11.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6 \cdot 10^{-233}:\\ \;\;\;\;\frac{c0}{\sqrt{-V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{1}{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 17.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -2 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -2e-310) (* c0 (sqrt (/ 1.0 l))) (/ c0 (sqrt (* l V)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -2e-310) {
		tmp = c0 * sqrt((1.0 / l));
	} else {
		tmp = c0 / sqrt((l * V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (a <= (-2d-310)) then
        tmp = c0 * sqrt((1.0d0 / l))
    else
        tmp = c0 / sqrt((l * v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -2e-310) {
		tmp = c0 * Math.sqrt((1.0 / l));
	} else {
		tmp = c0 / Math.sqrt((l * V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -2e-310:
		tmp = c0 * math.sqrt((1.0 / l))
	else:
		tmp = c0 / math.sqrt((l * V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -2e-310)
		tmp = Float64(c0 * sqrt(Float64(1.0 / l)));
	else
		tmp = Float64(c0 / sqrt(Float64(l * V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (A <= -2e-310)
		tmp = c0 * sqrt((1.0 / l));
	else
		tmp = c0 / sqrt((l * V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -2e-310], N[(c0 * N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -1.999999999999994e-310

   1. Initial program 74.3%

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

   3. Taylor expanded in c0 around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. rem-square-sqrt N/A

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

      4. unpow2 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

   5. Simplified 7.0%

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

   6. Taylor expanded in V around inf

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

      1. *-lft-identity N/A

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

      2. metadata-eval N/A

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

      3. rem-square-sqrt N/A

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

      4. distribute-lft-neg-out N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-neg-out N/A

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

      14. distribute-rgt-neg-out N/A

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

      15. distribute-rgt-neg-in N/A

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

      16. associate-*r* N/A

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

      17. rem-square-sqrt N/A

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

      18. neg-mul-1 N/A

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

      19. remove-double-neg N/A

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

      20. lower-sqrt.f64 N/A

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

   8. Simplified 7.9%

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

   if -1.999999999999994e-310 < A

   1. Initial program 68.0%

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

   3. Taylor expanded in c0 around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. rem-square-sqrt N/A

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

      4. unpow2 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

   5. Simplified 13.0%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 13.0

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

   7. Applied egg-rr 13.0%

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

4. Final simplification 10.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 14.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-311}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{-1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= l -2e-311) (* c0 (sqrt (/ -1.0 l))) (* c0 (sqrt (/ 1.0 l)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -2e-311) {
		tmp = c0 * sqrt((-1.0 / l));
	} else {
		tmp = c0 * sqrt((1.0 / l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-2d-311)) then
        tmp = c0 * sqrt(((-1.0d0) / l))
    else
        tmp = c0 * sqrt((1.0d0 / l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -2e-311) {
		tmp = c0 * Math.sqrt((-1.0 / l));
	} else {
		tmp = c0 * Math.sqrt((1.0 / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if l <= -2e-311:
		tmp = c0 * math.sqrt((-1.0 / l))
	else:
		tmp = c0 * math.sqrt((1.0 / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (l <= -2e-311)
		tmp = Float64(c0 * sqrt(Float64(-1.0 / l)));
	else
		tmp = Float64(c0 * sqrt(Float64(1.0 / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (l <= -2e-311)
		tmp = c0 * sqrt((-1.0 / l));
	else
		tmp = c0 * sqrt((1.0 / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[l, -2e-311], N[(c0 * N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.9999999999999e-311

   1. Initial program 70.8%

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

   3. Taylor expanded in c0 around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. rem-square-sqrt N/A

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

      4. unpow2 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

   5. Simplified 8.2%

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

      1. associate-/l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 8.2

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

   7. Applied egg-rr 8.2%

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

   8. Taylor expanded in V around -inf

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

      1. lower-/.f64 14.5

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

   10. Simplified 14.5%

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

   if -1.9999999999999e-311 < l

   1. Initial program 72.0%

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

   3. Taylor expanded in c0 around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. rem-square-sqrt N/A

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

      4. unpow2 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

   5. Simplified 11.7%

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

   6. Taylor expanded in V around inf

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

      1. *-lft-identity N/A

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

      2. metadata-eval N/A

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

      3. rem-square-sqrt N/A

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

      4. distribute-lft-neg-out N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-neg-out N/A

