Henrywood and Agarwal, Equation (3)

Percentage Accurate: 74.1% → 88.0%

Time: 7.3s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.1% 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: 88.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-302}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-V\right) \cdot \ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{-238}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) (- INFINITY))
   (* (/ (sqrt (/ (- A) l)) (sqrt (- V))) c0)
   (if (<= (* l V) -5e-302)
     (* (/ (sqrt (- A)) (sqrt (* (- V) l))) c0)
     (if (<= (* l V) 2e-238)
       (* (sqrt (/ (/ A V) l)) c0)
       (/ (* (sqrt A) c0) (sqrt (* l V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = (sqrt((-A / l)) / sqrt(-V)) * c0;
	} else if ((l * V) <= -5e-302) {
		tmp = (sqrt(-A) / sqrt((-V * l))) * c0;
	} else if ((l * V) <= 2e-238) {
		tmp = sqrt(((A / V) / l)) * c0;
	} else {
		tmp = (sqrt(A) * c0) / sqrt((l * V));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = (Math.sqrt((-A / l)) / Math.sqrt(-V)) * c0;
	} else if ((l * V) <= -5e-302) {
		tmp = (Math.sqrt(-A) / Math.sqrt((-V * l))) * c0;
	} else if ((l * V) <= 2e-238) {
		tmp = Math.sqrt(((A / V) / l)) * c0;
	} else {
		tmp = (Math.sqrt(A) * c0) / Math.sqrt((l * V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = (math.sqrt((-A / l)) / math.sqrt(-V)) * c0
	elif (l * V) <= -5e-302:
		tmp = (math.sqrt(-A) / math.sqrt((-V * l))) * c0
	elif (l * V) <= 2e-238:
		tmp = math.sqrt(((A / V) / l)) * c0
	else:
		tmp = (math.sqrt(A) * c0) / math.sqrt((l * V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(Float64(-A) / l)) / sqrt(Float64(-V))) * c0);
	elseif (Float64(l * V) <= -5e-302)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-V) * l))) * c0);
	elseif (Float64(l * V) <= 2e-238)
		tmp = Float64(sqrt(Float64(Float64(A / V) / l)) * c0);
	else
		tmp = Float64(Float64(sqrt(A) * c0) / sqrt(Float64(l * V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = (sqrt((-A / l)) / sqrt(-V)) * c0;
	elseif ((l * V) <= -5e-302)
		tmp = (sqrt(-A) / sqrt((-V * l))) * c0;
	elseif ((l * V) <= 2e-238)
		tmp = sqrt(((A / V) / l)) * c0;
	else
		tmp = (sqrt(A) * c0) / sqrt((l * V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], N[(N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -5e-302], N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-V) * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 2e-238], N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision], N[(N[(N[Sqrt[A], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 36.6%

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

      1. lift-/.f64 N/A

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

      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 62.0

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

   4. Applied rewrites 62.0%

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

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. associate-*l/ N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. un-div-inv N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-neg.f64 34.2

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

   6. Applied rewrites 34.2%

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

   if -inf.0 < (*.f64 V l) < -5.00000000000000033e-302

   1. Initial program 89.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/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 82.2

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

   4. Applied rewrites 82.2%

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

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 99.4

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

   6. Applied rewrites 99.4%

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

   if -5.00000000000000033e-302 < (*.f64 V l) < 2e-238

   1. Initial program 54.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/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 73.2

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

   4. Applied rewrites 73.2%

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

   if 2e-238 < (*.f64 V l)

   1. Initial program 86.0%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. *-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 90.7

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

      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 90.7

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

   4. Applied rewrites 90.7%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-302}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-V\right) \cdot \ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{-238}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;t\_0 \leq 10^{+293}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \end{array} \end{array} \]

