Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.7% → 91.2%

Time: 10.9s

Alternatives: 9

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: 73.7% 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: 91.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\frac{\sqrt{-V}}{\sqrt{\frac{A}{-\ell}}}}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-294}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-319}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+281}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ c0 (/ (sqrt (- V)) (sqrt (/ A (- l)))))))
   (if (<= (* V l) (- INFINITY))
     t_0
     (if (<= (* V l) -1e-294)
       (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
       (if (<= (* V l) 4e-319)
         t_0
         (if (<= (* V l) 2e+281)
           (* c0 (/ (sqrt A) (sqrt (* V l))))
           (* c0 (sqrt (/ (/ A V) l)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 / (sqrt(-V) / sqrt((A / -l)));
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((V * l) <= -1e-294) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 4e-319) {
		tmp = t_0;
	} else if ((V * l) <= 2e+281) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 / (Math.sqrt(-V) / Math.sqrt((A / -l)));
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if ((V * l) <= -1e-294) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 4e-319) {
		tmp = t_0;
	} else if ((V * l) <= 2e+281) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 / (math.sqrt(-V) / math.sqrt((A / -l)))
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = t_0
	elif (V * l) <= -1e-294:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 4e-319:
		tmp = t_0
	elif (V * l) <= 2e+281:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 / Float64(sqrt(Float64(-V)) / sqrt(Float64(A / Float64(-l)))))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(V * l) <= -1e-294)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 4e-319)
		tmp = t_0;
	elseif (Float64(V * l) <= 2e+281)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 / (sqrt(-V) / sqrt((A / -l)));
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = t_0;
	elseif ((V * l) <= -1e-294)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 4e-319)
		tmp = t_0;
	elseif ((V * l) <= 2e+281)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 / N[(N[Sqrt[(-V)], $MachinePrecision] / N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -1e-294], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e-319], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], 2e+281], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -inf.0 or -1.00000000000000002e-294 < (*.f64 V l) < 4.0000049e-319

   1. Initial program 42.4%

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

      1. clear-num 42.4%

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

      2. associate-/r/ 40.7%

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

      3. associate-/r* 40.7%

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

   4. Applied egg-rr 40.7%

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

      1. *-commutative 40.7%

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

      2. associate-/l/ 40.7%

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

      3. div-inv 42.4%

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

      4. frac-2neg 42.4%

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

      5. distribute-lft-neg-out 42.4%

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

      6. *-commutative 42.4%

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

      7. sqrt-undiv 31.4%

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

      8. clear-num 31.4%

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

      9. un-div-inv 31.4%

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

      10. sqrt-undiv 42.4%

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

      11. add-sqr-sqrt 21.6%

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

      12. sqrt-unprod 21.7%

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

      13. sqr-neg 21.7%

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

      14. sqrt-unprod 0.1%

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

      15. add-sqr-sqrt 0.2%

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

      16. add-sqr-sqrt 0.1%

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

      17. sqrt-unprod 9.8%

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

      18. sqr-neg 9.8%

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

      19. sqrt-unprod 11.0%

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

      20. add-sqr-sqrt 42.4%

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

   6. Applied egg-rr 42.4%

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

      1. associate-/l* 72.5%

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

   8. Simplified 72.5%

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

      1. clear-num 72.5%

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

      2. div-inv 72.5%

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

      3. frac-2neg 72.5%

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

      4. sqrt-div 54.5%

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

      5. distribute-neg-frac2 54.5%

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

   10. Applied egg-rr 54.5%

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

       1. distribute-frac-neg2 54.5%

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

       2. distribute-frac-neg 54.5%

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

   12. Simplified 54.5%

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

   if -inf.0 < (*.f64 V l) < -1.00000000000000002e-294

   1. Initial program 88.9%

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

      1. frac-2neg 88.9%

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

      2. sqrt-div 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

   4. Applied egg-rr 99.4%

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

   if 4.0000049e-319 < (*.f64 V l) < 2.0000000000000001e281

   1. Initial program 87.4%

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

      1. sqrt-div 99.5%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.5%

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

      2. *-rgt-identity 99.5%

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

   6. Simplified 99.5%

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

   if 2.0000000000000001e281 < (*.f64 V l)

