Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.3% → 89.5%

Time: 8.2s

Alternatives: 18

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.3% 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: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{-V}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}} \cdot c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-243}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{-1}{V}}{\ell}} \cdot \sqrt{-A}\right) \cdot c0\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{-\ell}{A}} \cdot t\_0}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+282}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{-\ell} \cdot t\_0}\\ \end{array} \end{array} \]

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))))
   (if (<= (* V l) (- INFINITY))
     (/ (* (sqrt (/ A V)) c0) (sqrt l))
     (if (<= (* V l) -2e-243)
       (* (* (sqrt (/ (/ -1.0 V) l)) (sqrt (- A))) c0)
       (if (<= (* V l) 0.0)
         (/ c0 (* (sqrt (/ (- l) A)) t_0))
         (if (<= (* V l) 1e+282)
           (* (/ (sqrt A) (sqrt (* V l))) c0)
           (/ (* (sqrt A) c0) (* (sqrt (- l)) 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);
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = (sqrt((A / V)) * c0) / sqrt(l);
	} else if ((V * l) <= -2e-243) {
		tmp = (sqrt(((-1.0 / V) / l)) * sqrt(-A)) * c0;
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (sqrt((-l / A)) * t_0);
	} else if ((V * l) <= 1e+282) {
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	} else {
		tmp = (sqrt(A) * c0) / (sqrt(-l) * t_0);
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt(-V);
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = (Math.sqrt((A / V)) * c0) / Math.sqrt(l);
	} else if ((V * l) <= -2e-243) {
		tmp = (Math.sqrt(((-1.0 / V) / l)) * Math.sqrt(-A)) * c0;
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (Math.sqrt((-l / A)) * t_0);
	} else if ((V * l) <= 1e+282) {
		tmp = (Math.sqrt(A) / Math.sqrt((V * l))) * c0;
	} else {
		tmp = (Math.sqrt(A) * c0) / (Math.sqrt(-l) * 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)
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = (math.sqrt((A / V)) * c0) / math.sqrt(l)
	elif (V * l) <= -2e-243:
		tmp = (math.sqrt(((-1.0 / V) / l)) * math.sqrt(-A)) * c0
	elif (V * l) <= 0.0:
		tmp = c0 / (math.sqrt((-l / A)) * t_0)
	elif (V * l) <= 1e+282:
		tmp = (math.sqrt(A) / math.sqrt((V * l))) * c0
	else:
		tmp = (math.sqrt(A) * c0) / (math.sqrt(-l) * 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))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(A / V)) * c0) / sqrt(l));
	elseif (Float64(V * l) <= -2e-243)
		tmp = Float64(Float64(sqrt(Float64(Float64(-1.0 / V) / l)) * sqrt(Float64(-A))) * c0);
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 / Float64(sqrt(Float64(Float64(-l) / A)) * t_0));
	elseif (Float64(V * l) <= 1e+282)
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(V * l))) * c0);
	else
		tmp = Float64(Float64(sqrt(A) * c0) / Float64(sqrt(Float64(-l)) * 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);
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = (sqrt((A / V)) * c0) / sqrt(l);
	elseif ((V * l) <= -2e-243)
		tmp = (sqrt(((-1.0 / V) / l)) * sqrt(-A)) * c0;
	elseif ((V * l) <= 0.0)
		tmp = c0 / (sqrt((-l / A)) * t_0);
	elseif ((V * l) <= 1e+282)
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	else
		tmp = (sqrt(A) * c0) / (sqrt(-l) * 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[(-V)], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-243], N[(N[(N[Sqrt[N[(N[(-1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[(N[Sqrt[N[((-l) / A), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+282], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], N[(N[(N[Sqrt[A], $MachinePrecision] * c0), $MachinePrecision] / N[(N[Sqrt[(-l)], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 35.0%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. sqrt-div N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 64.4

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

   4. Applied rewrites 64.4%

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

   if -inf.0 < (*.f64 V l) < -1.99999999999999999e-243

   1. Initial program 86.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 79.0

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

   4. Applied rewrites 79.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. associate-/l* N/A

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

      7. sqrt-prod N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. distribute-neg-frac2 N/A

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

      16. remove-double-neg N/A

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

      17. lower-/.f64 99.4

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

   6. Applied rewrites 99.4%

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

   if -1.99999999999999999e-243 < (*.f64 V l) < 0.0

   1. Initial program 52.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 52.4

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 52.4

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

   4. Applied rewrites 52.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. associate-/r/ N/A

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

      10. sqrt-prod N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. frac-2neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-neg.f64 43.8

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

   6. Applied rewrites 43.8%

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

   if 0.0 < (*.f64 V l) < 1.00000000000000003e282

   1. Initial program 85.8%

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 98.3

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

   4. Applied rewrites 98.3%

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

   if 1.00000000000000003e282 < (*.f64 V l)

   1. Initial program 36.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 42.4

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 42.4

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

   4. Applied rewrites 42.4%

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

      1. lift-sqrt.f64 N/A

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

      2. remove-double-neg N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. distribute-lft-neg-in N/A

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

      6. sqrt-prod N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-neg.f64 70.2

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

   6. Applied rewrites 70.2%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}} \cdot c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-243}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{-1}{V}}{\ell}} \cdot \sqrt{-A}\right) \cdot c0\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{-\ell}{A}} \cdot \sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+282}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{-\ell} \cdot \sqrt{-V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V \cdot \ell}} \cdot c0\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+229}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* (sqrt (/ A (* V l))) c0)))
   (if (<= t_0 4e-307)
     (* (sqrt (/ (/ A l) V)) c0)
     (if (<= t_0 5e+229) t_0 (/ c0 (sqrt (* (/ l A) V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / (V * l))) * c0;
	double tmp;
	if (t_0 <= 4e-307) {
		tmp = sqrt(((A / l) / V)) * c0;
	} else if (t_0 <= 5e+229) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt(((l / A) * V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a / (v * l))) * c0
    if (t_0 <= 4d-307) then
        tmp = sqrt(((a / l) / v)) * c0
    else if (t_0 <= 5d+229) then
        tmp = t_0
    else
        tmp = c0 / sqrt(((l / a) * v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / (V * l))) * c0;
	double tmp;
	if (t_0 <= 4e-307) {
		tmp = Math.sqrt(((A / l) / V)) * c0;
	} else if (t_0 <= 5e+229) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt(((l / A) * V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / (V * l))) * c0
	tmp = 0
	if t_0 <= 4e-307:
		tmp = math.sqrt(((A / l) / V)) * c0
	elif t_0 <= 5e+229:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt(((l / A) * V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(sqrt(Float64(A / Float64(V * l))) * c0)
	tmp = 0.0
	if (t_0 <= 4e-307)
		tmp = Float64(sqrt(Float64(Float64(A / l) / V)) * c0);
	elseif (t_0 <= 5e+229)
		tmp = t_0;
	else
		tmp = Float64(c0 / sqrt(Float64(Float64(l / A) * V)));
	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 * l))) * c0;
	tmp = 0.0;
	if (t_0 <= 4e-307)
		tmp = sqrt(((A / l) / V)) * c0;
	elseif (t_0 <= 5e+229)
		tmp = t_0;
	else
		tmp = c0 / sqrt(((l / A) * V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-307], N[(N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[t$95$0, 5e+229], t$95$0, N[(c0 / N[Sqrt[N[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 3.99999999999999964e-307

   1. Initial program 71.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 67.4

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

   4. Applied rewrites 67.4%

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

   if 3.99999999999999964e-307 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 5.0000000000000005e229

   1. Initial program 98.9%

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

   if 5.0000000000000005e229 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 56.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 56.2

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 56.2

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

   4. Applied rewrites 56.2%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{\ell \cdot V}{A}}}} \]
   5. 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. lower-*.f64 N/A