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 (* l V))) c0)))
   (if (<= t_0 0.0)
     (* (sqrt (/ (/ A l) V)) c0)
     (if (<= t_0 1e+293) 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 / (l * V))) * c0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = sqrt(((A / l) / V)) * c0;
	} else if (t_0 <= 1e+293) {
		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 / (l * v))) * c0
    if (t_0 <= 0.0d0) then
        tmp = sqrt(((a / l) / v)) * c0
    else if (t_0 <= 1d+293) 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 / (l * V))) * c0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.sqrt(((A / l) / V)) * c0;
	} else if (t_0 <= 1e+293) {
		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 / (l * V))) * c0
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.sqrt(((A / l) / V)) * c0
	elif t_0 <= 1e+293:
		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(l * V))) * c0)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(A / l) / V)) * c0);
	elseif (t_0 <= 1e+293)
		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 / (l * V))) * c0;
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = sqrt(((A / l) / V)) * c0;
	elseif (t_0 <= 1e+293)
		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[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[t$95$0, 1e+293], 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)))) < 0.0

   1. Initial program 74.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/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 72.1

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

   4. Applied rewrites 72.1%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 9.9999999999999992e292

   1. Initial program 98.5%

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

   if 9.9999999999999992e292 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 60.9%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 63.5

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

      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 63.5

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

   4. Applied rewrites 63.5%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{\ell \cdot V}{A}}}} \]
   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 77.8

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

   6. Applied rewrites 77.8%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0 \leq 0:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0 \leq 10^{+293}:\\ \;\;\;\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ t_1 := \sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-246}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* l V))) c0)) (t_1 (* (sqrt (/ (/ A V) l)) c0)))
   (if (<= t_0 5e-246) t_1 (if (<= t_0 2e+302) 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 / (l * V))) * c0;
	double t_1 = sqrt(((A / V) / l)) * c0;
	double tmp;
	if (t_0 <= 5e-246) {
		tmp = t_1;
	} else if (t_0 <= 2e+302) {
		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 / (l * v))) * c0
    t_1 = sqrt(((a / v) / l)) * c0
    if (t_0 <= 5d-246) then
        tmp = t_1
    else if (t_0 <= 2d+302) 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 / (l * V))) * c0;
	double t_1 = Math.sqrt(((A / V) / l)) * c0;
	double tmp;
	if (t_0 <= 5e-246) {
		tmp = t_1;
	} else if (t_0 <= 2e+302) {
		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 / (l * V))) * c0
	t_1 = math.sqrt(((A / V) / l)) * c0
	tmp = 0
	if t_0 <= 5e-246:
		tmp = t_1
	elif t_0 <= 2e+302:
		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(l * V))) * c0)
	t_1 = Float64(sqrt(Float64(Float64(A / V) / l)) * c0)
	tmp = 0.0
	if (t_0 <= 5e-246)
		tmp = t_1;
	elseif (t_0 <= 2e+302)
		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 / (l * V))) * c0;
	t_1 = sqrt(((A / V) / l)) * c0;
	tmp = 0.0;
	if (t_0 <= 5e-246)
		tmp = t_1;
	elseif (t_0 <= 2e+302)
		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[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-246], t$95$1, If[LessEqual[t$95$0, 2e+302], t$95$0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.9999999999999997e-246 or 2.0000000000000002e302 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 72.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 73.1

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

   4. Applied rewrites 73.1%

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

   if 4.9999999999999997e-246 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.0000000000000002e302

