Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.6% → 90.7%

Time: 8.0s

Alternatives: 15

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: 73.6% 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, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{-A}\\ \mathbf{if}\;\ell \cdot V \leq -1 \cdot 10^{+307}:\\ \;\;\;\;\frac{t\_0 \cdot \frac{c0}{\sqrt{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;\ell \cdot V \leq -4 \cdot 10^{-202}:\\ \;\;\;\;c0 \cdot \frac{t\_0}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;\ell \cdot V \leq 4 \cdot 10^{+259}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{1}{\ell \cdot V}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

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))))
   (if (<= (* l V) -1e+307)
     (/ (* t_0 (/ c0 (sqrt l))) (sqrt (- V)))
     (if (<= (* l V) -4e-202)
       (* c0 (/ t_0 (sqrt (* V (- l)))))
       (if (<= (* l V) 0.0)
         (/ c0 (sqrt (* V (/ l A))))
         (if (<= (* l V) 4e+259)
           (* c0 (* (sqrt A) (sqrt (/ 1.0 (* l V)))))
           (* c0 (sqrt (/ (/ A 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);
	double tmp;
	if ((l * V) <= -1e+307) {
		tmp = (t_0 * (c0 / sqrt(l))) / sqrt(-V);
	} else if ((l * V) <= -4e-202) {
		tmp = c0 * (t_0 / sqrt((V * -l)));
	} else if ((l * V) <= 0.0) {
		tmp = c0 / sqrt((V * (l / A)));
	} else if ((l * V) <= 4e+259) {
		tmp = c0 * (sqrt(A) * sqrt((1.0 / (l * V))));
	} else {
		tmp = c0 * sqrt(((A / 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)
    if ((l * v) <= (-1d+307)) then
        tmp = (t_0 * (c0 / sqrt(l))) / sqrt(-v)
    else if ((l * v) <= (-4d-202)) then
        tmp = c0 * (t_0 / sqrt((v * -l)))
    else if ((l * v) <= 0.0d0) then
        tmp = c0 / sqrt((v * (l / a)))
    else if ((l * v) <= 4d+259) then
        tmp = c0 * (sqrt(a) * sqrt((1.0d0 / (l * v))))
    else
        tmp = c0 * sqrt(((a / 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);
	double tmp;
	if ((l * V) <= -1e+307) {
		tmp = (t_0 * (c0 / Math.sqrt(l))) / Math.sqrt(-V);
	} else if ((l * V) <= -4e-202) {
		tmp = c0 * (t_0 / Math.sqrt((V * -l)));
	} else if ((l * V) <= 0.0) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else if ((l * V) <= 4e+259) {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt((1.0 / (l * V))));
	} else {
		tmp = c0 * Math.sqrt(((A / 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)
	tmp = 0
	if (l * V) <= -1e+307:
		tmp = (t_0 * (c0 / math.sqrt(l))) / math.sqrt(-V)
	elif (l * V) <= -4e-202:
		tmp = c0 * (t_0 / math.sqrt((V * -l)))
	elif (l * V) <= 0.0:
		tmp = c0 / math.sqrt((V * (l / A)))
	elif (l * V) <= 4e+259:
		tmp = c0 * (math.sqrt(A) * math.sqrt((1.0 / (l * V))))
	else:
		tmp = c0 * math.sqrt(((A / 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))
	tmp = 0.0
	if (Float64(l * V) <= -1e+307)
		tmp = Float64(Float64(t_0 * Float64(c0 / sqrt(l))) / sqrt(Float64(-V)));
	elseif (Float64(l * V) <= -4e-202)
		tmp = Float64(c0 * Float64(t_0 / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(l * V) <= 0.0)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	elseif (Float64(l * V) <= 4e+259)
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(1.0 / Float64(l * V)))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / 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);
	tmp = 0.0;
	if ((l * V) <= -1e+307)
		tmp = (t_0 * (c0 / sqrt(l))) / sqrt(-V);
	elseif ((l * V) <= -4e-202)
		tmp = c0 * (t_0 / sqrt((V * -l)));
	elseif ((l * V) <= 0.0)
		tmp = c0 / sqrt((V * (l / A)));
	elseif ((l * V) <= 4e+259)
		tmp = c0 * (sqrt(A) * sqrt((1.0 / (l * V))));
	else
		tmp = c0 * sqrt(((A / 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[(-A)], $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], -1e+307], N[(N[(t$95$0 * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -4e-202], N[(c0 * N[(t$95$0 / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 4e+259], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -9.99999999999999986e306

   1. Initial program 42.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 61.9

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

   4. Applied rewrites 61.9%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. pow1/2 N/A

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

      6. associate-*r/ N/A

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

      7. pow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. sqrt-div N/A

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

      10. metadata-eval N/A

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

      11. div-inv N/A

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

      12. associate-/r* N/A

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

      13. lift-/.f64 N/A

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

      14. sqrt-div N/A

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

      15. lift-sqrt.f64 N/A

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

      16. associate-/l* N/A

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

      17. sqrt-prod N/A

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

      18. lift-*.f64 N/A

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

      19. lift-sqrt.f64 N/A

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

   6. Applied rewrites 99.6%

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

   if -9.99999999999999986e306 < (*.f64 V l) < -4.0000000000000001e-202

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 77.2

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

   4. Applied rewrites 77.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r/ N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-2neg N/A