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

      5. lower-/.f64 65.3

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

   6. Applied rewrites 65.3%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\frac{A}{V \cdot \ell}} \cdot c0 \leq 4 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;\sqrt{\frac{A}{V \cdot \ell}} \cdot c0 \leq 5 \cdot 10^{+229}:\\ \;\;\;\;\sqrt{\frac{A}{V \cdot \ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V \cdot \ell}} \cdot c0\\ t_1 := \sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-307}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+229}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* (sqrt (/ A (* V l))) c0)) (t_1 (* (sqrt (/ (/ A l) V)) c0)))
   (if (<= t_0 4e-307) t_1 (if (<= t_0 5e+229) t_0 t_1))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / (V * l))) * c0;
	double t_1 = sqrt(((A / l) / V)) * c0;
	double tmp;
	if (t_0 <= 4e-307) {
		tmp = t_1;
	} else if (t_0 <= 5e+229) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((a / (v * l))) * c0
    t_1 = sqrt(((a / l) / v)) * c0
    if (t_0 <= 4d-307) then
        tmp = t_1
    else if (t_0 <= 5d+229) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / (V * l))) * c0;
	double t_1 = Math.sqrt(((A / l) / V)) * c0;
	double tmp;
	if (t_0 <= 4e-307) {
		tmp = t_1;
	} else if (t_0 <= 5e+229) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / (V * l))) * c0
	t_1 = math.sqrt(((A / l) / V)) * c0
	tmp = 0
	if t_0 <= 4e-307:
		tmp = t_1
	elif t_0 <= 5e+229:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(sqrt(Float64(A / Float64(V * l))) * c0)
	t_1 = Float64(sqrt(Float64(Float64(A / l) / V)) * c0)
	tmp = 0.0
	if (t_0 <= 4e-307)
		tmp = t_1;
	elseif (t_0 <= 5e+229)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((A / (V * l))) * c0;
	t_1 = sqrt(((A / l) / V)) * c0;
	tmp = 0.0;
	if (t_0 <= 4e-307)
		tmp = t_1;
	elseif (t_0 <= 5e+229)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-307], t$95$1, If[LessEqual[t$95$0, 5e+229], t$95$0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 3.99999999999999964e-307 or 5.0000000000000005e229 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 68.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 67.0

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

   4. Applied rewrites 67.0%

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

   if 3.99999999999999964e-307 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 5.0000000000000005e229

   1. Initial program 98.9%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\frac{A}{V \cdot \ell}} \cdot c0 \leq 4 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;\sqrt{\frac{A}{V \cdot \ell}} \cdot c0 \leq 5 \cdot 10^{+229}:\\ \;\;\;\;\sqrt{\frac{A}{V \cdot \ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}} \cdot c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-243}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{-1}{V}}{\ell}} \cdot \sqrt{-A}\right) \cdot c0\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{-\ell}{A}} \cdot \sqrt{-V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \end{array} \end{array} \]

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))
   (/ (* (sqrt (/ A V)) c0) (sqrt l))
   (if (<= (* V l) -2e-243)
     (* (* (sqrt (/ (/ -1.0 V) l)) (sqrt (- A))) c0)
     (if (<= (* V l) 0.0)
       (/ c0 (* (sqrt (/ (- l) A)) (sqrt (- V))))
       (* (/ (sqrt A) (sqrt (* V l))) c0)))))

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 = (sqrt((A / V)) * c0) / sqrt(l);
	} else if ((V * l) <= -2e-243) {
		tmp = (sqrt(((-1.0 / V) / l)) * sqrt(-A)) * c0;
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (sqrt((-l / A)) * sqrt(-V));
	} else {
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	}
	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 = (Math.sqrt((A / V)) * c0) / Math.sqrt(l);
	} else if ((V * l) <= -2e-243) {
		tmp = (Math.sqrt(((-1.0 / V) / l)) * Math.sqrt(-A)) * c0;
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (Math.sqrt((-l / A)) * Math.sqrt(-V));
	} else {
		tmp = (Math.sqrt(A) / Math.sqrt((V * l))) * c0;
	}
	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 = (math.sqrt((A / V)) * c0) / math.sqrt(l)
	elif (V * l) <= -2e-243:
		tmp = (math.sqrt(((-1.0 / V) / l)) * math.sqrt(-A)) * c0
	elif (V * l) <= 0.0:
		tmp = c0 / (math.sqrt((-l / A)) * math.sqrt(-V))
	else:
		tmp = (math.sqrt(A) / math.sqrt((V * l))) * c0
	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(Float64(sqrt(Float64(A / V)) * c0) / sqrt(l));
	elseif (Float64(V * l) <= -2e-243)
		tmp = Float64(Float64(sqrt(Float64(Float64(-1.0 / V) / l)) * sqrt(Float64(-A))) * c0);
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 / Float64(sqrt(Float64(Float64(-l) / A)) * sqrt(Float64(-V))));
	else
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(V * l))) * c0);
	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 = (sqrt((A / V)) * c0) / sqrt(l);
	elseif ((V * l) <= -2e-243)
		tmp = (sqrt(((-1.0 / V) / l)) * sqrt(-A)) * c0;
	elseif ((V * l) <= 0.0)
		tmp = c0 / (sqrt((-l / A)) * sqrt(-V));
	else
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	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[(N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-243], N[(N[(N[Sqrt[N[(N[(-1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[(N[Sqrt[N[((-l) / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 35.0%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. sqrt-div N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 64.4

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

   4. Applied rewrites 64.4%

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

   if -inf.0 < (*.f64 V l) < -1.99999999999999999e-243

   1. Initial program 86.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 79.0

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

   4. Applied rewrites 79.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. associate-/l* N/A

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

      7. sqrt-prod N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. distribute-neg-frac2 N/A

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

      16. remove-double-neg N/A

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

      17. lower-/.f64 99.4

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

   6. Applied rewrites 99.4%

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

   if -1.99999999999999999e-243 < (*.f64 V l) < 0.0

   1. Initial program 52.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 52.4

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 52.4

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

   4. Applied rewrites 52.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. associate-/r/ N/A

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

      10. sqrt-prod N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. frac-2neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-neg.f64 43.8

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

   6. Applied rewrites 43.8%

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

   if 0.0 < (*.f64 V l)

   1. Initial program 78.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 90.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 90.0

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

   4. Applied rewrites 90.0%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}} \cdot c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-243}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{-1}{V}}{\ell}} \cdot \sqrt{-A}\right) \cdot c0\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{-\ell}{A}} \cdot \sqrt{-V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\sqrt{\frac{-\ell}{A}} \cdot \sqrt{-V}}\\ \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+219}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-243}:\\ \;\;\;\;\frac{c0}{\sqrt{\left(-\ell\right) \cdot V}} \cdot \sqrt{-A}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \end{array} \end{array} \]

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) A)) (sqrt (- V))))))
   (if (<= (* V l) -1e+219)
     t_0
     (if (<= (* V l) -2e-243)
       (* (/ c0 (sqrt (* (- l) V))) (sqrt (- A)))
       (if (<= (* V l) 0.0) t_0 (* (/ (sqrt A) (sqrt (* V l))) c0))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 / (sqrt((-l / A)) * sqrt(-V));
	double tmp;
	if ((V * l) <= -1e+219) {
		tmp = t_0;
	} else if ((V * l) <= -2e-243) {
		tmp = (c0 / sqrt((-l * V))) * sqrt(-A);
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	}
	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 / a)) * sqrt(-v))
    if ((v * l) <= (-1d+219)) then
        tmp = t_0
    else if ((v * l) <= (-2d-243)) then
        tmp = (c0 / sqrt((-l * v))) * sqrt(-a)
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else
        tmp = (sqrt(a) / sqrt((v * l))) * c0
    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 / A)) * Math.sqrt(-V));
	double tmp;
	if ((V * l) <= -1e+219) {
		tmp = t_0;
	} else if ((V * l) <= -2e-243) {
		tmp = (c0 / Math.sqrt((-l * V))) * Math.sqrt(-A);
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = (Math.sqrt(A) / Math.sqrt((V * l))) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 / (math.sqrt((-l / A)) * math.sqrt(-V))
	tmp = 0
	if (V * l) <= -1e+219:
		tmp = t_0
	elif (V * l) <= -2e-243:
		tmp = (c0 / math.sqrt((-l * V))) * math.sqrt(-A)
	elif (V * l) <= 0.0:
		tmp = t_0
	else:
		tmp = (math.sqrt(A) / math.sqrt((V * l))) * c0
	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(Float64(-l) / A)) * sqrt(Float64(-V))))
	tmp = 0.0
	if (Float64(V * l) <= -1e+219)
		tmp = t_0;
	elseif (Float64(V * l) <= -2e-243)
		tmp = Float64(Float64(c0 / sqrt(Float64(Float64(-l) * V))) * sqrt(Float64(-A)));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(V * l))) * c0);
	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 / A)) * sqrt(-V));
	tmp = 0.0;
	if ((V * l) <= -1e+219)
		tmp = t_0;
	elseif ((V * l) <= -2e-243)
		tmp = (c0 / sqrt((-l * V))) * sqrt(-A);
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	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[((-l) / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -1e+219], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -2e-243], N[(N[(c0 / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -9.99999999999999965e218 or -1.99999999999999999e-243 < (*.f64 V l) < 0.0

