Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.3% → 90.7%

Time: 11.0s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.7% accurate, 0.3× speedup

Math

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

FPCore

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

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(-A) / sqrt(-V)) / sqrt(l));
	} else {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Fortran

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

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -2e-310) {
		tmp = c0 * ((Math.sqrt(-A) / Math.sqrt(-V)) / Math.sqrt(l));
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	}
	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(-A) / math.sqrt(-V)) / math.sqrt(l))
	else:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	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 * Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(-V))) / sqrt(l)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if A < -1.999999999999994e-310

   1. Initial program 75.6%

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

      1. associate-/r* 76.8%

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

      2. sqrt-div 47.0%

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

      3. associate-*r/ 44.0%

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

   4. Applied egg-rr 44.0%

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

      1. associate-/l* 47.0%

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

   6. Simplified 47.0%

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

      1. frac-2neg 47.0%

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

      2. sqrt-div 53.0%

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

   8. Applied egg-rr 53.0%

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

   if -1.999999999999994e-310 < A

   1. Initial program 75.3%

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

      1. pow1/2 75.3%

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

      2. div-inv 75.3%

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

      3. unpow-prod-down 88.1%

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

      4. pow1/2 88.1%

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

   4. Applied egg-rr 88.1%

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

      1. unpow1/2 88.1%

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

      2. associate-/r* 88.4%

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

   6. Simplified 88.4%

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

Alternative 2: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-272}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 10^{+185}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 4e-272)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 1e+185) t_0 (* c0 (pow (* l (/ V A)) -0.5))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 4e-272) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 1e+185) {
		tmp = t_0;
	} else {
		tmp = c0 * pow((l * (V / A)), -0.5);
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 4d-272) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 1d+185) then
        tmp = t_0
    else
        tmp = c0 * ((l * (v / a)) ** (-0.5d0))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 4e-272) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 1e+185) {
		tmp = t_0;
	} else {
		tmp = c0 * Math.pow((l * (V / A)), -0.5);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 4e-272:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 1e+185:
		tmp = t_0
	else:
		tmp = c0 * math.pow((l * (V / A)), -0.5)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 4e-272)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 1e+185)
		tmp = t_0;
	else
		tmp = Float64(c0 * (Float64(l * Float64(V / A)) ^ -0.5));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 4e-272)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 1e+185)
		tmp = t_0;
	else
		tmp = c0 * ((l * (V / A)) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-272], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+185], t$95$0, N[(c0 * N[Power[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 3.99999999999999972e-272

   1. Initial program 71.6%

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

      1. associate-/r* 75.8%

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

   3. Simplified 75.8%

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

   if 3.99999999999999972e-272 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 9.9999999999999998e184

   1. Initial program 99.7%

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

   if 9.9999999999999998e184 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 64.3%

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

      1. associate-/r* 67.6%

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

      2. div-inv 67.6%

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

   4. Applied egg-rr 67.6%

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

      1. associate-*l/ 66.5%

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

      2. div-inv 66.5%

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

      3. frac-2neg 66.5%

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

      4. sqrt-div 44.0%

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

   6. Applied egg-rr 44.0%

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

      1. distribute-neg-frac2 44.0%

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

   8. Simplified 44.0%

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

      1. clear-num 43.9%

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

      2. sqrt-undiv 68.0%

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

      3. add-sqr-sqrt 38.8%

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

      4. sqrt-unprod 34.9%

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

      5. sqr-neg 34.9%

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

      6. sqrt-unprod 0.1%

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

      7. add-sqr-sqrt 0.1%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 27.8%

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

      10. sqr-neg 27.8%

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

      11. sqrt-unprod 34.8%

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

      12. add-sqr-sqrt 68.0%

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

      13. un-div-inv 67.5%

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

      14. clear-num 67.5%

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

      15. associate-*r/ 65.9%

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

      16. associate-*l/ 68.5%

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

      17. *-commutative 68.5%

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

      18. pow1/2 68.5%

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

   10. Applied egg-rr 68.5%

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

Alternative 3: 75.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-272} \lor \neg \left(t\_0 \leq 10^{+121}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (or (<= t_0 4e-272) (not (<= t_0 1e+121)))
     (* c0 (sqrt (/ (/ A V) l)))
     t_0)))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 4e-272) || !(t_0 <= 1e+121)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if ((t_0 <= 4d-272) .or. (.not. (t_0 <= 1d+121))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 4e-272) || !(t_0 <= 1e+121)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if (t_0 <= 4e-272) or not (t_0 <= 1e+121):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if ((t_0 <= 4e-272) || !(t_0 <= 1e+121))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if ((t_0 <= 4e-272) || ~((t_0 <= 1e+121)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 4e-272], N[Not[LessEqual[t$95$0, 1e+121]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 3.99999999999999972e-272 or 1.00000000000000004e121 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 70.2%