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

      14. distribute-rgt-neg-out N/A

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

      15. distribute-rgt-neg-in N/A

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

      16. associate-*r* N/A

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

      17. rem-square-sqrt N/A

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

      18. neg-mul-1 N/A

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

      19. remove-double-neg N/A

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

      20. lower-sqrt.f64 N/A

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

   8. Simplified 14.7%

      \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{1}{\ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 74.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.4%

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

3. Final simplification 71.4%

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

Alternative 17: 14.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if V < -9.999999999999969e-311

   1. Initial program 68.3%

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

   3. Taylor expanded in c0 around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. rem-square-sqrt N/A

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

      4. unpow2 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

   5. Simplified 8.6%

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

   6. Taylor expanded in l around -inf

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

      1. lower-/.f64 12.8

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

   8. Simplified 12.8%

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

      1. frac-2neg N/A

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

      2. metadata-eval N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\mathsf{neg}\left(V\right)}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\mathsf{neg}\left(V\right)}}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{\color{blue}{\sqrt{\mathsf{neg}\left(V\right)}}} \]

      8. lower-neg.f64 12.8

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

   10. Applied egg-rr 12.8%

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

   if -9.999999999999969e-311 < V

   1. Initial program 74.0%

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

   3. Taylor expanded in c0 around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. rem-square-sqrt N/A

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

      4. unpow2 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

   5. Simplified 10.8%

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

   6. Taylor expanded in l around -inf

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

      1. lower-/.f64 0.0

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

   8. Simplified 0.0%

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

      1. frac-2neg N/A

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

      2. metadata-eval N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\mathsf{neg}\left(V\right)}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\mathsf{neg}\left(V\right)}}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{\color{blue}{\sqrt{\mathsf{neg}\left(V\right)}}} \]

      8. lower-neg.f64 0.0

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

   10. Applied egg-rr 0.0%

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

       1. lift-neg.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\mathsf{neg}\left(V\right)}}} \]

       2. pow1/2 N/A

          \[\leadsto \frac{c0}{\color{blue}{{\left(\mathsf{neg}\left(V\right)\right)}^{\frac{1}{2}}}} \]

       3. sqr-pow N/A

          \[\leadsto \frac{c0}{\color{blue}{{\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}} \]

       4. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{c0}{{\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{{\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}} \]

       5. sqr-pow N/A

          \[\leadsto \frac{\frac{c0}{{\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{\color{blue}{{\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{\frac{1}{2}}{2}}{2}\right)} \cdot {\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{\frac{1}{2}}{2}}{2}\right)}}} \]

       6. pow-prod-down N/A

          \[\leadsto \frac{\frac{c0}{{\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{\color{blue}{{\left(\left(\mathsf{neg}\left(V\right)\right) \cdot \left(\mathsf{neg}\left(V\right)\right)\right)}^{\left(\frac{\frac{\frac{1}{2}}{2}}{2}\right)}}} \]

       7. lift-neg.f64 N/A

          \[\leadsto \frac{\frac{c0}{{\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{{\left(\color{blue}{\left(\mathsf{neg}\left(V\right)\right)} \cdot \left(\mathsf{neg}\left(V\right)\right)\right)}^{\left(\frac{\frac{\frac{1}{2}}{2}}{2}\right)}} \]

       8. lift-neg.f64 N/A

          \[\leadsto \frac{\frac{c0}{{\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{{\left(\left(\mathsf{neg}\left(V\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(V\right)\right)}\right)}^{\left(\frac{\frac{\frac{1}{2}}{2}}{2}\right)}} \]