   1. Initial program 53.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 53.1

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 53.1

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

   4. Applied rewrites 53.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. associate-/r/ N/A

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

      10. sqrt-prod N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. frac-2neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-neg.f64 43.4

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

   6. Applied rewrites 43.4%

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

   if -9.99999999999999965e218 < (*.f64 V l) < -1.99999999999999999e-243

   1. Initial program 89.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 89.4

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 89.4

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

   4. Applied rewrites 89.4%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. sqrt-div N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-neg.f64 99.3

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

   6. Applied rewrites 99.3%

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

   if 0.0 < (*.f64 V l)

   1. Initial program 78.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 90.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 90.0

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

   4. Applied rewrites 90.0%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+219}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{-\ell}{A}} \cdot \sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-243}:\\ \;\;\;\;\frac{c0}{\sqrt{\left(-\ell\right) \cdot V}} \cdot \sqrt{-A}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{-\ell}{A}} \cdot \sqrt{-V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}} \cdot c0\\ \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+219}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-243}:\\ \;\;\;\;\frac{c0}{\sqrt{\left(-\ell\right) \cdot V}} \cdot \sqrt{-A}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \end{array} \end{array} \]

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) l)) (sqrt (- V))) c0)))
   (if (<= (* V l) -1e+219)
     t_0
     (if (<= (* V l) -2e-243)
       (* (/ c0 (sqrt (* (- l) V))) (sqrt (- A)))
       (if (<= (* V l) 0.0) t_0 (* (/ (sqrt A) (sqrt (* V l))) c0))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = (sqrt((-A / l)) / sqrt(-V)) * c0;
	double tmp;
	if ((V * l) <= -1e+219) {
		tmp = t_0;
	} else if ((V * l) <= -2e-243) {
		tmp = (c0 / sqrt((-l * V))) * sqrt(-A);
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	}
	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 / l)) / sqrt(-v)) * c0
    if ((v * l) <= (-1d+219)) then
        tmp = t_0
    else if ((v * l) <= (-2d-243)) then
        tmp = (c0 / sqrt((-l * v))) * sqrt(-a)
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else
        tmp = (sqrt(a) / sqrt((v * l))) * c0
    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 / l)) / Math.sqrt(-V)) * c0;
	double tmp;
	if ((V * l) <= -1e+219) {
		tmp = t_0;
	} else if ((V * l) <= -2e-243) {
		tmp = (c0 / Math.sqrt((-l * V))) * Math.sqrt(-A);
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = (Math.sqrt(A) / Math.sqrt((V * l))) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = (math.sqrt((-A / l)) / math.sqrt(-V)) * c0
	tmp = 0
	if (V * l) <= -1e+219:
		tmp = t_0
	elif (V * l) <= -2e-243:
		tmp = (c0 / math.sqrt((-l * V))) * math.sqrt(-A)
	elif (V * l) <= 0.0:
		tmp = t_0
	else:
		tmp = (math.sqrt(A) / math.sqrt((V * l))) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(Float64(sqrt(Float64(Float64(-A) / l)) / sqrt(Float64(-V))) * c0)
	tmp = 0.0
	if (Float64(V * l) <= -1e+219)
		tmp = t_0;
	elseif (Float64(V * l) <= -2e-243)
		tmp = Float64(Float64(c0 / sqrt(Float64(Float64(-l) * V))) * sqrt(Float64(-A)));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(V * l))) * c0);
	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 / l)) / sqrt(-V)) * c0;
	tmp = 0.0;
	if ((V * l) <= -1e+219)
		tmp = t_0;
	elseif ((V * l) <= -2e-243)
		tmp = (c0 / sqrt((-l * V))) * sqrt(-A);
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -1e+219], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -2e-243], N[(N[(c0 / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -9.99999999999999965e218 or -1.99999999999999999e-243 < (*.f64 V l) < 0.0

   1. Initial program 53.1%

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

      1. lift-/.f64 N/A

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

      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 71.3

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

   4. Applied rewrites 71.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lift-sqrt.f64 N/A

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

      5. div-inv N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. sqrt-div N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

      12. inv-pow N/A

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

      13. lift-sqrt.f64 N/A

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

      14. sqrt-pow2 N/A

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

      15. sqrt-pow1 N/A

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

      16. inv-pow N/A

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

      17. sqrt-prod N/A

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

      18. *-commutative N/A

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

      19. lower-sqrt.f64 N/A

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

      20. un-div-inv N/A

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

      21. lower-/.f64 N/A

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

      22. lower-neg.f64 N/A

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

      23. lower-sqrt.f64 N/A

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

      24. lower-neg.f64 43.3

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

   6. Applied rewrites 43.3%

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

   if -9.99999999999999965e218 < (*.f64 V l) < -1.99999999999999999e-243

   1. Initial program 89.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 89.4

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 89.4

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

   4. Applied rewrites 89.4%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. sqrt-div N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-neg.f64 99.3

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

   6. Applied rewrites 99.3%

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

   if 0.0 < (*.f64 V l)

   1. Initial program 78.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 90.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 90.0

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

   4. Applied rewrites 90.0%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+219}:\\ \;\;\;\;\frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-243}:\\ \;\;\;\;\frac{c0}{\sqrt{\left(-\ell\right) \cdot V}} \cdot \sqrt{-A}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ t_1 := \sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+289}:\\ \;\;\;\;\sqrt{t\_0} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))) (t_1 (* (sqrt (/ (/ A V) l)) c0)))
   (if (<= t_0 0.0) t_1 (if (<= t_0 4e+289) (* (sqrt t_0) c0) t_1))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double t_1 = sqrt(((A / V) / l)) * c0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 4e+289) {
		tmp = sqrt(t_0) * c0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a / (v * l)
    t_1 = sqrt(((a / v) / l)) * c0
    if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 4d+289) then
        tmp = sqrt(t_0) * c0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double t_1 = Math.sqrt(((A / V) / l)) * c0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 4e+289) {
		tmp = Math.sqrt(t_0) * c0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	t_1 = math.sqrt(((A / V) / l)) * c0
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 4e+289:
		tmp = math.sqrt(t_0) * c0
	else:
		tmp = t_1
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	t_1 = Float64(sqrt(Float64(Float64(A / V) / l)) * c0)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 4e+289)
		tmp = Float64(sqrt(t_0) * c0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	t_1 = sqrt(((A / V) / l)) * c0;
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 4e+289)
		tmp = sqrt(t_0) * c0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 4e+289], N[(N[Sqrt[t$95$0], $MachinePrecision] * c0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 4.0000000000000002e289 < (/.f64 A (*.f64 V l))

   1. Initial program 42.0%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/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 53.7

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

   4. Applied rewrites 53.7%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.0000000000000002e289

   1. Initial program 98.8%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 4 \cdot 10^{+289}:\\ \;\;\;\;\sqrt{\frac{A}{V \cdot \ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}} \cdot c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-243}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{\left(-\ell\right) \cdot V}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-323}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \end{array} \end{array} \]

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))
   (/ (* (sqrt (/ A V)) c0) (sqrt l))
   (if (<= (* V l) -2e-243)
     (/ (* (sqrt (- A)) c0) (sqrt (* (- l) V)))
     (if (<= (* V l) 5e-323)
       (/ c0 (sqrt (/ V (/ A l))))
       (* (/ (sqrt A) (sqrt (* V l))) c0)))))

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 = (sqrt((A / V)) * c0) / sqrt(l);
	} else if ((V * l) <= -2e-243) {
		tmp = (sqrt(-A) * c0) / sqrt((-l * V));
	} else if ((V * l) <= 5e-323) {
		tmp = c0 / sqrt((V / (A / l)));
	} else {
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	}
	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 = (Math.sqrt((A / V)) * c0) / Math.sqrt(l);
	} else if ((V * l) <= -2e-243) {
		tmp = (Math.sqrt(-A) * c0) / Math.sqrt((-l * V));
	} else if ((V * l) <= 5e-323) {
		tmp = c0 / Math.sqrt((V / (A / l)));
	} else {
		tmp = (Math.sqrt(A) / Math.sqrt((V * l))) * c0;
	}
	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 = (math.sqrt((A / V)) * c0) / math.sqrt(l)
	elif (V * l) <= -2e-243:
		tmp = (math.sqrt(-A) * c0) / math.sqrt((-l * V))
	elif (V * l) <= 5e-323:
		tmp = c0 / math.sqrt((V / (A / l)))
	else:
		tmp = (math.sqrt(A) / math.sqrt((V * l))) * c0
	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(Float64(sqrt(Float64(A / V)) * c0) / sqrt(l));
	elseif (Float64(V * l) <= -2e-243)
		tmp = Float64(Float64(sqrt(Float64(-A)) * c0) / sqrt(Float64(Float64(-l) * V)));
	elseif (Float64(V * l) <= 5e-323)
		tmp = Float64(c0 / sqrt(Float64(V / Float64(A / l))));
	else
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(V * l))) * c0);
	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 = (sqrt((A / V)) * c0) / sqrt(l);
	elseif ((V * l) <= -2e-243)
		tmp = (sqrt(-A) * c0) / sqrt((-l * V));
	elseif ((V * l) <= 5e-323)
		tmp = c0 / sqrt((V / (A / l)));
	else
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	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[(N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-243], N[(N[(N[Sqrt[(-A)], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-323], N[(c0 / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 35.0%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. sqrt-div N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 64.4