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

      1. associate-/r* 74.1%

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

   3. Simplified 74.1%

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

   if 3.99999999999999972e-272 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.00000000000000004e121

   1. Initial program 99.7%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 4 \cdot 10^{-272} \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{+121}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-272}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+157}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 4e-272)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 2e+157) t_0 (/ c0 (sqrt (* l (/ V A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 4e-272) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 2e+157) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt((l * (V / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 4d-272) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 2d+157) then
        tmp = t_0
    else
        tmp = c0 / sqrt((l * (v / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 4e-272) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 2e+157) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt((l * (V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 4e-272:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 2e+157:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt((l * (V / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 4e-272)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 2e+157)
		tmp = t_0;
	else
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 4e-272)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 2e+157)
		tmp = t_0;
	else
		tmp = c0 / sqrt((l * (V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-272], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+157], t$95$0, N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 3.99999999999999972e-272

   1. Initial program 71.6%

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

      1. associate-/r* 75.8%

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

   3. Simplified 75.8%

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

   if 3.99999999999999972e-272 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.99999999999999997e157

   1. Initial program 99.7%

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

   if 1.99999999999999997e157 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 65.6%

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

      1. associate-/r* 68.8%

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

      2. div-inv 68.7%

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

   4. Applied egg-rr 68.7%

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

      1. un-div-inv 68.8%

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

      2. sqrt-undiv 47.0%

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

      3. clear-num 47.0%

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

      4. un-div-inv 47.0%

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

      5. sqrt-undiv 69.7%

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

   6. Applied egg-rr 69.7%

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

      1. associate-/r/ 66.9%

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

      2. associate-*l/ 67.0%

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

      3. associate-*r/ 69.6%

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

   8. Simplified 69.6%

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

Alternative 5: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-272}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+224}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 4e-272)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 2e+224) t_0 (/ c0 (sqrt (* V (/ l A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 4e-272) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 2e+224) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 4d-272) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 2d+224) then
        tmp = t_0
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 4e-272) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 2e+224) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 4e-272:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 2e+224:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 4e-272)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 2e+224)
		tmp = t_0;
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 4e-272)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 2e+224)
		tmp = t_0;
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-272], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+224], t$95$0, N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 3.99999999999999972e-272

   1. Initial program 71.6%

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

      1. associate-/r* 75.8%

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

   3. Simplified 75.8%

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

   if 3.99999999999999972e-272 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.99999999999999994e224

   1. Initial program 99.7%

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

   if 1.99999999999999994e224 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 58.5%

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

      1. associate-/r* 64.4%

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

      2. div-inv 64.4%

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

   4. Applied egg-rr 64.4%

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

      1. un-div-inv 64.4%

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

      2. sqrt-undiv 48.4%

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

      3. clear-num 48.4%

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

      4. un-div-inv 48.4%

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

      5. sqrt-undiv 65.4%

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

   6. Applied egg-rr 65.4%

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

      1. associate-/r/ 66.1%

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

      2. *-commutative 66.1%

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

   8. Simplified 66.1%

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

Alternative 6: 88.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-225}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-288}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (/ c0 (* (sqrt l) (sqrt (/ V A))))
   (if (<= (* V l) -2e-225)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 1e-288)
       (* c0 (/ 1.0 (sqrt (* V (/ l A)))))
       (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	} else if ((V * l) <= -2e-225) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 1e-288) {
		tmp = c0 * (1.0 / sqrt((V * (l / A))));
	} else {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 / (Math.sqrt(l) * Math.sqrt((V / A)));
	} else if ((V * l) <= -2e-225) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 1e-288) {
		tmp = c0 * (1.0 / Math.sqrt((V * (l / A))));
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 / (math.sqrt(l) * math.sqrt((V / A)))
	elif (V * l) <= -2e-225:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 1e-288:
		tmp = c0 * (1.0 / math.sqrt((V * (l / A))))
	else:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))));
	elseif (Float64(V * l) <= -2e-225)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 1e-288)
		tmp = Float64(c0 * Float64(1.0 / sqrt(Float64(V * Float64(l / A)))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	elseif ((V * l) <= -2e-225)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 1e-288)
		tmp = c0 * (1.0 / sqrt((V * (l / A))));
	else
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-225], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-288], N[(c0 * N[(1.0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 25.6%