   1. Initial program 19.4%

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

      1. associate-/r* 67.4%

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

   3. Simplified 67.4%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{-V}}{\sqrt{\frac{A}{-\ell}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-294}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-319}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{-V}}{\sqrt{\frac{A}{-\ell}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+281}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 4 \cdot 10^{+276}\right):\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (or (<= t_0 0.0) (not (<= t_0 4e+276)))
     (/ c0 (sqrt (* V (/ l A))))
     t_0)))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 4e+276)) {
		tmp = c0 / sqrt((V * (l / A)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 4d+276))) then
        tmp = c0 / sqrt((v * (l / a)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 4e+276)) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 4e+276):
		tmp = c0 / math.sqrt((V * (l / A)))
	else:
		tmp = t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 4e+276))
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 4e+276)))
		tmp = c0 / sqrt((V * (l / A)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 4e+276]], $MachinePrecision]], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 0.0 or 4.0000000000000002e276 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 67.2%

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

      1. clear-num 66.3%

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

      2. associate-/r/ 66.7%

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

      3. associate-/r* 67.3%

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

   4. Applied egg-rr 67.3%

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

      1. *-commutative 67.3%

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

      2. associate-/l/ 66.7%

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

      3. div-inv 67.2%

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

      4. frac-2neg 67.2%

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

      5. distribute-lft-neg-out 67.2%

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

      6. *-commutative 67.2%

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

      7. sqrt-undiv 41.2%

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

      8. clear-num 41.1%

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

      9. un-div-inv 41.2%

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

      10. sqrt-undiv 66.9%

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

      11. add-sqr-sqrt 33.7%

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

      12. sqrt-unprod 22.3%

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

      13. sqr-neg 22.3%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 0.1%

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

      16. add-sqr-sqrt 0.0%

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

      17. sqrt-unprod 22.4%

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

      18. sqr-neg 22.4%

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

      19. sqrt-unprod 31.5%

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

      20. add-sqr-sqrt 66.9%

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

   6. Applied egg-rr 66.9%

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

      1. associate-/l* 72.5%

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

   8. Simplified 72.5%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.0000000000000002e276

   1. Initial program 98.6%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0 \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 4 \cdot 10^{+276}\right):\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 4 \cdot 10^{+231}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (or (<= t_0 0.0) (not (<= t_0 4e+231)))
     (* c0 (sqrt (/ (/ A V) l)))
     t_0)))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 4e+231)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 4d+231))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 4e+231)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 4e+231):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 4e+231))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 4e+231)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 4e+231]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 0.0 or 4.0000000000000002e231 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 67.9%

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

      1. associate-/r* 74.3%

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

   3. Simplified 74.3%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.0000000000000002e231

   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 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0 \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 4 \cdot 10^{+231}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-269}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+276}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 4e-269)
     (/ c0 (sqrt (* l (/ V A))))
     (if (<= t_0 4e+276) t_0 (/ c0 (sqrt (* V (/ l A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 4e-269) {
		tmp = c0 / sqrt((l * (V / A)));
	} else if (t_0 <= 4e+276) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 4d-269) then
        tmp = c0 / sqrt((l * (v / a)))
    else if (t_0 <= 4d+276) then
        tmp = t_0
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 4e-269) {
		tmp = c0 / Math.sqrt((l * (V / A)));
	} else if (t_0 <= 4e+276) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 4e-269:
		tmp = c0 / math.sqrt((l * (V / A)))
	elif t_0 <= 4e+276:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 4e-269)
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	elseif (t_0 <= 4e+276)
		tmp = t_0;
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 4e-269)
		tmp = c0 / sqrt((l * (V / A)));
	elseif (t_0 <= 4e+276)
		tmp = t_0;
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-269], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+276], t$95$0, N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 3.9999999999999998e-269