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

   4. Applied rewrites 64.4%

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

   if -inf.0 < (*.f64 V l) < -1.99999999999999999e-243

   1. Initial program 86.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 86.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 86.9

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

   4. Applied rewrites 86.9%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. sqrt-prod N/A

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

      7. lift-sqrt.f64 N/A

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

      8. pow1/2 N/A

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

      9. associate-/r* N/A

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

      10. lift-/.f64 N/A

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

      11. un-div-inv N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. pow1/2 N/A

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

      15. sqrt-div N/A

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

      16. metadata-eval N/A

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

      17. clear-num N/A

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

      18. lift-/.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-/.f64 N/A

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

   6. Applied rewrites 97.6%

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

   if -1.99999999999999999e-243 < (*.f64 V l) < 4.94066e-323

   1. Initial program 51.0%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 51.0

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 51.0

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

   4. Applied rewrites 51.0%

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

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

      4. associate-*l/ N/A

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

      5. /-rgt-identity N/A

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

      6. metadata-eval N/A

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

      7. clear-num N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. associate-/l/ N/A

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

      11. *-commutative N/A

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

      12. div-inv N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 70.1

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

   6. Applied rewrites 70.1%

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

   if 4.94066e-323 < (*.f64 V l)

   1. Initial program 79.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 90.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 90.8

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

   4. Applied rewrites 90.8%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}} \cdot c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-243}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{\left(-\ell\right) \cdot V}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-323}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-243}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{\left(-\ell\right) \cdot V}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-323}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \end{array} \end{array} \]

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 (/ V A)) (sqrt l)))
   (if (<= (* V l) -2e-243)
     (/ (* (sqrt (- A)) c0) (sqrt (* (- l) V)))
     (if (<= (* V l) 5e-323)
       (/ c0 (sqrt (/ V (/ A l))))
       (* (/ (sqrt A) (sqrt (* V l))) c0)))))

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((V / A)) * sqrt(l));
	} else if ((V * l) <= -2e-243) {
		tmp = (sqrt(-A) * c0) / sqrt((-l * V));
	} else if ((V * l) <= 5e-323) {
		tmp = c0 / sqrt((V / (A / l)));
	} else {
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	}
	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((V / A)) * Math.sqrt(l));
	} else if ((V * l) <= -2e-243) {
		tmp = (Math.sqrt(-A) * c0) / Math.sqrt((-l * V));
	} else if ((V * l) <= 5e-323) {
		tmp = c0 / Math.sqrt((V / (A / l)));
	} else {
		tmp = (Math.sqrt(A) / Math.sqrt((V * l))) * c0;
	}
	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((V / A)) * math.sqrt(l))
	elif (V * l) <= -2e-243:
		tmp = (math.sqrt(-A) * c0) / math.sqrt((-l * V))
	elif (V * l) <= 5e-323:
		tmp = c0 / math.sqrt((V / (A / l)))
	else:
		tmp = (math.sqrt(A) / math.sqrt((V * l))) * c0
	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(Float64(V / A)) * sqrt(l)));
	elseif (Float64(V * l) <= -2e-243)
		tmp = Float64(Float64(sqrt(Float64(-A)) * c0) / sqrt(Float64(Float64(-l) * V)));
	elseif (Float64(V * l) <= 5e-323)
		tmp = Float64(c0 / sqrt(Float64(V / Float64(A / l))));
	else
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(V * l))) * c0);
	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((V / A)) * sqrt(l));
	elseif ((V * l) <= -2e-243)
		tmp = (sqrt(-A) * c0) / sqrt((-l * V));
	elseif ((V * l) <= 5e-323)
		tmp = c0 / sqrt((V / (A / l)));
	else
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	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[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-243], N[(N[(N[Sqrt[(-A)], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-323], N[(c0 / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 35.0%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 35.0

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 35.0

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

   4. Applied rewrites 35.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. sqrt-prod N/A

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

      6. lift-sqrt.f64 N/A

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

      7. pow1/2 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 64.3

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

   6. Applied rewrites 64.3%

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

   if -inf.0 < (*.f64 V l) < -1.99999999999999999e-243

   1. Initial program 86.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 86.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 86.9

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

   4. Applied rewrites 86.9%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. sqrt-prod N/A

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

      7. lift-sqrt.f64 N/A

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

      8. pow1/2 N/A

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

      9. associate-/r* N/A

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

      10. lift-/.f64 N/A

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

      11. un-div-inv N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. pow1/2 N/A

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

      15. sqrt-div N/A

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

      16. metadata-eval N/A

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

      17. clear-num N/A

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

      18. lift-/.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-/.f64 N/A

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

   6. Applied rewrites 97.6%

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

   if -1.99999999999999999e-243 < (*.f64 V l) < 4.94066e-323

   1. Initial program 51.0%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 51.0

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 51.0

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

   4. Applied rewrites 51.0%

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

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

      4. associate-*l/ N/A

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

      5. /-rgt-identity N/A

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

      6. metadata-eval N/A

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

      7. clear-num N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. associate-/l/ N/A

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

      11. *-commutative N/A

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

      12. div-inv N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 70.1

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

   6. Applied rewrites 70.1%

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

   if 4.94066e-323 < (*.f64 V l)

   1. Initial program 79.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 90.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 90.8

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

   4. Applied rewrites 90.8%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-243}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{\left(-\ell\right) \cdot V}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-323}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-243}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{\left(-\ell\right) \cdot V}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-323}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \end{array} \end{array} \]

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))
   (* (sqrt (/ A V)) (/ c0 (sqrt l)))
   (if (<= (* V l) -2e-243)
     (/ (* (sqrt (- A)) c0) (sqrt (* (- l) V)))
     (if (<= (* V l) 5e-323)
       (/ c0 (sqrt (/ V (/ A l))))
       (* (/ (sqrt A) (sqrt (* V l))) c0)))))

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 = sqrt((A / V)) * (c0 / sqrt(l));
	} else if ((V * l) <= -2e-243) {
		tmp = (sqrt(-A) * c0) / sqrt((-l * V));
	} else if ((V * l) <= 5e-323) {
		tmp = c0 / sqrt((V / (A / l)));
	} else {
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	}
	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 = Math.sqrt((A / V)) * (c0 / Math.sqrt(l));
	} else if ((V * l) <= -2e-243) {
		tmp = (Math.sqrt(-A) * c0) / Math.sqrt((-l * V));
	} else if ((V * l) <= 5e-323) {
		tmp = c0 / Math.sqrt((V / (A / l)));
	} else {
		tmp = (Math.sqrt(A) / Math.sqrt((V * l))) * c0;
	}
	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 = math.sqrt((A / V)) * (c0 / math.sqrt(l))
	elif (V * l) <= -2e-243:
		tmp = (math.sqrt(-A) * c0) / math.sqrt((-l * V))
	elif (V * l) <= 5e-323:
		tmp = c0 / math.sqrt((V / (A / l)))
	else:
		tmp = (math.sqrt(A) / math.sqrt((V * l))) * c0
	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(sqrt(Float64(A / V)) * Float64(c0 / sqrt(l)));
	elseif (Float64(V * l) <= -2e-243)
		tmp = Float64(Float64(sqrt(Float64(-A)) * c0) / sqrt(Float64(Float64(-l) * V)));
	elseif (Float64(V * l) <= 5e-323)
		tmp = Float64(c0 / sqrt(Float64(V / Float64(A / l))));
	else
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(V * l))) * c0);
	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 = sqrt((A / V)) * (c0 / sqrt(l));
	elseif ((V * l) <= -2e-243)
		tmp = (sqrt(-A) * c0) / sqrt((-l * V));
	elseif ((V * l) <= 5e-323)
		tmp = c0 / sqrt((V / (A / l)));
	else
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	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[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-243], N[(N[(N[Sqrt[(-A)], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-323], N[(c0 / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 35.0%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. sqrt-div N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 62.7