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

      1. associate-/r* 73.3%

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

      2. div-inv 73.4%

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

   4. Applied egg-rr 73.4%

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

      1. un-div-inv 73.3%

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

      2. sqrt-undiv 44.5%

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

      3. clear-num 44.4%

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

      4. un-div-inv 44.7%

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

      5. sqrt-undiv 68.1%

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

   6. Applied egg-rr 68.1%

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

      1. associate-/r/ 68.2%

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

      2. associate-*l/ 25.6%

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

      3. associate-*r/ 68.2%

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

   8. Simplified 68.2%

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

      1. add-sqr-sqrt 27.7%

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

      2. sqrt-prod 5.6%

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

      3. times-frac 5.6%

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

   10. Applied egg-rr 5.6%

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

       1. associate-*l/ 5.6%

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

       2. associate-*r/ 5.6%

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

       3. rem-square-sqrt 44.8%

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

       4. associate-/r* 44.9%

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

   12. Simplified 44.9%

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

   if -inf.0 < (*.f64 V l) < -1.9999999999999999e-225

   1. Initial program 90.1%

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

      1. frac-2neg 90.1%

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

      2. sqrt-div 99.5%

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

      3. *-commutative 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

   4. Applied egg-rr 99.5%

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

   if -1.9999999999999999e-225 < (*.f64 V l) < 1.00000000000000006e-288

   1. Initial program 59.9%

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

      1. associate-/r* 76.5%

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

      2. clear-num 76.4%

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

      3. sqrt-div 76.5%

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

      4. metadata-eval 76.5%

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

      5. clear-num 76.5%

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

      6. associate-/r* 59.8%

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

      7. clear-num 59.8%

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

      8. associate-/l* 76.5%

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

   4. Applied egg-rr 76.5%

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

   if 1.00000000000000006e-288 < (*.f64 V l)

   1. Initial program 77.8%

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

      1. pow1/2 77.8%

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

      2. div-inv 77.8%

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

      3. unpow-prod-down 92.8%

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

      4. pow1/2 92.8%

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

   4. Applied egg-rr 92.8%

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

      1. unpow1/2 92.8%

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

      2. associate-/r* 93.2%

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

   6. Simplified 93.2%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-225}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-288}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+150}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-124}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ c0 (* (sqrt l) (sqrt (/ V A))))))
   (if (<= (* V l) -1e+150)
     t_0
     (if (<= (* V l) -2e-124)
       (* c0 (sqrt (/ A (* V l))))
       (if (<= (* V l) 0.0) t_0 (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 / (sqrt(l) * sqrt((V / A)));
	double tmp;
	if ((V * l) <= -1e+150) {
		tmp = t_0;
	} else if ((V * l) <= -2e-124) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 / (sqrt(l) * sqrt((v / a)))
    if ((v * l) <= (-1d+150)) then
        tmp = t_0
    else if ((v * l) <= (-2d-124)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else
        tmp = c0 * (sqrt(a) * sqrt(((1.0d0 / v) / l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 / (Math.sqrt(l) * Math.sqrt((V / A)));
	double tmp;
	if ((V * l) <= -1e+150) {
		tmp = t_0;
	} else if ((V * l) <= -2e-124) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 / (math.sqrt(l) * math.sqrt((V / A)))
	tmp = 0
	if (V * l) <= -1e+150:
		tmp = t_0
	elif (V * l) <= -2e-124:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 0.0:
		tmp = t_0
	else:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))))
	tmp = 0.0
	if (Float64(V * l) <= -1e+150)
		tmp = t_0;
	elseif (Float64(V * l) <= -2e-124)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 / (sqrt(l) * sqrt((V / A)));
	tmp = 0.0;
	if ((V * l) <= -1e+150)
		tmp = t_0;
	elseif ((V * l) <= -2e-124)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -1e+150], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -2e-124], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -9.99999999999999981e149 or -1.99999999999999987e-124 < (*.f64 V l) < -0.0