   1. Initial program 67.8%

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

      1. clear-num 66.8%

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

      2. associate-/r/ 67.5%

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

      3. associate-/r* 68.1%

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

   4. Applied egg-rr 68.1%

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

      1. *-commutative 68.1%

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

      2. associate-/l/ 67.5%

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

      3. div-inv 67.8%

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

      4. frac-2neg 67.8%

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

      5. distribute-lft-neg-out 67.8%

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

      6. *-commutative 67.8%

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

      7. sqrt-undiv 40.2%

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

      8. clear-num 40.2%

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

      9. un-div-inv 40.2%

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

      10. sqrt-undiv 67.5%

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

      11. add-sqr-sqrt 34.4%

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

      12. sqrt-unprod 20.8%

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

      13. sqr-neg 20.8%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 0.0%

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

      16. add-sqr-sqrt 0.0%

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

      17. sqrt-unprod 21.8%

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

      18. sqr-neg 21.8%

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

      19. sqrt-unprod 32.7%

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

      20. add-sqr-sqrt 67.5%

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

   6. Applied egg-rr 67.5%

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

      1. associate-*l/ 71.6%

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

      2. *-commutative 71.6%

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

   8. Simplified 71.6%

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

   if 3.9999999999999998e-269 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.0000000000000002e276

   1. Initial program 98.5%

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

   if 4.0000000000000002e276 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 65.9%

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

      1. clear-num 65.9%

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

      2. associate-/r/ 64.7%

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

      3. associate-/r* 64.7%

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

   4. Applied egg-rr 64.7%

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

      1. *-commutative 64.7%

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

      2. associate-/l/ 64.7%

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

      3. div-inv 65.9%

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

      4. frac-2neg 65.9%

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

      5. distribute-lft-neg-out 65.9%

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

      6. *-commutative 65.9%

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

      7. sqrt-undiv 47.9%

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

      8. clear-num 47.8%

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

      9. un-div-inv 48.0%

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

      10. sqrt-undiv 65.9%

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

      11. add-sqr-sqrt 30.1%

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

      12. sqrt-unprod 30.2%

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

      13. sqr-neg 30.2%

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

      14. sqrt-unprod 0.1%

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

      15. add-sqr-sqrt 0.1%

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

      16. add-sqr-sqrt 0.1%

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

      17. sqrt-unprod 25.6%

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

      18. sqr-neg 25.6%

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

      19. sqrt-unprod 25.5%

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

      20. add-sqr-sqrt 65.9%

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

   6. Applied egg-rr 65.9%

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

      1. associate-/l* 76.5%

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

   8. Simplified 76.5%

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

Alternative 5: 90.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-270}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V} \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+281}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (/ c0 (* (sqrt (/ V A)) (sqrt l)))
   (if (<= (* V l) -2e-270)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 4e-319)
       (* c0 (sqrt (* (/ A V) (/ 1.0 l))))
       (if (<= (* V l) 2e+281)
         (* c0 (/ (sqrt A) (sqrt (* V l))))
         (* c0 (sqrt (/ (/ A V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	} else if ((V * l) <= -2e-270) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 4e-319) {
		tmp = c0 * sqrt(((A / V) * (1.0 / l)));
	} else if ((V * l) <= 2e+281) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	} else if ((V * l) <= -2e-270) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 4e-319) {
		tmp = c0 * Math.sqrt(((A / V) * (1.0 / l)));
	} else if ((V * l) <= 2e+281) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 / (math.sqrt((V / A)) * math.sqrt(l))
	elif (V * l) <= -2e-270:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 4e-319:
		tmp = c0 * math.sqrt(((A / V) * (1.0 / l)))
	elif (V * l) <= 2e+281:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * sqrt(l)));
	elseif (Float64(V * l) <= -2e-270)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 4e-319)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) * Float64(1.0 / l))));
	elseif (Float64(V * l) <= 2e+281)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	elseif ((V * l) <= -2e-270)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 4e-319)
		tmp = c0 * sqrt(((A / V) * (1.0 / l)));
	elseif ((V * l) <= 2e+281)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(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-270], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e-319], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e+281], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 36.3%