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

   4. Applied rewrites 62.7%

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

   if -inf.0 < (*.f64 V l) < -1.99999999999999999e-243

   1. Initial program 86.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 86.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 86.9

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

   4. Applied rewrites 86.9%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. sqrt-prod N/A

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

      7. lift-sqrt.f64 N/A

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

      8. pow1/2 N/A

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

      9. associate-/r* N/A

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

      10. lift-/.f64 N/A

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

      11. un-div-inv N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. pow1/2 N/A

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

      15. sqrt-div N/A

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

      16. metadata-eval N/A

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

      17. clear-num N/A

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

      18. lift-/.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-/.f64 N/A

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

   6. Applied rewrites 97.6%

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

   if -1.99999999999999999e-243 < (*.f64 V l) < 4.94066e-323

   1. Initial program 51.0%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 51.0

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 51.0

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

   4. Applied rewrites 51.0%

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

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

      4. associate-*l/ N/A

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

      5. /-rgt-identity N/A

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

      6. metadata-eval N/A

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

      7. clear-num N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. associate-/l/ N/A

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

      11. *-commutative N/A

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

      12. div-inv N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 70.1

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

   6. Applied rewrites 70.1%

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

   if 4.94066e-323 < (*.f64 V l)

   1. Initial program 79.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 90.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 90.8

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

   4. Applied rewrites 90.8%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-243}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{\left(-\ell\right) \cdot V}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-323}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-243}:\\ \;\;\;\;\frac{c0}{\sqrt{\left(-\ell\right) \cdot V}} \cdot \sqrt{-A}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-323}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \end{array} \end{array} \]

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 A) V)))
   (if (<= (* V l) -2e-243)
     (* (/ c0 (sqrt (* (- l) V))) (sqrt (- A)))
     (if (<= (* V l) 5e-323)
       (/ c0 (sqrt (/ V (/ A l))))
       (* (/ (sqrt A) (sqrt (* V l))) c0)))))

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 / A) * V));
	} else if ((V * l) <= -2e-243) {
		tmp = (c0 / sqrt((-l * V))) * sqrt(-A);
	} else if ((V * l) <= 5e-323) {
		tmp = c0 / sqrt((V / (A / l)));
	} else {
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	}
	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 / A) * V));
	} else if ((V * l) <= -2e-243) {
		tmp = (c0 / Math.sqrt((-l * V))) * Math.sqrt(-A);
	} else if ((V * l) <= 5e-323) {
		tmp = c0 / Math.sqrt((V / (A / l)));
	} else {
		tmp = (Math.sqrt(A) / Math.sqrt((V * l))) * c0;
	}
	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 / A) * V))
	elif (V * l) <= -2e-243:
		tmp = (c0 / math.sqrt((-l * V))) * math.sqrt(-A)
	elif (V * l) <= 5e-323:
		tmp = c0 / math.sqrt((V / (A / l)))
	else:
		tmp = (math.sqrt(A) / math.sqrt((V * l))) * c0
	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 / sqrt(Float64(Float64(l / A) * V)));
	elseif (Float64(V * l) <= -2e-243)
		tmp = Float64(Float64(c0 / sqrt(Float64(Float64(-l) * V))) * sqrt(Float64(-A)));
	elseif (Float64(V * l) <= 5e-323)
		tmp = Float64(c0 / sqrt(Float64(V / Float64(A / l))));
	else
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(V * l))) * c0);
	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 / A) * V));
	elseif ((V * l) <= -2e-243)
		tmp = (c0 / sqrt((-l * V))) * sqrt(-A);
	elseif ((V * l) <= 5e-323)
		tmp = c0 / sqrt((V / (A / l)));
	else
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	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[Sqrt[N[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-243], N[(N[(c0 / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-323], N[(c0 / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 35.0%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 35.0

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 35.0

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

   4. Applied rewrites 35.0%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{\ell \cdot V}{A}}}} \]
   5. 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. lower-*.f64 N/A

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

      5. lower-/.f64 72.7

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

   6. Applied rewrites 72.7%

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

   if -inf.0 < (*.f64 V l) < -1.99999999999999999e-243

   1. Initial program 86.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 86.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 86.9

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

   4. Applied rewrites 86.9%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. sqrt-div N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-neg.f64 97.5

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

   6. Applied rewrites 97.5%

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

   if -1.99999999999999999e-243 < (*.f64 V l) < 4.94066e-323

   1. Initial program 51.0%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 51.0

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 51.0

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

   4. Applied rewrites 51.0%

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

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

      4. associate-*l/ N/A

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

      5. /-rgt-identity N/A

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

      6. metadata-eval N/A

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

      7. clear-num N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. associate-/l/ N/A

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

      11. *-commutative N/A

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

      12. div-inv N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 70.1

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

   6. Applied rewrites 70.1%

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

   if 4.94066e-323 < (*.f64 V l)

   1. Initial program 79.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 90.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 90.8

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

   4. Applied rewrites 90.8%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-243}:\\ \;\;\;\;\frac{c0}{\sqrt{\left(-\ell\right) \cdot V}} \cdot \sqrt{-A}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-323}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 82.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-183}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-323}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \end{array} \end{array} \]

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) -5e+197)
   (* (sqrt (/ (/ A l) V)) c0)
   (if (<= (* V l) -1e-183)
     (/ c0 (sqrt (/ (* V l) A)))
     (if (<= (* V l) 5e-323)
       (/ c0 (sqrt (/ V (/ A l))))
       (* (/ (sqrt A) (sqrt (* V l))) c0)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+197) {
		tmp = sqrt(((A / l) / V)) * c0;
	} else if ((V * l) <= -1e-183) {
		tmp = c0 / sqrt(((V * l) / A));
	} else if ((V * l) <= 5e-323) {
		tmp = c0 / sqrt((V / (A / l)));
	} else {
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-5d+197)) then
        tmp = sqrt(((a / l) / v)) * c0
    else if ((v * l) <= (-1d-183)) then
        tmp = c0 / sqrt(((v * l) / a))
    else if ((v * l) <= 5d-323) then
        tmp = c0 / sqrt((v / (a / l)))
    else
        tmp = (sqrt(a) / sqrt((v * l))) * c0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+197) {
		tmp = Math.sqrt(((A / l) / V)) * c0;
	} else if ((V * l) <= -1e-183) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else if ((V * l) <= 5e-323) {
		tmp = c0 / Math.sqrt((V / (A / l)));
	} else {
		tmp = (Math.sqrt(A) / Math.sqrt((V * l))) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -5e+197:
		tmp = math.sqrt(((A / l) / V)) * c0
	elif (V * l) <= -1e-183:
		tmp = c0 / math.sqrt(((V * l) / A))
	elif (V * l) <= 5e-323:
		tmp = c0 / math.sqrt((V / (A / l)))
	else:
		tmp = (math.sqrt(A) / math.sqrt((V * l))) * c0
	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) <= -5e+197)
		tmp = Float64(sqrt(Float64(Float64(A / l) / V)) * c0);
	elseif (Float64(V * l) <= -1e-183)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	elseif (Float64(V * l) <= 5e-323)
		tmp = Float64(c0 / sqrt(Float64(V / Float64(A / l))));
	else
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(V * l))) * c0);
	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) <= -5e+197)
		tmp = sqrt(((A / l) / V)) * c0;
	elseif ((V * l) <= -1e-183)
		tmp = c0 / sqrt(((V * l) / A));
	elseif ((V * l) <= 5e-323)
		tmp = c0 / sqrt((V / (A / l)));
	else
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	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], -5e+197], N[(N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-183], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e-323], N[(c0 / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -5.00000000000000009e197

   1. Initial program 55.0%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 68.1

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

   4. Applied rewrites 68.1%

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

   if -5.00000000000000009e197 < (*.f64 V l) < -1.00000000000000001e-183

   1. Initial program 95.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 95.8

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 95.8

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

   4. Applied rewrites 95.8%

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

   if -1.00000000000000001e-183 < (*.f64 V l) < 4.94066e-323

   1. Initial program 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 51.4

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 51.4

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

   4. Applied rewrites 51.4%

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

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

      4. associate-*l/ N/A

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

      5. /-rgt-identity N/A

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

      6. metadata-eval N/A

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

      7. clear-num N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. associate-/l/ N/A