   1. Initial program 57.7%

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

      1. associate-/r* 73.2%

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

      2. div-inv 73.1%

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

   4. Applied egg-rr 73.1%

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

      1. un-div-inv 73.2%

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

      2. sqrt-undiv 46.5%

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

      3. clear-num 46.5%

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

      4. un-div-inv 46.5%

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

      5. sqrt-undiv 72.6%

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

   6. Applied egg-rr 72.6%

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

      1. associate-/r/ 73.6%

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

      2. associate-*l/ 56.7%

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

      3. associate-*r/ 72.6%

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

   8. Simplified 72.6%

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

      1. add-sqr-sqrt 35.1%

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

      2. sqrt-prod 20.8%

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

      3. times-frac 20.8%

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

   10. Applied egg-rr 20.8%

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

       1. associate-*l/ 20.8%

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

       2. associate-*r/ 20.8%

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

       3. rem-square-sqrt 44.8%

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

       4. associate-/r* 47.0%

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

   12. Simplified 47.0%

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

   if -9.99999999999999981e149 < (*.f64 V l) < -1.99999999999999987e-124

   1. Initial program 95.9%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 78.4%

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

      1. pow1/2 78.4%

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

      2. div-inv 78.4%

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

      3. unpow-prod-down 93.0%

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

      4. pow1/2 93.0%

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

   4. Applied egg-rr 93.0%

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

      1. unpow1/2 93.0%

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

      2. associate-/r* 93.3%

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

   6. Simplified 93.3%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+150}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-124}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+150}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-124}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{\left(V \cdot \ell\right)}^{-0.5} \cdot \left(c0 \cdot \sqrt{A}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ c0 (* (sqrt l) (sqrt (/ V A))))))
   (if (<= (* V l) -1e+150)
     t_0
     (if (<= (* V l) -2e-124)
       (* c0 (sqrt (/ A (* V l))))
       (if (<= (* V l) 0.0) t_0 (* (pow (* V l) -0.5) (* c0 (sqrt A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 / (sqrt(l) * sqrt((V / A)));
	double tmp;
	if ((V * l) <= -1e+150) {
		tmp = t_0;
	} else if ((V * l) <= -2e-124) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = pow((V * l), -0.5) * (c0 * sqrt(A));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 / (sqrt(l) * sqrt((v / a)))
    if ((v * l) <= (-1d+150)) then
        tmp = t_0
    else if ((v * l) <= (-2d-124)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else
        tmp = ((v * l) ** (-0.5d0)) * (c0 * sqrt(a))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 / (Math.sqrt(l) * Math.sqrt((V / A)));
	double tmp;
	if ((V * l) <= -1e+150) {
		tmp = t_0;
	} else if ((V * l) <= -2e-124) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = Math.pow((V * l), -0.5) * (c0 * Math.sqrt(A));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 / (math.sqrt(l) * math.sqrt((V / A)))
	tmp = 0
	if (V * l) <= -1e+150:
		tmp = t_0
	elif (V * l) <= -2e-124:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 0.0:
		tmp = t_0
	else:
		tmp = math.pow((V * l), -0.5) * (c0 * math.sqrt(A))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))))
	tmp = 0.0
	if (Float64(V * l) <= -1e+150)
		tmp = t_0;
	elseif (Float64(V * l) <= -2e-124)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = Float64((Float64(V * l) ^ -0.5) * Float64(c0 * sqrt(A)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 / (sqrt(l) * sqrt((V / A)));
	tmp = 0.0;
	if ((V * l) <= -1e+150)
		tmp = t_0;
	elseif ((V * l) <= -2e-124)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = ((V * l) ^ -0.5) * (c0 * sqrt(A));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -1e+150], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -2e-124], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, N[(N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision] * N[(c0 * N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -9.99999999999999981e149 or -1.99999999999999987e-124 < (*.f64 V l) < -0.0