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

      1. clear-num 36.3%

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

      2. associate-/r/ 36.3%

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

      3. associate-/r* 36.3%

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

   4. Applied egg-rr 36.3%

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

      1. *-commutative 36.3%

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

      2. associate-/l/ 36.3%

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

      3. div-inv 36.3%

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

      4. frac-2neg 36.3%

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

      5. distribute-lft-neg-out 36.3%

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

      6. *-commutative 36.3%

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

      7. sqrt-undiv 36.3%

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

      8. clear-num 36.3%

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

      9. un-div-inv 36.3%

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

      10. sqrt-undiv 36.3%

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

      11. add-sqr-sqrt 11.1%

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

      12. sqrt-unprod 11.1%

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

      13. sqr-neg 11.1%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 0.0%

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

      16. add-sqr-sqrt 0.0%

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

      17. sqrt-unprod 0.0%

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

      18. sqr-neg 0.0%

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

      19. sqrt-unprod 0.0%

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

      20. add-sqr-sqrt 36.3%

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

   6. Applied egg-rr 36.3%

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

      1. associate-/l* 68.2%

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

   8. Simplified 68.2%

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

      1. clear-num 68.1%

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

      2. div-inv 68.0%

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

      3. associate-/r/ 68.2%

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

      4. sqrt-prod 52.2%

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

   10. Applied egg-rr 52.2%

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

   if -inf.0 < (*.f64 V l) < -2.0000000000000001e-270

   1. Initial program 88.7%

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

      1. frac-2neg 88.7%

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

      2. sqrt-div 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

   4. Applied egg-rr 99.4%

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

   if -2.0000000000000001e-270 < (*.f64 V l) < 4.0000049e-319

   1. Initial program 52.8%

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

      1. associate-/r* 78.8%

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

      2. div-inv 78.8%

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

   4. Applied egg-rr 78.8%

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

   if 4.0000049e-319 < (*.f64 V l) < 2.0000000000000001e281

   1. Initial program 87.4%

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

      1. sqrt-div 99.5%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.5%

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

      2. *-rgt-identity 99.5%

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

   6. Simplified 99.5%

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

   if 2.0000000000000001e281 < (*.f64 V l)