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

      11. *-commutative N/A

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

      12. div-inv N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 66.7

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

   6. Applied rewrites 66.7%

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

   if 4.94066e-323 < (*.f64 V l)

   1. Initial program 79.2%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 90.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 90.8

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

   4. Applied rewrites 90.8%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-183}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-323}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 86.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-243}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{\left(-\ell\right) \cdot V}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \end{array} \end{array} \]

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 A) V)))))
   (if (<= (* V l) (- INFINITY))
     t_0
     (if (<= (* V l) -2e-243)
       (/ (* (sqrt (- A)) c0) (sqrt (* (- l) V)))
       (if (<= (* V l) 0.0) t_0 (* (/ (sqrt A) (sqrt (* V l))) c0))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 / sqrt(((l / A) * V));
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((V * l) <= -2e-243) {
		tmp = (sqrt(-A) * c0) / sqrt((-l * V));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 / Math.sqrt(((l / A) * V));
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if ((V * l) <= -2e-243) {
		tmp = (Math.sqrt(-A) * c0) / Math.sqrt((-l * V));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = (Math.sqrt(A) / Math.sqrt((V * l))) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 / math.sqrt(((l / A) * V))
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = t_0
	elif (V * l) <= -2e-243:
		tmp = (math.sqrt(-A) * c0) / math.sqrt((-l * V))
	elif (V * l) <= 0.0:
		tmp = t_0
	else:
		tmp = (math.sqrt(A) / math.sqrt((V * l))) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 / sqrt(Float64(Float64(l / A) * V)))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(V * l) <= -2e-243)
		tmp = Float64(Float64(sqrt(Float64(-A)) * c0) / sqrt(Float64(Float64(-l) * V)));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(V * l))) * c0);
	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 / A) * V));
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = t_0;
	elseif ((V * l) <= -2e-243)
		tmp = (sqrt(-A) * c0) / sqrt((-l * V));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	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[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -2e-243], N[(N[(N[Sqrt[(-A)], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[N[((-l) * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -inf.0 or -1.99999999999999999e-243 < (*.f64 V l) < 0.0

   1. Initial program 47.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 47.1

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 47.1

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

   4. Applied rewrites 47.1%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{\ell \cdot V}{A}}}} \]
   5. 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. lower-*.f64 N/A

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

      5. lower-/.f64 72.3

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

   6. Applied rewrites 72.3%

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

   if -inf.0 < (*.f64 V l) < -1.99999999999999999e-243

   1. Initial program 86.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 86.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 86.9

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

   4. Applied rewrites 86.9%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. sqrt-prod N/A

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

      7. lift-sqrt.f64 N/A

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

      8. pow1/2 N/A

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

      9. associate-/r* N/A

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

      10. lift-/.f64 N/A

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

      11. un-div-inv N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. pow1/2 N/A

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

      15. sqrt-div N/A

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

      16. metadata-eval N/A

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

      17. clear-num N/A

          \[\leadsto \frac{c0}{\sqrt{\ell}} \cdot \sqrt{\color{blue}{\frac{A}{V}}} \]

      18. lift-/.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\ell}} \cdot \sqrt{\color{blue}{\frac{A}{V}}} \]

      19. *-commutative N/A

          \[\leadsto \color{blue}{\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}} \]

      20. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{A}{V}} \cdot \color{blue}{\frac{c0}{\sqrt{\ell}}} \]

   6. Applied rewrites 97.6%

      \[\leadsto \color{blue}{\frac{\sqrt{-A} \cdot c0}{\sqrt{\left(-V\right) \cdot \ell}}} \]

   if 0.0 < (*.f64 V l)

   1. Initial program 78.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 90.0

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\color{blue}{\sqrt{V \cdot \ell}}} \]

      7. lift-*.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell}}} \]

      8. *-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

      9. lower-*.f64 90.0

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

   4. Applied rewrites 90.0%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\ell \cdot V}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-243}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{\left(-\ell\right) \cdot V}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 82.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-140}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \end{array} \end{array} \]

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) -5e+197)
   (* (sqrt (/ (/ A l) V)) c0)
   (if (<= (* V l) -2e-140)
     (/ c0 (sqrt (/ (* V l) A)))
     (if (<= (* V l) 0.0)
       (/ c0 (sqrt (* (/ l A) V)))
       (* (/ (sqrt A) (sqrt (* V l))) c0)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+197) {
		tmp = sqrt(((A / l) / V)) * c0;
	} else if ((V * l) <= -2e-140) {
		tmp = c0 / sqrt(((V * l) / A));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / sqrt(((l / A) * V));
	} else {
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-5d+197)) then
        tmp = sqrt(((a / l) / v)) * c0
    else if ((v * l) <= (-2d-140)) then
        tmp = c0 / sqrt(((v * l) / a))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 / sqrt(((l / a) * v))
    else
        tmp = (sqrt(a) / sqrt((v * l))) * c0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+197) {
		tmp = Math.sqrt(((A / l) / V)) * c0;
	} else if ((V * l) <= -2e-140) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / Math.sqrt(((l / A) * V));
	} else {
		tmp = (Math.sqrt(A) / Math.sqrt((V * l))) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -5e+197:
		tmp = math.sqrt(((A / l) / V)) * c0
	elif (V * l) <= -2e-140:
		tmp = c0 / math.sqrt(((V * l) / A))
	elif (V * l) <= 0.0:
		tmp = c0 / math.sqrt(((l / A) * V))
	else:
		tmp = (math.sqrt(A) / math.sqrt((V * l))) * c0
	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) <= -5e+197)
		tmp = Float64(sqrt(Float64(Float64(A / l) / V)) * c0);
	elseif (Float64(V * l) <= -2e-140)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 / sqrt(Float64(Float64(l / A) * V)));
	else
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(V * l))) * c0);
	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) <= -5e+197)
		tmp = sqrt(((A / l) / V)) * c0;
	elseif ((V * l) <= -2e-140)
		tmp = c0 / sqrt(((V * l) / A));
	elseif ((V * l) <= 0.0)
		tmp = c0 / sqrt(((l / A) * V));
	else
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	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], -5e+197], N[(N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-140], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[Sqrt[N[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -5.00000000000000009e197

   1. Initial program 55.0%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 68.1

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{\ell}}}{V}} \]

   4. Applied rewrites 68.1%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{\ell}}{V}}} \]

   if -5.00000000000000009e197 < (*.f64 V l) < -2e-140

   1. Initial program 95.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \]

      3. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      4. clear-num N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\frac{V \cdot \ell}{A}}}} \]

      5. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      6. metadata-eval N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{1}}{\sqrt{\frac{V \cdot \ell}{A}}} \]

      7. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      10. lower-/.f64 95.4

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V \cdot \ell}{A}}}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell}}{A}}} \]

      12. *-commutative N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\ell \cdot V}}{A}}} \]

      13. lower-*.f64 95.4

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\ell \cdot V}}{A}}} \]

   4. Applied rewrites 95.4%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{\ell \cdot V}{A}}}} \]

   if -2e-140 < (*.f64 V l) < 0.0

   1. Initial program 57.8%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \]

      3. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      4. clear-num N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\frac{V \cdot \ell}{A}}}} \]

      5. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      6. metadata-eval N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{1}}{\sqrt{\frac{V \cdot \ell}{A}}} \]

      7. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      10. lower-/.f64 57.8

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V \cdot \ell}{A}}}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell}}{A}}} \]

      12. *-commutative N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\ell \cdot V}}{A}}} \]

      13. lower-*.f64 57.8

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\ell \cdot V}}{A}}} \]

   4. Applied rewrites 57.8%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{\ell \cdot V}{A}}}} \]
   5. 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. lower-*.f64 N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\ell}{A} \cdot V}}} \]

      5. lower-/.f64 71.7

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\ell}{A}} \cdot V}} \]

   6. Applied rewrites 71.7%

      \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\ell}{A} \cdot V}}} \]

   if 0.0 < (*.f64 V l)

   1. Initial program 78.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 90.0

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\color{blue}{\sqrt{V \cdot \ell}}} \]

      7. lift-*.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell}}} \]

      8. *-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

      9. lower-*.f64 90.0

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

   4. Applied rewrites 90.0%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\ell \cdot V}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-140}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 90.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c0}{\sqrt{-V} \cdot \frac{\sqrt{\ell}}{\sqrt{-A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \end{array} \end{array} \]