   1. Initial program 98.5%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0 \leq 5 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{elif}\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\sqrt{t\_0} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A} \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* l V))))
   (if (<= t_0 0.0)
     (* (sqrt (/ (/ A l) V)) c0)
     (if (<= t_0 5e+305) (* (sqrt t_0) c0) (/ c0 (sqrt (* (/ V A) l)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = sqrt(((A / l) / V)) * c0;
	} else if (t_0 <= 5e+305) {
		tmp = sqrt(t_0) * c0;
	} else {
		tmp = c0 / sqrt(((V / A) * l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (l * v)
    if (t_0 <= 0.0d0) then
        tmp = sqrt(((a / l) / v)) * c0
    else if (t_0 <= 5d+305) then
        tmp = sqrt(t_0) * c0
    else
        tmp = c0 / sqrt(((v / a) * l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.sqrt(((A / l) / V)) * c0;
	} else if (t_0 <= 5e+305) {
		tmp = Math.sqrt(t_0) * c0;
	} else {
		tmp = c0 / Math.sqrt(((V / A) * l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (l * V)
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.sqrt(((A / l) / V)) * c0
	elif t_0 <= 5e+305:
		tmp = math.sqrt(t_0) * c0
	else:
		tmp = c0 / math.sqrt(((V / A) * l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(l * V))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(A / l) / V)) * c0);
	elseif (t_0 <= 5e+305)
		tmp = Float64(sqrt(t_0) * c0);
	else
		tmp = Float64(c0 / sqrt(Float64(Float64(V / A) * l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (l * V);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = sqrt(((A / l) / V)) * c0;
	elseif (t_0 <= 5e+305)
		tmp = sqrt(t_0) * c0;
	else
		tmp = c0 / sqrt(((V / A) * l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[t$95$0, 5e+305], N[(N[Sqrt[t$95$0], $MachinePrecision] * c0), $MachinePrecision], N[(c0 / N[Sqrt[N[(N[(V / A), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 48.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/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 60.4

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

   4. Applied rewrites 60.4%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 5.00000000000000009e305

   1. Initial program 99.1%

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

   if 5.00000000000000009e305 < (/.f64 A (*.f64 V l))

   1. Initial program 55.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 58.0

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

      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 58.0

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

   4. Applied rewrites 58.0%

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

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 70.3

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

   6. Applied rewrites 70.3%

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

4. Final simplification 85.3%

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

Alternative 5: 80.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ t_1 := \sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+283}:\\ \;\;\;\;\sqrt{t\_0} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* l V))) (t_1 (* (sqrt (/ (/ A l) V)) c0)))
   (if (<= t_0 0.0) t_1 (if (<= t_0 2e+283) (* (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 / (l * V);
	double t_1 = sqrt(((A / l) / V)) * c0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+283) {
		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 / (l * v)
    t_1 = sqrt(((a / l) / v)) * c0
    if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 2d+283) 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 / (l * V);
	double t_1 = Math.sqrt(((A / l) / V)) * c0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+283) {
		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 / (l * V)
	t_1 = math.sqrt(((A / l) / V)) * c0
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 2e+283:
		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(l * V))
	t_1 = Float64(sqrt(Float64(Float64(A / l) / V)) * c0)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+283)
		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 / (l * V);
	t_1 = sqrt(((A / l) / V)) * c0;
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+283)
		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[(l * V), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 2e+283], 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 1.99999999999999991e283 < (/.f64 A (*.f64 V l))