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

      10. *-lft-identity N/A

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

      11. sqrt-div N/A

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

      12. pow1/2 N/A

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

      13. lower-/.f64 N/A

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

      14. pow1/2 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. lower-*.f64 N/A

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

      22. lower-neg.f64 99.5

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

   6. Applied rewrites 99.5%

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

   if -4.0000000000000001e-202 < (*.f64 V l) < -0.0

   1. Initial program 45.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 61.0

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

   4. Applied rewrites 61.0%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. sqrt-div N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. associate-/r/ N/A

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

      11. sqrt-div N/A

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

      12. lift-/.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lower-/.f64 45.5

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 45.5

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

   6. Applied rewrites 45.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 61.1

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

   8. Applied rewrites 61.1%

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

   if -0.0 < (*.f64 V l) < 4e259

   1. Initial program 91.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. sqrt-prod N/A

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

      6. pow1/2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 99.4

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

   4. Applied rewrites 99.4%

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

   if 4e259 < (*.f64 V l)

   1. Initial program 55.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 77.1

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

   4. Applied rewrites 77.1%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -1 \cdot 10^{+307}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot \frac{c0}{\sqrt{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;\ell \cdot V \leq -4 \cdot 10^{-202}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;\ell \cdot V \leq 4 \cdot 10^{+259}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{1}{\ell \cdot V}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;\ell \cdot V \leq -4 \cdot 10^{-202}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;\ell \cdot V \leq 4 \cdot 10^{+259}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{1}{\ell \cdot V}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

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 V) (- INFINITY))
   (/ c0 (* (sqrt l) (sqrt (/ V A))))
   (if (<= (* l V) -4e-202)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* l V) 0.0)
       (/ c0 (sqrt (* V (/ l A))))
       (if (<= (* l V) 4e+259)
         (* c0 (* (sqrt A) (sqrt (/ 1.0 (* l V)))))
         (* c0 (sqrt (/ (/ A V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	} else if ((l * V) <= -4e-202) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((l * V) <= 0.0) {
		tmp = c0 / sqrt((V * (l / A)));
	} else if ((l * V) <= 4e+259) {
		tmp = c0 * (sqrt(A) * sqrt((1.0 / (l * V))));
	} else {
		tmp = c0 * sqrt(((A / 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 ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 / (Math.sqrt(l) * Math.sqrt((V / A)));
	} else if ((l * V) <= -4e-202) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((l * V) <= 0.0) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else if ((l * V) <= 4e+259) {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt((1.0 / (l * V))));
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = c0 / (math.sqrt(l) * math.sqrt((V / A)))
	elif (l * V) <= -4e-202:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (l * V) <= 0.0:
		tmp = c0 / math.sqrt((V * (l / A)))
	elif (l * V) <= 4e+259:
		tmp = c0 * (math.sqrt(A) * math.sqrt((1.0 / (l * V))))
	else:
		tmp = c0 * math.sqrt(((A / 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(l * V) <= Float64(-Inf))
		tmp = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))));
	elseif (Float64(l * V) <= -4e-202)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(l * V) <= 0.0)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	elseif (Float64(l * V) <= 4e+259)
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(1.0 / Float64(l * V)))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / 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 ((l * V) <= -Inf)
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	elseif ((l * V) <= -4e-202)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((l * V) <= 0.0)
		tmp = c0 / sqrt((V * (l / A)));
	elseif ((l * V) <= 4e+259)
		tmp = c0 * (sqrt(A) * sqrt((1.0 / (l * V))));
	else
		tmp = c0 * sqrt(((A / 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[(l * V), $MachinePrecision], (-Infinity)], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -4e-202], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 4e+259], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 36.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 61.7

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

   4. Applied rewrites 61.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. sqrt-div N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. associate-/r/ N/A

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

      11. sqrt-div N/A

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

      12. lift-/.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lower-/.f64 36.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 36.3

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

   6. Applied rewrites 36.3%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. unpow-prod-down N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 82.2

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

   8. Applied rewrites 82.2%

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

   if -inf.0 < (*.f64 V l) < -4.0000000000000001e-202

   1. Initial program 87.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 79.1

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

   4. Applied rewrites 79.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r/ N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-2neg N/A

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

      10. *-lft-identity N/A

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

      11. sqrt-div N/A

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

      12. pow1/2 N/A

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

      13. lower-/.f64 N/A

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

      14. pow1/2 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

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

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

      21. lower-*.f64 N/A

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

      22. lower-neg.f64 99.4

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

   6. Applied rewrites 99.4%

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

   if -4.0000000000000001e-202 < (*.f64 V l) < -0.0

   1. Initial program 48.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 66.9

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

   4. Applied rewrites 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. sqrt-div N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. associate-/r/ N/A

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

      11. sqrt-div N/A

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

      12. lift-/.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lower-/.f64 49.4

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 49.4

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

   6. Applied rewrites 49.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 67.8

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

   8. Applied rewrites 67.8%

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

   if -0.0 < (*.f64 V l) < 4e259

   1. Initial program 86.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. sqrt-prod N/A

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

      6. pow1/2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 98.5

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

   4. Applied rewrites 98.5%

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

   if 4e259 < (*.f64 V l)

   1. Initial program 43.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 64.4

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

   4. Applied rewrites 64.4%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;\ell \cdot V \leq -4 \cdot 10^{-202}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;\ell \cdot V \leq 4 \cdot 10^{+259}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{1}{\ell \cdot V}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024229 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))