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 -5e-310)
   (/ c0 (* (sqrt (- V)) (/ (sqrt l) (sqrt (- A)))))
   (* (/ (sqrt A) (sqrt (* V l))) c0)))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = c0 / (sqrt(-V) * (sqrt(l) / sqrt(-A)));
	} else {
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	}
	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 <= (-5d-310)) then
        tmp = c0 / (sqrt(-v) * (sqrt(l) / sqrt(-a)))
    else
        tmp = (sqrt(a) / sqrt((v * l))) * c0
    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 <= -5e-310) {
		tmp = c0 / (Math.sqrt(-V) * (Math.sqrt(l) / Math.sqrt(-A)));
	} else {
		tmp = (Math.sqrt(A) / Math.sqrt((V * l))) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -5e-310:
		tmp = c0 / (math.sqrt(-V) * (math.sqrt(l) / math.sqrt(-A)))
	else:
		tmp = (math.sqrt(A) / math.sqrt((V * l))) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -5e-310)
		tmp = Float64(c0 / Float64(sqrt(Float64(-V)) * Float64(sqrt(l) / sqrt(Float64(-A)))));
	else
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(V * l))) * c0);
	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 <= -5e-310)
		tmp = c0 / (sqrt(-V) * (sqrt(l) / sqrt(-A)));
	else
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	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, -5e-310], N[(c0 / N[(N[Sqrt[(-V)], $MachinePrecision] * N[(N[Sqrt[l], $MachinePrecision] / N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -4.999999999999985e-310

   1. Initial program 76.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \]

      3. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      4. clear-num N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\frac{V \cdot \ell}{A}}}} \]

      5. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      6. metadata-eval N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{1}}{\sqrt{\frac{V \cdot \ell}{A}}} \]

      7. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      10. lower-/.f64 76.6

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V \cdot \ell}{A}}}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell}}{A}}} \]

      12. *-commutative N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\ell \cdot V}}{A}}} \]

      13. lower-*.f64 76.6

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\ell \cdot V}}{A}}} \]

   4. Applied rewrites 76.6%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{\ell \cdot V}{A}}}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{\ell \cdot V}{A}}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\ell \cdot V}{A}}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\ell \cdot V}}{A}}} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\ell}{A} \cdot V}}} \]

      5. associate-/r/ N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\ell}{\frac{A}{V}}}}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{c0}{\sqrt{\frac{\ell}{\color{blue}{\frac{A}{V}}}}} \]

      7. sqrt-div N/A

         \[\leadsto \frac{c0}{\color{blue}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{c0}{\frac{\color{blue}{\sqrt{\ell}}}{\sqrt{\frac{A}{V}}}} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\color{blue}{\frac{A}{V}}}}} \]

      10. frac-2neg N/A

          \[\leadsto \frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\color{blue}{\frac{\mathsf{neg}\left(A\right)}{\mathsf{neg}\left(V\right)}}}}} \]

      11. sqrt-div N/A

          \[\leadsto \frac{c0}{\frac{\sqrt{\ell}}{\color{blue}{\frac{\sqrt{\mathsf{neg}\left(A\right)}}{\sqrt{\mathsf{neg}\left(V\right)}}}}} \]

      12. associate-/r/ N/A

          \[\leadsto \frac{c0}{\color{blue}{\frac{\sqrt{\ell}}{\sqrt{\mathsf{neg}\left(A\right)}} \cdot \sqrt{\mathsf{neg}\left(V\right)}}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{c0}{\color{blue}{\frac{\sqrt{\ell}}{\sqrt{\mathsf{neg}\left(A\right)}} \cdot \sqrt{\mathsf{neg}\left(V\right)}}} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{c0}{\color{blue}{\frac{\sqrt{\ell}}{\sqrt{\mathsf{neg}\left(A\right)}}} \cdot \sqrt{\mathsf{neg}\left(V\right)}} \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \frac{c0}{\frac{\sqrt{\ell}}{\color{blue}{\sqrt{\mathsf{neg}\left(A\right)}}} \cdot \sqrt{\mathsf{neg}\left(V\right)}} \]

      16. lower-neg.f64 N/A

          \[\leadsto \frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\color{blue}{-A}}} \cdot \sqrt{\mathsf{neg}\left(V\right)}} \]

      17. lower-sqrt.f64 N/A

          \[\leadsto \frac{c0}{\frac{\sqrt{\ell}}{\sqrt{-A}} \cdot \color{blue}{\sqrt{\mathsf{neg}\left(V\right)}}} \]

      18. lower-neg.f64 41.6

          \[\leadsto \frac{c0}{\frac{\sqrt{\ell}}{\sqrt{-A}} \cdot \sqrt{\color{blue}{-V}}} \]

   6. Applied rewrites 41.6%

      \[\leadsto \frac{c0}{\color{blue}{\frac{\sqrt{\ell}}{\sqrt{-A}} \cdot \sqrt{-V}}} \]

   if -4.999999999999985e-310 < A

   1. Initial program 75.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. 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 85.8

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\color{blue}{\sqrt{V \cdot \ell}}} \]

      7. lift-*.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell}}} \]

      8. *-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

      9. lower-*.f64 85.8

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

   4. Applied rewrites 85.8%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\ell \cdot V}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c0}{\sqrt{-V} \cdot \frac{\sqrt{\ell}}{\sqrt{-A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{-V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \end{array} \end{array} \]

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 -5e-310)
   (* (/ (sqrt (- A)) (sqrt (- V))) (/ c0 (sqrt l)))
   (* (/ (sqrt A) (sqrt (* V l))) c0)))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = (sqrt(-A) / sqrt(-V)) * (c0 / sqrt(l));
	} else {
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	}
	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 <= (-5d-310)) then
        tmp = (sqrt(-a) / sqrt(-v)) * (c0 / sqrt(l))
    else
        tmp = (sqrt(a) / sqrt((v * l))) * c0
    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 <= -5e-310) {
		tmp = (Math.sqrt(-A) / Math.sqrt(-V)) * (c0 / Math.sqrt(l));
	} else {
		tmp = (Math.sqrt(A) / Math.sqrt((V * l))) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -5e-310:
		tmp = (math.sqrt(-A) / math.sqrt(-V)) * (c0 / math.sqrt(l))
	else:
		tmp = (math.sqrt(A) / math.sqrt((V * l))) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -5e-310)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(-V))) * Float64(c0 / sqrt(l)));
	else
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(V * l))) * c0);
	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 <= -5e-310)
		tmp = (sqrt(-A) / sqrt(-V)) * (c0 / sqrt(l));
	else
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	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, -5e-310], N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -4.999999999999985e-310

   1. Initial program 76.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \]

      3. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      4. lift-*.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{V \cdot \ell}}} \]

      5. associate-/r* N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \]

      6. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{c0}{\sqrt{\ell}}} \cdot \sqrt{\frac{A}{V}} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{c0}{\color{blue}{\sqrt{\ell}}} \cdot \sqrt{\frac{A}{V}} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\ell}} \cdot \color{blue}{\sqrt{\frac{A}{V}}} \]

      13. lower-/.f64 37.7

          \[\leadsto \frac{c0}{\sqrt{\ell}} \cdot \sqrt{\color{blue}{\frac{A}{V}}} \]

   4. Applied rewrites 37.7%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{c0}{\sqrt{\ell}} \cdot \color{blue}{\sqrt{\frac{A}{V}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{c0}{\sqrt{\ell}} \cdot \sqrt{\color{blue}{\frac{A}{V}}} \]

      3. frac-2neg N/A

         \[\leadsto \frac{c0}{\sqrt{\ell}} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(A\right)}{\mathsf{neg}\left(V\right)}}} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{\color{blue}{-A}}{\mathsf{neg}\left(V\right)}} \]

      5. sqrt-div N/A

         \[\leadsto \frac{c0}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{-A}}{\sqrt{\mathsf{neg}\left(V\right)}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{c0}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{-A}}{\sqrt{\mathsf{neg}\left(V\right)}}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{\sqrt{\ell}} \cdot \frac{\color{blue}{\sqrt{-A}}}{\sqrt{\mathsf{neg}\left(V\right)}} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{\sqrt{\ell}} \cdot \frac{\sqrt{-A}}{\color{blue}{\sqrt{\mathsf{neg}\left(V\right)}}} \]

      9. lower-neg.f64 41.5

         \[\leadsto \frac{c0}{\sqrt{\ell}} \cdot \frac{\sqrt{-A}}{\sqrt{\color{blue}{-V}}} \]

   6. Applied rewrites 41.5%

      \[\leadsto \frac{c0}{\sqrt{\ell}} \cdot \color{blue}{\frac{\sqrt{-A}}{\sqrt{-V}}} \]

   if -4.999999999999985e-310 < A

   1. Initial program 75.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. 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 85.8