   1. Initial program 54.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 66.3

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

   4. Applied rewrites 66.3%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 1.99999999999999991e283

   1. Initial program 99.1%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 0:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;\frac{A}{\ell \cdot V} \leq 2 \cdot 10^{+283}:\\ \;\;\;\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-302}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-V\right) \cdot \ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{-238}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) (- INFINITY))
   (* (sqrt (/ A V)) (/ c0 (sqrt l)))
   (if (<= (* l V) -5e-302)
     (* (/ (sqrt (- A)) (sqrt (* (- V) l))) c0)
     (if (<= (* l V) 2e-238)
       (* (sqrt (/ (/ A V) l)) c0)
       (/ (* (sqrt A) c0) (sqrt (* l V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	} else if ((l * V) <= -5e-302) {
		tmp = (sqrt(-A) / sqrt((-V * l))) * c0;
	} else if ((l * V) <= 2e-238) {
		tmp = sqrt(((A / V) / l)) * c0;
	} else {
		tmp = (sqrt(A) * c0) / sqrt((l * V));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((A / V)) * (c0 / Math.sqrt(l));
	} else if ((l * V) <= -5e-302) {
		tmp = (Math.sqrt(-A) / Math.sqrt((-V * l))) * c0;
	} else if ((l * V) <= 2e-238) {
		tmp = Math.sqrt(((A / V) / l)) * c0;
	} else {
		tmp = (Math.sqrt(A) * c0) / Math.sqrt((l * V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -math.inf:
		tmp = math.sqrt((A / V)) * (c0 / math.sqrt(l))
	elif (l * V) <= -5e-302:
		tmp = (math.sqrt(-A) / math.sqrt((-V * l))) * c0
	elif (l * V) <= 2e-238:
		tmp = math.sqrt(((A / V) / l)) * c0
	else:
		tmp = (math.sqrt(A) * c0) / math.sqrt((l * V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(A / V)) * Float64(c0 / sqrt(l)));
	elseif (Float64(l * V) <= -5e-302)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-V) * l))) * c0);
	elseif (Float64(l * V) <= 2e-238)
		tmp = Float64(sqrt(Float64(Float64(A / V) / l)) * c0);
	else
		tmp = Float64(Float64(sqrt(A) * c0) / sqrt(Float64(l * V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -Inf)
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	elseif ((l * V) <= -5e-302)
		tmp = (sqrt(-A) / sqrt((-V * l))) * c0;
	elseif ((l * V) <= 2e-238)
		tmp = sqrt(((A / V) / l)) * c0;
	else
		tmp = (sqrt(A) * c0) / sqrt((l * V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -5e-302], N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-V) * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 2e-238], N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision], N[(N[(N[Sqrt[A], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 36.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 33.9

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

   4. Applied rewrites 33.9%

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

   if -inf.0 < (*.f64 V l) < -5.00000000000000033e-302

   1. Initial program 89.4%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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-/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 82.2

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

   4. Applied rewrites 82.2%

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

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 99.4

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

   6. Applied rewrites 99.4%

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

   if -5.00000000000000033e-302 < (*.f64 V l) < 2e-238

   1. Initial program 54.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/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 73.2

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

   4. Applied rewrites 73.2%

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

   if 2e-238 < (*.f64 V l)

   1. Initial program 86.0%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. *-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 90.7

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

      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 90.7

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

   4. Applied rewrites 90.7%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-302}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-V\right) \cdot \ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{-238}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -5 \cdot 10^{-302}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-V\right) \cdot \ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{-238}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) -5e-302)
   (* (/ (sqrt (- A)) (sqrt (* (- V) l))) c0)
   (if (<= (* l V) 2e-238)
     (* (sqrt (/ (/ A V) l)) c0)
     (/ (* (sqrt A) c0) (sqrt (* l V))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -5e-302) {
		tmp = (sqrt(-A) / sqrt((-V * l))) * c0;
	} else if ((l * V) <= 2e-238) {
		tmp = sqrt(((A / V) / l)) * c0;
	} else {
		tmp = (sqrt(A) * c0) / sqrt((l * V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((l * v) <= (-5d-302)) then
        tmp = (sqrt(-a) / sqrt((-v * l))) * c0
    else if ((l * v) <= 2d-238) then
        tmp = sqrt(((a / v) / l)) * c0
    else
        tmp = (sqrt(a) * c0) / sqrt((l * v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -5e-302) {
		tmp = (Math.sqrt(-A) / Math.sqrt((-V * l))) * c0;
	} else if ((l * V) <= 2e-238) {
		tmp = Math.sqrt(((A / V) / l)) * c0;
	} else {
		tmp = (Math.sqrt(A) * c0) / Math.sqrt((l * V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -5e-302:
		tmp = (math.sqrt(-A) / math.sqrt((-V * l))) * c0
	elif (l * V) <= 2e-238:
		tmp = math.sqrt(((A / V) / l)) * c0
	else:
		tmp = (math.sqrt(A) * c0) / math.sqrt((l * V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= -5e-302)
		tmp = Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(Float64(-V) * l))) * c0);
	elseif (Float64(l * V) <= 2e-238)
		tmp = Float64(sqrt(Float64(Float64(A / V) / l)) * c0);
	else
		tmp = Float64(Float64(sqrt(A) * c0) / sqrt(Float64(l * V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -5e-302)
		tmp = (sqrt(-A) / sqrt((-V * l))) * c0;
	elseif ((l * V) <= 2e-238)
		tmp = sqrt(((A / V) / l)) * c0;
	else
		tmp = (sqrt(A) * c0) / sqrt((l * V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], -5e-302], N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[((-V) * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 2e-238], N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision], N[(N[(N[Sqrt[A], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -5.00000000000000033e-302