       9. sqr-neg N/A

          \[\leadsto \frac{\frac{c0}{{\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{{\color{blue}{\left(V \cdot V\right)}}^{\left(\frac{\frac{\frac{1}{2}}{2}}{2}\right)}} \]

       10. pow-prod-down N/A

           \[\leadsto \frac{\frac{c0}{{\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{\color{blue}{{V}^{\left(\frac{\frac{\frac{1}{2}}{2}}{2}\right)} \cdot {V}^{\left(\frac{\frac{\frac{1}{2}}{2}}{2}\right)}}} \]

       11. sqr-pow N/A

           \[\leadsto \frac{\frac{c0}{{\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{\color{blue}{{V}^{\left(\frac{\frac{1}{2}}{2}\right)}}} \]

       12. sqr-pow N/A

           \[\leadsto \frac{\frac{c0}{\color{blue}{{\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{\frac{1}{2}}{2}}{2}\right)} \cdot {\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{\frac{1}{2}}{2}}{2}\right)}}}}{{V}^{\left(\frac{\frac{1}{2}}{2}\right)}} \]

       13. pow-prod-down N/A

           \[\leadsto \frac{\frac{c0}{\color{blue}{{\left(\left(\mathsf{neg}\left(V\right)\right) \cdot \left(\mathsf{neg}\left(V\right)\right)\right)}^{\left(\frac{\frac{\frac{1}{2}}{2}}{2}\right)}}}}{{V}^{\left(\frac{\frac{1}{2}}{2}\right)}} \]

       14. lift-neg.f64 N/A

           \[\leadsto \frac{\frac{c0}{{\left(\color{blue}{\left(\mathsf{neg}\left(V\right)\right)} \cdot \left(\mathsf{neg}\left(V\right)\right)\right)}^{\left(\frac{\frac{\frac{1}{2}}{2}}{2}\right)}}}{{V}^{\left(\frac{\frac{1}{2}}{2}\right)}} \]

       15. lift-neg.f64 N/A

           \[\leadsto \frac{\frac{c0}{{\left(\left(\mathsf{neg}\left(V\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(V\right)\right)}\right)}^{\left(\frac{\frac{\frac{1}{2}}{2}}{2}\right)}}}{{V}^{\left(\frac{\frac{1}{2}}{2}\right)}} \]

       16. sqr-neg N/A

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

       17. pow-prod-down N/A

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

       18. sqr-pow N/A

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

   12. Applied egg-rr 17.8%

       \[\leadsto \color{blue}{\frac{c0}{\sqrt{V}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 3.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.4%

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

3. Taylor expanded in c0 around 0

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

   1. remove-double-neg N/A

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

   2. mul-1-neg N/A

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

   3. rem-square-sqrt N/A

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

   4. unpow2 N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. distribute-rgt-neg-in N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. *-commutative N/A

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

   17. associate-*r* N/A

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

5. Simplified 9.8%

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

6. Taylor expanded in l around -inf

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

   1. lower-/.f64 5.9

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

8. Simplified 5.9%

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

   1. frac-2neg N/A

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

   2. metadata-eval N/A

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

   3. sqrt-div N/A

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

   4. metadata-eval N/A

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

   5. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\mathsf{neg}\left(V\right)}}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\mathsf{neg}\left(V\right)}}} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \frac{c0}{\color{blue}{\sqrt{\mathsf{neg}\left(V\right)}}} \]

   8. lower-neg.f64 5.9

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

10. Applied egg-rr 5.9%

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

    1. lift-neg.f64 N/A

       \[\leadsto \frac{c0}{\sqrt{\color{blue}{\mathsf{neg}\left(V\right)}}} \]

    2. pow1/2 N/A

       \[\leadsto \frac{c0}{\color{blue}{{\left(\mathsf{neg}\left(V\right)\right)}^{\frac{1}{2}}}} \]

    3. sqr-pow N/A

       \[\leadsto \frac{c0}{\color{blue}{{\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}} \]