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\color{blue}{\sqrt{V \cdot \ell}}} \]

      7. lift-*.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell}}} \]

      8. *-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

      9. lower-*.f64 85.8

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

   4. Applied rewrites 85.8%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\ell \cdot V}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{-V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 89.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{-V} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \end{array} \end{array} \]

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 -5e-310)
   (/ (* (sqrt (- A)) c0) (* (sqrt (- V)) (sqrt l)))
   (* (/ (sqrt A) (sqrt (* V l))) c0)))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = (sqrt(-A) * c0) / (sqrt(-V) * sqrt(l));
	} else {
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	}
	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 <= (-5d-310)) then
        tmp = (sqrt(-a) * c0) / (sqrt(-v) * sqrt(l))
    else
        tmp = (sqrt(a) / sqrt((v * l))) * c0
    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 <= -5e-310) {
		tmp = (Math.sqrt(-A) * c0) / (Math.sqrt(-V) * Math.sqrt(l));
	} else {
		tmp = (Math.sqrt(A) / Math.sqrt((V * l))) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -5e-310:
		tmp = (math.sqrt(-A) * c0) / (math.sqrt(-V) * math.sqrt(l))
	else:
		tmp = (math.sqrt(A) / math.sqrt((V * l))) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -5e-310)
		tmp = Float64(Float64(sqrt(Float64(-A)) * c0) / Float64(sqrt(Float64(-V)) * sqrt(l)));
	else
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(V * l))) * c0);
	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 <= -5e-310)
		tmp = (sqrt(-A) * c0) / (sqrt(-V) * sqrt(l));
	else
		tmp = (sqrt(A) / sqrt((V * l))) * c0;
	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, -5e-310], N[(N[(N[Sqrt[(-A)], $MachinePrecision] * c0), $MachinePrecision] / N[(N[Sqrt[(-V)], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -4.999999999999985e-310

   1. Initial program 76.5%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\sqrt{\frac{A}{V \cdot \ell}}} \]

      3. lift-/.f64 N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V \cdot \ell}}} \]

      4. clear-num N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\frac{V \cdot \ell}{A}}}} \]

      5. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      6. metadata-eval N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{1}}{\sqrt{\frac{V \cdot \ell}{A}}} \]

      7. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{\color{blue}{\sqrt{\frac{V \cdot \ell}{A}}}} \]

      10. lower-/.f64 76.6

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V \cdot \ell}{A}}}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell}}{A}}} \]

      12. *-commutative N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\ell \cdot V}}{A}}} \]

      13. lower-*.f64 76.6

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\ell \cdot V}}{A}}} \]

   4. Applied rewrites 76.6%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{\ell \cdot V}{A}}}} \]
   5. 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. *-commutative N/A

         \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell}}{A}}} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V}{A} \cdot \ell}}} \]

      5. /-rgt-identity N/A

         \[\leadsto \frac{c0}{\sqrt{\frac{V}{A} \cdot \color{blue}{\frac{\ell}{1}}}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{c0}{\sqrt{\frac{V}{A} \cdot \frac{\ell}{\color{blue}{\mathsf{neg}\left(-1\right)}}}} \]

      7. clear-num N/A

         \[\leadsto \frac{c0}{\sqrt{\frac{V}{A} \cdot \color{blue}{\frac{1}{\frac{\mathsf{neg}\left(-1\right)}{\ell}}}}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{c0}{\sqrt{\frac{V}{A} \cdot \frac{1}{\frac{\color{blue}{1}}{\ell}}}} \]

      9. div-inv N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\frac{V}{A}}{\frac{1}{\ell}}}}} \]

      10. associate-/l/ N/A

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V}{\frac{1}{\ell} \cdot A}}}} \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{V}{\color{blue}{A \cdot \frac{1}{\ell}}}}} \]

      12. div-inv N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{V}{\color{blue}{\frac{A}{\ell}}}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V}{\frac{A}{\ell}}}}} \]

      14. lower-/.f64 75.9

          \[\leadsto \frac{c0}{\sqrt{\frac{V}{\color{blue}{\frac{A}{\ell}}}}} \]

   6. Applied rewrites 75.9%

      \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V}{\frac{A}{\ell}}}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}} \]

      2. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{1 \cdot c0}}{\sqrt{\frac{V}{\frac{A}{\ell}}}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{1 \cdot c0}{\color{blue}{\sqrt{\frac{V}{\frac{A}{\ell}}}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{1 \cdot c0}{\sqrt{\color{blue}{\frac{V}{\frac{A}{\ell}}}}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{1 \cdot c0}{\sqrt{\frac{V}{\color{blue}{\frac{A}{\ell}}}}} \]

      6. associate-/r/ N/A

         \[\leadsto \frac{1 \cdot c0}{\sqrt{\color{blue}{\frac{V}{A} \cdot \ell}}} \]

      7. sqrt-prod N/A

         \[\leadsto \frac{1 \cdot c0}{\color{blue}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{1 \cdot c0}{\sqrt{\frac{V}{A}} \cdot \color{blue}{\sqrt{\ell}}} \]

      9. frac-times N/A

         \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{V}{A}}} \cdot \frac{c0}{\sqrt{\ell}}} \]

      10. frac-2neg N/A

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{\mathsf{neg}\left(V\right)}{\mathsf{neg}\left(A\right)}}}} \cdot \frac{c0}{\sqrt{\ell}} \]

      11. lift-neg.f64 N/A

          \[\leadsto \frac{1}{\sqrt{\frac{\mathsf{neg}\left(V\right)}{\color{blue}{-A}}}} \cdot \frac{c0}{\sqrt{\ell}} \]

      12. sqrt-div N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\mathsf{neg}\left(V\right)}}{\sqrt{-A}}}} \cdot \frac{c0}{\sqrt{\ell}} \]

      13. clear-num N/A

          \[\leadsto \color{blue}{\frac{\sqrt{-A}}{\sqrt{\mathsf{neg}\left(V\right)}}} \cdot \frac{c0}{\sqrt{\ell}} \]

      14. frac-times N/A

          \[\leadsto \color{blue}{\frac{\sqrt{-A} \cdot c0}{\sqrt{\mathsf{neg}\left(V\right)} \cdot \sqrt{\ell}}} \]

      15. *-commutative N/A

          \[\leadsto \frac{\color{blue}{c0 \cdot \sqrt{-A}}}{\sqrt{\mathsf{neg}\left(V\right)} \cdot \sqrt{\ell}} \]

      16. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{-A}}{\sqrt{\mathsf{neg}\left(V\right)} \cdot \sqrt{\ell}}} \]

      17. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\sqrt{-A} \cdot c0}}{\sqrt{\mathsf{neg}\left(V\right)} \cdot \sqrt{\ell}} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{-A} \cdot c0}}{\sqrt{\mathsf{neg}\left(V\right)} \cdot \sqrt{\ell}} \]

      19. lower-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{-A}} \cdot c0}{\sqrt{\mathsf{neg}\left(V\right)} \cdot \sqrt{\ell}} \]

      20. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{-A} \cdot c0}{\color{blue}{\sqrt{\mathsf{neg}\left(V\right)} \cdot \sqrt{\ell}}} \]

   8. Applied rewrites 41.6%

      \[\leadsto \color{blue}{\frac{\sqrt{-A} \cdot c0}{\sqrt{-V} \cdot \sqrt{\ell}}} \]

   if -4.999999999999985e-310 < A

   1. Initial program 75.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. 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 85.8

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\color{blue}{\sqrt{V \cdot \ell}}} \]

      7. lift-*.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell}}} \]

      8. *-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

      9. lower-*.f64 85.8

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\ell \cdot V}}} \]

   4. Applied rewrites 85.8%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\ell \cdot V}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-A} \cdot c0}{\sqrt{-V} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 73.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \sqrt{\frac{A}{V \cdot \ell}} \cdot c0 \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l) :precision binary64 (* (sqrt (/ A (* V l))) c0))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	return sqrt((A / (V * l))) * c0;
}

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 = sqrt((a / (v * l))) * c0
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	return Math.sqrt((A / (V * l))) * c0;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	return math.sqrt((A / (V * l))) * c0

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	return Float64(sqrt(Float64(A / Float64(V * l))) * c0)
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp = code(c0, A, V, l)
	tmp = sqrt((A / (V * l))) * c0;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := N[(N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]

Derivation

1. Initial program 75.9%

   \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
2. Add Preprocessing

3. Final simplification 75.9%

   \[\leadsto \sqrt{\frac{A}{V \cdot \ell}} \cdot c0 \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024276 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))