Henrywood and Agarwal, Equation (3)

Percentage Accurate: 74.4% → 90.9%

Time: 7.7s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.4% 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: 90.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{V \cdot \frac{\ell}{A}}\\ \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{-315}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{-V} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{1}{t\_0}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+305}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (* V (/ l A)))))
   (if (<= (* V l) -1e-315)
     (/ (* (sqrt (- A)) c0) (* (sqrt (- V)) (sqrt l)))
     (if (<= (* V l) 0.0)
       (* c0 (/ 1.0 t_0))
       (if (<= (* V l) 1e+305)
         (* c0 (/ (sqrt A) (sqrt (* V l))))
         (/ c0 t_0))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((V * (l / A)));
	double tmp;
	if ((V * l) <= -1e-315) {
		tmp = (sqrt(-A) * c0) / (sqrt(-V) * sqrt(l));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (1.0 / t_0);
	} else if ((V * l) <= 1e+305) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 / 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 = sqrt((v * (l / a)))
    if ((v * l) <= (-1d-315)) then
        tmp = (sqrt(-a) * c0) / (sqrt(-v) * sqrt(l))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 * (1.0d0 / t_0)
    else if ((v * l) <= 1d+305) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 / 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 = Math.sqrt((V * (l / A)));
	double tmp;
	if ((V * l) <= -1e-315) {
		tmp = (Math.sqrt(-A) * c0) / (Math.sqrt(-V) * Math.sqrt(l));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (1.0 / t_0);
	} else if ((V * l) <= 1e+305) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 / t_0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((V * (l / A)))
	tmp = 0
	if (V * l) <= -1e-315:
		tmp = (math.sqrt(-A) * c0) / (math.sqrt(-V) * math.sqrt(l))
	elif (V * l) <= 0.0:
		tmp = c0 * (1.0 / t_0)
	elif (V * l) <= 1e+305:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 / t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(V * Float64(l / A)))
	tmp = 0.0
	if (Float64(V * l) <= -1e-315)
		tmp = Float64(Float64(sqrt(Float64(-A)) * c0) / Float64(sqrt(Float64(-V)) * sqrt(l)));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * Float64(1.0 / t_0));
	elseif (Float64(V * l) <= 1e+305)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 / t_0);
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((V * (l / A)));
	tmp = 0.0;
	if ((V * l) <= -1e-315)
		tmp = (sqrt(-A) * c0) / (sqrt(-V) * sqrt(l));
	elseif ((V * l) <= 0.0)
		tmp = c0 * (1.0 / t_0);
	elseif ((V * l) <= 1e+305)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 / 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[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -1e-315], N[(N[(N[Sqrt[(-A)], $MachinePrecision] * c0), $MachinePrecision] / N[(N[Sqrt[(-V)], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+305], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -9.999999985e-316

   1. Initial program 79.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. 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 78.2

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

   4. Applied rewrites 78.2%

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

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. times-frac N/A

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

      12. *-commutative N/A

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

      13. frac-times N/A

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

      14. lift-sqrt.f64 N/A

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

      15. sqrt-div N/A

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

      16. frac-2neg N/A

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

      17. sqrt-div N/A

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

      18. frac-2neg N/A

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

      19. frac-times N/A

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

      20. lower-/.f64 N/A

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

   6. Applied rewrites 96.6%

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

   if -9.999999985e-316 < (*.f64 V l) < -0.0

   1. Initial program 34.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 64.4

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

   4. Applied rewrites 64.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lift-/.f64 N/A

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

      9. sqrt-unprod N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. sqrt-unprod N/A

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

      16. lift-/.f64 N/A

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

      17. associate-/r/ N/A

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

      18. lift-/.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-/r/ N/A

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

      22. associate-*l/ N/A

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

      23. lift-*.f64 N/A

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

      24. lower-/.f64 34.5

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

      25. lift-*.f64 N/A

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

      26. *-commutative N/A

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

      27. lower-*.f64 34.5

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

   6. Applied rewrites 34.5%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. inv-pow N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow-prod-down N/A