   1. Initial program 57.7%

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

      1. associate-/r* 73.2%

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

      2. div-inv 73.1%

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

   4. Applied egg-rr 73.1%

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

      1. un-div-inv 73.2%

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

      2. sqrt-undiv 46.5%

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

      3. clear-num 46.5%

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

      4. un-div-inv 46.5%

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

      5. sqrt-undiv 72.6%

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

   6. Applied egg-rr 72.6%

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

      1. associate-/r/ 73.6%

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

      2. associate-*l/ 56.7%

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

      3. associate-*r/ 72.6%

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

   8. Simplified 72.6%

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

      1. add-sqr-sqrt 35.1%

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

      2. sqrt-prod 20.8%

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

      3. times-frac 20.8%

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

   10. Applied egg-rr 20.8%

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

       1. associate-*l/ 20.8%

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

       2. associate-*r/ 20.8%

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

       3. rem-square-sqrt 44.8%

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

       4. associate-/r* 47.0%

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

   12. Simplified 47.0%

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

   if -9.99999999999999981e149 < (*.f64 V l) < -1.99999999999999987e-124

   1. Initial program 95.9%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 78.4%

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

      1. *-commutative 78.4%

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

      2. sqrt-div 93.0%

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

      3. associate-*l/ 90.5%

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

   4. Applied egg-rr 90.5%

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

      1. clear-num 90.4%

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

      2. associate-/r/ 90.5%

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

      3. pow1/2 90.5%

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

      4. pow-flip 90.7%

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

      5. metadata-eval 90.7%

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

      6. *-commutative 90.7%

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

   6. Applied egg-rr 90.7%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+150}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-124}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(V \cdot \ell\right)}^{-0.5} \cdot \left(c0 \cdot \sqrt{A}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+150}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-124}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ c0 (* (sqrt l) (sqrt (/ V A))))))
   (if (<= (* V l) -1e+150)
     t_0
     (if (<= (* V l) -2e-124)
       (* c0 (sqrt (/ A (* V l))))
       (if (<= (* V l) 0.0) t_0 (/ (* c0 (sqrt A)) (sqrt (* V l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 / (sqrt(l) * sqrt((V / A)));
	double tmp;
	if ((V * l) <= -1e+150) {
		tmp = t_0;
	} else if ((V * l) <= -2e-124) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = (c0 * sqrt(A)) / sqrt((V * l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 / (sqrt(l) * sqrt((v / a)))
    if ((v * l) <= (-1d+150)) then
        tmp = t_0
    else if ((v * l) <= (-2d-124)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else
        tmp = (c0 * sqrt(a)) / sqrt((v * l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 / (Math.sqrt(l) * Math.sqrt((V / A)));
	double tmp;
	if ((V * l) <= -1e+150) {
		tmp = t_0;
	} else if ((V * l) <= -2e-124) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = (c0 * Math.sqrt(A)) / Math.sqrt((V * l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 / (math.sqrt(l) * math.sqrt((V / A)))
	tmp = 0
	if (V * l) <= -1e+150:
		tmp = t_0
	elif (V * l) <= -2e-124:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 0.0:
		tmp = t_0
	else:
		tmp = (c0 * math.sqrt(A)) / math.sqrt((V * l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))))
	tmp = 0.0
	if (Float64(V * l) <= -1e+150)
		tmp = t_0;
	elseif (Float64(V * l) <= -2e-124)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(c0 * sqrt(A)) / sqrt(Float64(V * l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 / (sqrt(l) * sqrt((V / A)));
	tmp = 0.0;
	if ((V * l) <= -1e+150)
		tmp = t_0;
	elseif ((V * l) <= -2e-124)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = (c0 * sqrt(A)) / sqrt((V * l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -1e+150], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -2e-124], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, N[(N[(c0 * N[Sqrt[A], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -9.99999999999999981e149 or -1.99999999999999987e-124 < (*.f64 V l) < -0.0