   1. Initial program 82.0%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 79.4

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

   4. Applied rewrites 79.4%

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

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 90.6

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

   6. Applied rewrites 90.6%

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

   if -5.00000000000000033e-302 < (*.f64 V l) < 2e-238

   1. Initial program 54.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/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 73.2

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

   4. Applied rewrites 73.2%

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

   if 2e-238 < (*.f64 V l)

   1. Initial program 86.0%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. *-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 90.7

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

      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 90.7

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

   4. Applied rewrites 90.7%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -5 \cdot 10^{-302}:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{\left(-V\right) \cdot \ell}} \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 2 \cdot 10^{-238}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c0}{\sqrt{-V} \cdot \sqrt{\frac{-\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{V} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= V -5e-310)
   (/ c0 (* (sqrt (- V)) (sqrt (/ (- l) A))))
   (/ (* (sqrt A) c0) (* (sqrt V) (sqrt l)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (V <= -5e-310) {
		tmp = c0 / (sqrt(-V) * sqrt((-l / A)));
	} else {
		tmp = (sqrt(A) * c0) / (sqrt(V) * sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (v <= (-5d-310)) then
        tmp = c0 / (sqrt(-v) * sqrt((-l / a)))
    else
        tmp = (sqrt(a) * c0) / (sqrt(v) * sqrt(l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (V <= -5e-310) {
		tmp = c0 / (Math.sqrt(-V) * Math.sqrt((-l / A)));
	} else {
		tmp = (Math.sqrt(A) * c0) / (Math.sqrt(V) * Math.sqrt(l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if V <= -5e-310:
		tmp = c0 / (math.sqrt(-V) * math.sqrt((-l / A)))
	else:
		tmp = (math.sqrt(A) * c0) / (math.sqrt(V) * math.sqrt(l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (V <= -5e-310)
		tmp = Float64(c0 / Float64(sqrt(Float64(-V)) * sqrt(Float64(Float64(-l) / A))));
	else
		tmp = Float64(Float64(sqrt(A) * c0) / Float64(sqrt(V) * sqrt(l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (V <= -5e-310)
		tmp = c0 / (sqrt(-V) * sqrt((-l / A)));
	else
		tmp = (sqrt(A) * c0) / (sqrt(V) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[V, -5e-310], N[(c0 / N[(N[Sqrt[(-V)], $MachinePrecision] * N[Sqrt[N[((-l) / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[A], $MachinePrecision] * c0), $MachinePrecision] / N[(N[Sqrt[V], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if V < -4.999999999999985e-310

   1. Initial program 78.7%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 79.8

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

      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 79.8

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

   4. Applied rewrites 79.8%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{\ell \cdot V}{A}}}} \]
   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 85.9

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

   6. Applied rewrites 85.9%

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

   if -4.999999999999985e-310 < V

   1. Initial program 81.0%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. *-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 47.8