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

      9. remove-double-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. lift-neg.f64 N/A

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

      13. sqr-neg N/A

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

      14. pow-prod-down N/A

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

      15. pow-prod-up N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. inv-pow N/A

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

      20. clear-num N/A

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

      21. clear-num N/A

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

      22. div-inv N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*l/ N/A

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

      25. lift-/.f64 N/A

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

   8. Applied rewrites 65.8%

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

   if -0.0 < (*.f64 V l) < 9.9999999999999994e304

   1. Initial program 80.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 98.3

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

   4. Applied rewrites 98.3%

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

   if 9.9999999999999994e304 < (*.f64 V l)

   1. Initial program 31.9%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 74.9

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

   4. Applied rewrites 74.9%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{\ell}}{V}}} \]
   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. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lift-/.f64 N/A

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

      10. sqrt-unprod N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. div-inv N/A

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

      14. lower-/.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. sqrt-unprod N/A

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

      18. lift-/.f64 N/A

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

      19. associate-/r/ N/A

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

      20. lift-/.f64 N/A

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

      21. lower-sqrt.f64 N/A

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

      22. lift-/.f64 N/A

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

      23. associate-/r/ N/A

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

      24. associate-*l/ N/A

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

   6. Applied rewrites 31.9%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{\ell \cdot V}{A}}}} \]
   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 75.0

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

   8. Applied rewrites 75.0%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{-315}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{-V} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+305}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+272}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{1}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

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 0.0)
     (* c0 (sqrt (/ (/ A l) V)))
     (if (<= t_0 2e+272)
       (* c0 (sqrt (* A (/ 1.0 (* V l)))))
       (/ 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 <= 0.0) {
		tmp = c0 * sqrt(((A / l) / V));
	} else if (t_0 <= 2e+272) {
		tmp = c0 * sqrt((A * (1.0 / (V * l))));
	} 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 <= 0.0d0) then
        tmp = c0 * sqrt(((a / l) / v))
    else if (t_0 <= 2d+272) then
        tmp = c0 * sqrt((a * (1.0d0 / (v * l))))
    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 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 2e+272) {
		tmp = c0 * Math.sqrt((A * (1.0 / (V * l))));
	} 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 <= 0.0:
		tmp = c0 * math.sqrt(((A / l) / V))
	elif t_0 <= 2e+272:
		tmp = c0 * math.sqrt((A * (1.0 / (V * l))))
	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 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 2e+272)
		tmp = Float64(c0 * sqrt(Float64(A * Float64(1.0 / Float64(V * l)))));
	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 <= 0.0)
		tmp = c0 * sqrt(((A / l) / V));
	elseif (t_0 <= 2e+272)
		tmp = c0 * sqrt((A * (1.0 / (V * l))));
	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, 0.0], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+272], N[(c0 * N[Sqrt[N[(A * N[(1.0 / N[(V * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 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)))) < 0.0

   1. Initial program 69.4%

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

      1. lift-/.f64 N/A

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

      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 71.4

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

   4. Applied rewrites 71.4%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.0000000000000001e272

   1. Initial program 98.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 98.3

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

   4. Applied rewrites 98.3%

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

   if 2.0000000000000001e272 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 48.0%

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

      1. lift-/.f64 N/A

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

      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 57.3

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

   4. Applied rewrites 57.3%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{\ell}}{V}}} \]
   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. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lift-/.f64 N/A

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

      10. sqrt-unprod N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. div-inv N/A

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

      14. lower-/.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. sqrt-unprod N/A

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

      18. lift-/.f64 N/A

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

      19. associate-/r/ N/A

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

      20. lift-/.f64 N/A

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

      21. lower-sqrt.f64 N/A

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

      22. lift-/.f64 N/A

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

      23. associate-/r/ N/A

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

      24. associate-*l/ N/A

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

   6. Applied rewrites 50.0%

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

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 59.6

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

   8. Applied rewrites 59.6%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{+272}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{1}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024222 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))