   1. Initial program 57.7%

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

      1. associate-/r* 73.2%

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

      2. div-inv 73.1%

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

   4. Applied egg-rr 73.1%

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

      1. un-div-inv 73.2%

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

      2. sqrt-undiv 46.5%

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

      3. clear-num 46.5%

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

      4. un-div-inv 46.5%

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

      5. sqrt-undiv 72.6%

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

   6. Applied egg-rr 72.6%

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

      1. associate-/r/ 73.6%

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

      2. associate-*l/ 56.7%

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

      3. associate-*r/ 72.6%

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

   8. Simplified 72.6%

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

      1. add-sqr-sqrt 35.1%

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

      2. sqrt-prod 20.8%

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

      3. times-frac 20.8%

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

   10. Applied egg-rr 20.8%

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

       1. associate-*l/ 20.8%

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

       2. associate-*r/ 20.8%

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

       3. rem-square-sqrt 44.8%

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

       4. associate-/r* 47.0%

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

   12. Simplified 47.0%

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

   if -9.99999999999999981e149 < (*.f64 V l) < -1.99999999999999987e-124

   1. Initial program 95.9%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 78.4%

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

      1. associate-/r* 73.4%

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

      2. div-inv 73.2%

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

   4. Applied egg-rr 73.2%

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

      1. associate-*l/ 75.8%

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

      2. div-inv 75.9%

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

      3. frac-2neg 75.9%

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

      4. sqrt-div 44.8%

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

   6. Applied egg-rr 44.8%

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

      1. distribute-neg-frac2 44.8%

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

   8. Simplified 44.8%

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

      1. clear-num 44.8%

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

      2. sqrt-undiv 76.4%

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

      3. add-sqr-sqrt 41.3%

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

      4. sqrt-unprod 35.3%

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

      5. sqr-neg 35.3%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 21.1%

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

      10. sqr-neg 21.1%

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

      11. sqrt-unprod 34.9%

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

      12. add-sqr-sqrt 76.4%

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

      13. un-div-inv 76.3%

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

      14. clear-num 76.4%

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

      15. associate-*r/ 78.8%

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

      16. associate-*l/ 73.6%

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

      17. *-commutative 73.6%

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

      18. pow1/2 73.6%

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

   10. Applied egg-rr 73.6%

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

   12. Applied egg-rr 90.5%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+150}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-124}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+150}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-124}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: 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 l) (sqrt (/ V A))))))
   (if (<= (* V l) -1e+150)
     t_0
     (if (<= (* V l) -2e-124)
       (* c0 (sqrt (/ A (* V l))))
       (if (<= (* V l) 0.0) t_0 (* (sqrt A) (/ c0 (sqrt (* V l)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 / (sqrt(l) * sqrt((V / A)));
	double tmp;
	if ((V * l) <= -1e+150) {
		tmp = t_0;
	} else if ((V * l) <= -2e-124) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 / (sqrt(l) * sqrt((v / a)))
    if ((v * l) <= (-1d+150)) then
        tmp = t_0
    else if ((v * l) <= (-2d-124)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else
        tmp = sqrt(a) * (c0 / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 / (Math.sqrt(l) * Math.sqrt((V / A)));
	double tmp;
	if ((V * l) <= -1e+150) {
		tmp = t_0;
	} else if ((V * l) <= -2e-124) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(A) * (c0 / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 / (math.sqrt(l) * math.sqrt((V / A)))
	tmp = 0
	if (V * l) <= -1e+150:
		tmp = t_0
	elif (V * l) <= -2e-124:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 0.0:
		tmp = t_0
	else:
		tmp = math.sqrt(A) * (c0 / math.sqrt((V * l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))))
	tmp = 0.0
	if (Float64(V * l) <= -1e+150)
		tmp = t_0;
	elseif (Float64(V * l) <= -2e-124)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = Float64(sqrt(A) * Float64(c0 / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 / (sqrt(l) * sqrt((V / A)));
	tmp = 0.0;
	if ((V * l) <= -1e+150)
		tmp = t_0;
	elseif ((V * l) <= -2e-124)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -1e+150], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -2e-124], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, N[(N[Sqrt[A], $MachinePrecision] * N[(c0 / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -9.99999999999999981e149 or -1.99999999999999987e-124 < (*.f64 V l) < -0.0