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

      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 47.8

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

   4. Applied rewrites 47.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 55.3

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

   6. Applied rewrites 55.3%

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

4. Final simplification 70.8%

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

Alternative 9: 83.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \leq -2.1 \cdot 10^{-297}:\\ \;\;\;\;\frac{c0}{\sqrt{-V} \cdot \sqrt{\frac{-\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= V -2.1e-297)
   (/ c0 (* (sqrt (- V)) (sqrt (/ (- l) A))))
   (* (sqrt (/ A V)) (/ c0 (sqrt l)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (V <= -2.1e-297) {
		tmp = c0 / (sqrt(-V) * sqrt((-l / A)));
	} else {
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (v <= (-2.1d-297)) then
        tmp = c0 / (sqrt(-v) * sqrt((-l / a)))
    else
        tmp = sqrt((a / v)) * (c0 / sqrt(l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (V <= -2.1e-297) {
		tmp = c0 / (Math.sqrt(-V) * Math.sqrt((-l / A)));
	} else {
		tmp = Math.sqrt((A / V)) * (c0 / Math.sqrt(l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if V <= -2.1e-297:
		tmp = c0 / (math.sqrt(-V) * math.sqrt((-l / A)))
	else:
		tmp = math.sqrt((A / V)) * (c0 / math.sqrt(l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (V <= -2.1e-297)
		tmp = Float64(c0 / Float64(sqrt(Float64(-V)) * sqrt(Float64(Float64(-l) / A))));
	else
		tmp = Float64(sqrt(Float64(A / V)) * Float64(c0 / sqrt(l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (V <= -2.1e-297)
		tmp = c0 / (sqrt(-V) * sqrt((-l / A)));
	else
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[V, -2.1e-297], N[(c0 / N[(N[Sqrt[(-V)], $MachinePrecision] * N[Sqrt[N[((-l) / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if V < -2.10000000000000013e-297

   1. Initial program 78.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 79.2

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

      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 79.2

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

   4. Applied rewrites 79.2%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{\ell \cdot V}{A}}}} \]
   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 86.2

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

   6. Applied rewrites 86.2%

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

   if -2.10000000000000013e-297 < V

   1. Initial program 81.6%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 45.5

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

   4. Applied rewrites 45.5%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \leq -2.1 \cdot 10^{-297}:\\ \;\;\;\;\frac{c0}{\sqrt{-V} \cdot \sqrt{\frac{-\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq 2 \cdot 10^{-238}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) 2e-238)
   (/ c0 (sqrt (* (/ l A) V)))
   (/ (* (sqrt A) c0) (sqrt (* l V)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= 2e-238) {
		tmp = c0 / sqrt(((l / A) * V));
	} else {
		tmp = (sqrt(A) * c0) / sqrt((l * V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((l * v) <= 2d-238) then
        tmp = c0 / sqrt(((l / a) * v))
    else
        tmp = (sqrt(a) * c0) / sqrt((l * v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= 2e-238) {
		tmp = c0 / Math.sqrt(((l / A) * V));
	} else {
		tmp = (Math.sqrt(A) * c0) / Math.sqrt((l * V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= 2e-238:
		tmp = c0 / math.sqrt(((l / A) * V))
	else:
		tmp = (math.sqrt(A) * c0) / math.sqrt((l * V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= 2e-238)
		tmp = Float64(c0 / sqrt(Float64(Float64(l / A) * V)));
	else
		tmp = Float64(Float64(sqrt(A) * c0) / sqrt(Float64(l * V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= 2e-238)
		tmp = c0 / sqrt(((l / A) * V));
	else
		tmp = (sqrt(A) * c0) / sqrt((l * V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], 2e-238], N[(c0 / N[Sqrt[N[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[A], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 V l) < 2e-238

   1. Initial program 75.0%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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 75.9

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

      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 75.9

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

   4. Applied rewrites 75.9%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{\ell \cdot V}{A}}}} \]
   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 73.1

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

   6. Applied rewrites 73.1%

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

   if 2e-238 < (*.f64 V l)

   1. Initial program 86.0%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. *-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 90.7

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

      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 90.7

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

   4. Applied rewrites 90.7%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq 2 \cdot 10^{-238}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.1% accurate, 1.0× speedup

Math

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

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 (* l V))) c0))

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 79.9%

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

3. Final simplification 79.9%

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

Reproduce

herbie shell --seed 2024273 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))