    4. associate-/r* N/A

       \[\leadsto \color{blue}{\frac{\frac{c0}{{\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{{\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}} \]

    5. sqr-pow N/A

       \[\leadsto \frac{\frac{c0}{{\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{\color{blue}{{\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{\frac{1}{2}}{2}}{2}\right)} \cdot {\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{\frac{1}{2}}{2}}{2}\right)}}} \]

    6. pow-prod-down N/A

       \[\leadsto \frac{\frac{c0}{{\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{\color{blue}{{\left(\left(\mathsf{neg}\left(V\right)\right) \cdot \left(\mathsf{neg}\left(V\right)\right)\right)}^{\left(\frac{\frac{\frac{1}{2}}{2}}{2}\right)}}} \]

    7. lift-neg.f64 N/A

       \[\leadsto \frac{\frac{c0}{{\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{{\left(\color{blue}{\left(\mathsf{neg}\left(V\right)\right)} \cdot \left(\mathsf{neg}\left(V\right)\right)\right)}^{\left(\frac{\frac{\frac{1}{2}}{2}}{2}\right)}} \]

    8. lift-neg.f64 N/A

       \[\leadsto \frac{\frac{c0}{{\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{{\left(\left(\mathsf{neg}\left(V\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(V\right)\right)}\right)}^{\left(\frac{\frac{\frac{1}{2}}{2}}{2}\right)}} \]

    9. sqr-neg N/A

       \[\leadsto \frac{\frac{c0}{{\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{{\color{blue}{\left(V \cdot V\right)}}^{\left(\frac{\frac{\frac{1}{2}}{2}}{2}\right)}} \]

    10. pow-prod-down N/A

        \[\leadsto \frac{\frac{c0}{{\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{\color{blue}{{V}^{\left(\frac{\frac{\frac{1}{2}}{2}}{2}\right)} \cdot {V}^{\left(\frac{\frac{\frac{1}{2}}{2}}{2}\right)}}} \]

    11. sqr-pow N/A

        \[\leadsto \frac{\frac{c0}{{\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{\color{blue}{{V}^{\left(\frac{\frac{1}{2}}{2}\right)}}} \]

    12. sqr-pow N/A

        \[\leadsto \frac{\frac{c0}{\color{blue}{{\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{\frac{1}{2}}{2}}{2}\right)} \cdot {\left(\mathsf{neg}\left(V\right)\right)}^{\left(\frac{\frac{\frac{1}{2}}{2}}{2}\right)}}}}{{V}^{\left(\frac{\frac{1}{2}}{2}\right)}} \]

    13. pow-prod-down N/A

        \[\leadsto \frac{\frac{c0}{\color{blue}{{\left(\left(\mathsf{neg}\left(V\right)\right) \cdot \left(\mathsf{neg}\left(V\right)\right)\right)}^{\left(\frac{\frac{\frac{1}{2}}{2}}{2}\right)}}}}{{V}^{\left(\frac{\frac{1}{2}}{2}\right)}} \]

    14. lift-neg.f64 N/A

        \[\leadsto \frac{\frac{c0}{{\left(\color{blue}{\left(\mathsf{neg}\left(V\right)\right)} \cdot \left(\mathsf{neg}\left(V\right)\right)\right)}^{\left(\frac{\frac{\frac{1}{2}}{2}}{2}\right)}}}{{V}^{\left(\frac{\frac{1}{2}}{2}\right)}} \]

    15. lift-neg.f64 N/A

        \[\leadsto \frac{\frac{c0}{{\left(\left(\mathsf{neg}\left(V\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(V\right)\right)}\right)}^{\left(\frac{\frac{\frac{1}{2}}{2}}{2}\right)}}}{{V}^{\left(\frac{\frac{1}{2}}{2}\right)}} \]

    16. sqr-neg N/A

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

    17. pow-prod-down N/A

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

    18. sqr-pow N/A

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

12. Applied egg-rr 9.6%

    \[\leadsto \color{blue}{\frac{c0}{\sqrt{V}}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))