   1. Initial program 57.7%

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

      1. associate-/r* 73.2%

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

      2. div-inv 73.1%

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

   4. Applied egg-rr 73.1%

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

      1. un-div-inv 73.2%

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

      2. sqrt-undiv 46.5%

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

      3. clear-num 46.5%

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

      4. un-div-inv 46.5%

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

      5. sqrt-undiv 72.6%

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

   6. Applied egg-rr 72.6%

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

      1. associate-/r/ 73.6%

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

      2. associate-*l/ 56.7%

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

      3. associate-*r/ 72.6%

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

   8. Simplified 72.6%

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

      1. add-sqr-sqrt 35.1%

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

      2. sqrt-prod 20.8%

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

      3. times-frac 20.8%

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

   10. Applied egg-rr 20.8%

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

       1. associate-*l/ 20.8%

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

       2. associate-*r/ 20.8%

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

       3. rem-square-sqrt 44.8%

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

       4. associate-/r* 47.0%

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

   12. Simplified 47.0%

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

   if -9.99999999999999981e149 < (*.f64 V l) < -1.99999999999999987e-124

   1. Initial program 95.9%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 78.4%

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

      1. sqrt-div 93.0%

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

      2. associate-*r/ 90.5%

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

   4. Applied egg-rr 90.5%

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

      1. *-commutative 90.5%

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

      2. associate-/l* 90.5%

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

   6. Simplified 90.5%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+150}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-124}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+150}:\\ \;\;\;\;c0 \cdot \frac{t\_0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-124}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t\_0 \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (/ A V))))
   (if (<= (* V l) -1e+150)
     (* c0 (/ t_0 (sqrt l)))
     (if (<= (* V l) -2e-124)
       (* c0 (sqrt (/ A (* V l))))
       (if (<= (* V l) 0.0)
         (* t_0 (/ c0 (sqrt l)))
         (* (sqrt A) (/ c0 (sqrt (* V l)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / V));
	double tmp;
	if ((V * l) <= -1e+150) {
		tmp = c0 * (t_0 / sqrt(l));
	} else if ((V * l) <= -2e-124) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0 * (c0 / sqrt(l));
	} else {
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a / v))
    if ((v * l) <= (-1d+150)) then
        tmp = c0 * (t_0 / sqrt(l))
    else if ((v * l) <= (-2d-124)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0 * (c0 / sqrt(l))
    else
        tmp = sqrt(a) * (c0 / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / V));
	double tmp;
	if ((V * l) <= -1e+150) {
		tmp = c0 * (t_0 / Math.sqrt(l));
	} else if ((V * l) <= -2e-124) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0 * (c0 / Math.sqrt(l));
	} else {
		tmp = Math.sqrt(A) * (c0 / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / V))
	tmp = 0
	if (V * l) <= -1e+150:
		tmp = c0 * (t_0 / math.sqrt(l))
	elif (V * l) <= -2e-124:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 0.0:
		tmp = t_0 * (c0 / math.sqrt(l))
	else:
		tmp = math.sqrt(A) * (c0 / math.sqrt((V * l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(A / V))
	tmp = 0.0
	if (Float64(V * l) <= -1e+150)
		tmp = Float64(c0 * Float64(t_0 / sqrt(l)));
	elseif (Float64(V * l) <= -2e-124)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(t_0 * Float64(c0 / sqrt(l)));
	else
		tmp = Float64(sqrt(A) * Float64(c0 / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((A / V));
	tmp = 0.0;
	if ((V * l) <= -1e+150)
		tmp = c0 * (t_0 / sqrt(l));
	elseif ((V * l) <= -2e-124)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 0.0)
		tmp = t_0 * (c0 / sqrt(l));
	else
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -1e+150], N[(c0 * N[(t$95$0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-124], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(t$95$0 * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[A], $MachinePrecision] * N[(c0 / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -9.99999999999999981e149

   1. Initial program 51.3%

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

      1. associate-/r* 72.7%

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

      2. sqrt-div 44.0%

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

      3. associate-*r/ 44.1%

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

   4. Applied egg-rr 44.1%

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

      1. associate-/l* 44.0%

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

   6. Simplified 44.0%

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

   if -9.99999999999999981e149 < (*.f64 V l) < -1.99999999999999987e-124

   1. Initial program 95.9%

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

   if -1.99999999999999987e-124 < (*.f64 V l) < -0.0

   1. Initial program 62.1%

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

      1. associate-/r* 73.4%

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

      2. sqrt-div 48.3%

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

      3. associate-*r/ 44.6%

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

   4. Applied egg-rr 44.6%

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

      1. *-commutative 44.6%

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

      2. associate-/l* 46.3%

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

   6. Simplified 46.3%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 78.4%

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

      1. sqrt-div 93.0%

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

      2. associate-*r/ 90.5%

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

   4. Applied egg-rr 90.5%

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

      1. *-commutative 90.5%

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

      2. associate-/l* 90.5%

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

   6. Simplified 90.5%

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

Alternative 12: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+150}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-124}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: 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 (<= (* V l) -1e+150)
     t_0
     (if (<= (* V l) -2e-124)
       (* c0 (sqrt (/ A (* V l))))
       (if (<= (* V l) 0.0) t_0 (* (sqrt A) (/ c0 (sqrt (* V l)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (sqrt((A / V)) / sqrt(l));
	double tmp;
	if ((V * l) <= -1e+150) {
		tmp = t_0;
	} else if ((V * l) <= -2e-124) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * (sqrt((a / v)) / sqrt(l))
    if ((v * l) <= (-1d+150)) then
        tmp = t_0
    else if ((v * l) <= (-2d-124)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else
        tmp = sqrt(a) * (c0 / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert 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 ((V * l) <= -1e+150) {
		tmp = t_0;
	} else if ((V * l) <= -2e-124) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(A) * (c0 / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	tmp = 0
	if (V * l) <= -1e+150:
		tmp = t_0
	elif (V * l) <= -2e-124:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 0.0:
		tmp = t_0
	else:
		tmp = math.sqrt(A) * (c0 / math.sqrt((V * l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)))
	tmp = 0.0
	if (Float64(V * l) <= -1e+150)
		tmp = t_0;
	elseif (Float64(V * l) <= -2e-124)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = Float64(sqrt(A) * Float64(c0 / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

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 ((V * l) <= -1e+150)
		tmp = t_0;
	elseif ((V * l) <= -2e-124)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -1e+150], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -2e-124], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, N[(N[Sqrt[A], $MachinePrecision] * N[(c0 / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -9.99999999999999981e149 or -1.99999999999999987e-124 < (*.f64 V l) < -0.0

   1. Initial program 57.7%

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

      1. associate-/r* 73.2%

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

      2. sqrt-div 46.5%

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

      3. associate-*r/ 44.4%

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

   4. Applied egg-rr 44.4%

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

      1. associate-/l* 46.5%

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

   6. Simplified 46.5%

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

   if -9.99999999999999981e149 < (*.f64 V l) < -1.99999999999999987e-124

   1. Initial program 95.9%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 78.4%

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

      1. sqrt-div 93.0%

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

      2. associate-*r/ 90.5%

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

   4. Applied egg-rr 90.5%

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

      1. *-commutative 90.5%

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

      2. associate-/l* 90.5%

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

   6. Simplified 90.5%

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

Alternative 13: 80.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.05 \cdot 10^{-268}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

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 1.05e-268)
   (/ c0 (sqrt (/ (* V l) A)))
   (* 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 <= 1.05e-268) {
		tmp = c0 / sqrt(((V * l) / A));
	} 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 <= 1.05d-268) then
        tmp = c0 / sqrt(((v * l) / a))
    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 <= 1.05e-268) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} 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 <= 1.05e-268:
		tmp = c0 / math.sqrt(((V * l) / A))
	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 <= 1.05e-268)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	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 <= 1.05e-268)
		tmp = c0 / sqrt(((V * l) / A));
	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, 1.05e-268], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $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 < 1.04999999999999999e-268

   1. Initial program 78.8%

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

      1. associate-/r* 77.8%

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

      2. div-inv 77.8%

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

   4. Applied egg-rr 77.8%

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

      1. un-div-inv 77.8%

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

      2. sqrt-undiv 5.4%

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

      3. clear-num 5.4%

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

      4. un-div-inv 5.4%

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

      5. sqrt-undiv 77.0%

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

   6. Applied egg-rr 77.0%

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

      1. associate-/r/ 75.4%

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

      2. associate-*l/ 78.2%

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

      3. associate-*r/ 77.3%

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

   8. Simplified 77.3%

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

      1. associate-*r/ 78.2%

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

   10. Applied egg-rr 78.2%

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

   if 1.04999999999999999e-268 < l

   1. Initial program 71.8%

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

      1. associate-/r* 72.4%

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

      2. sqrt-div 82.5%

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

      3. associate-*r/ 79.2%

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

   4. Applied egg-rr 79.2%

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

      1. associate-/l* 82.5%

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

   6. Simplified 82.5%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.05 \cdot 10^{-268}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 73.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.4%

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

Reproduce

herbie shell --seed 2024129 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))