   1. Initial program 19.4%

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

      1. associate-/r* 67.4%

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

   3. Simplified 67.4%

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

Alternative 6: 86.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+143}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-164}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-319}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\ell \cdot \frac{V}{A}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+281}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -2e+143)
   (/ c0 (* (sqrt (/ V A)) (sqrt l)))
   (if (<= (* V l) -2e-164)
     (* c0 (sqrt (/ A (* V l))))
     (if (<= (* V l) 4e-319)
       (/ 1.0 (/ (sqrt (* l (/ V A))) c0))
       (if (<= (* V l) 2e+281)
         (* c0 (/ (sqrt A) (sqrt (* V l))))
         (* c0 (sqrt (/ (/ A V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+143) {
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	} else if ((V * l) <= -2e-164) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 4e-319) {
		tmp = 1.0 / (sqrt((l * (V / A))) / c0);
	} else if ((V * l) <= 2e+281) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-2d+143)) then
        tmp = c0 / (sqrt((v / a)) * sqrt(l))
    else if ((v * l) <= (-2d-164)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 4d-319) then
        tmp = 1.0d0 / (sqrt((l * (v / a))) / c0)
    else if ((v * l) <= 2d+281) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 * sqrt(((a / v) / l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+143) {
		tmp = c0 / (Math.sqrt((V / A)) * Math.sqrt(l));
	} else if ((V * l) <= -2e-164) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 4e-319) {
		tmp = 1.0 / (Math.sqrt((l * (V / A))) / c0);
	} else if ((V * l) <= 2e+281) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -2e+143:
		tmp = c0 / (math.sqrt((V / A)) * math.sqrt(l))
	elif (V * l) <= -2e-164:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 4e-319:
		tmp = 1.0 / (math.sqrt((l * (V / A))) / c0)
	elif (V * l) <= 2e+281:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -2e+143)
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) * sqrt(l)));
	elseif (Float64(V * l) <= -2e-164)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 4e-319)
		tmp = Float64(1.0 / Float64(sqrt(Float64(l * Float64(V / A))) / c0));
	elseif (Float64(V * l) <= 2e+281)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -2e+143)
		tmp = c0 / (sqrt((V / A)) * sqrt(l));
	elseif ((V * l) <= -2e-164)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 4e-319)
		tmp = 1.0 / (sqrt((l * (V / A))) / c0);
	elseif ((V * l) <= 2e+281)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -2e+143], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-164], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e-319], N[(1.0 / N[(N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e+281], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -2e143

   1. Initial program 62.9%

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

      1. clear-num 61.1%

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

      2. associate-/r/ 62.9%

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

      3. associate-/r* 62.9%

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

   4. Applied egg-rr 62.9%

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

      1. *-commutative 62.9%

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

      2. associate-/l/ 62.9%

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

      3. div-inv 62.9%

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

      4. frac-2neg 62.9%

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

      5. distribute-lft-neg-out 62.9%

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

      6. *-commutative 62.9%

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

      7. sqrt-undiv 74.0%

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

      8. clear-num 73.7%

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

      9. un-div-inv 73.8%

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

      10. sqrt-undiv 61.1%

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

      11. add-sqr-sqrt 23.7%

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

      12. sqrt-unprod 16.4%

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

      13. sqr-neg 16.4%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 0.0%

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

      16. add-sqr-sqrt 0.0%

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

      17. sqrt-unprod 0.1%

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

      18. sqr-neg 0.1%

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

      19. sqrt-unprod 0.0%

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

      20. add-sqr-sqrt 61.1%

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

   6. Applied egg-rr 61.1%

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

      1. associate-/l* 74.0%

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

   8. Simplified 74.0%

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

      1. clear-num 74.0%

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

      2. div-inv 73.9%

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

      3. associate-/r/ 72.1%

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

      4. sqrt-prod 53.6%

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

   10. Applied egg-rr 53.6%

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

   if -2e143 < (*.f64 V l) < -1.99999999999999992e-164

   1. Initial program 97.7%

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

   if -1.99999999999999992e-164 < (*.f64 V l) < 4.0000049e-319

   1. Initial program 61.4%

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

      1. clear-num 61.5%

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

      2. associate-/r/ 59.5%

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

      3. associate-/r* 59.5%

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

   4. Applied egg-rr 59.5%

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

      1. clear-num 59.5%

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

      2. associate-/r/ 59.5%

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

   6. Applied egg-rr 59.5%

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

      1. un-div-inv 59.5%

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

   8. Applied egg-rr 59.5%

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

      1. add-sqr-sqrt 59.5%

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

      2. associate-*r* 59.5%

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

      3. pow1/2 59.5%

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

      4. sqrt-pow1 59.5%

         \[\leadsto \left(c0 \cdot \color{blue}{{\left(\frac{\frac{1}{\ell}}{V} \cdot A\right)}^{\left(\frac{0.5}{2}\right)}}\right) \cdot \sqrt{\sqrt{\frac{\frac{1}{\ell}}{V} \cdot A}} \]

      5. *-commutative 59.5%

         \[\leadsto \left(c0 \cdot {\color{blue}{\left(A \cdot \frac{\frac{1}{\ell}}{V}\right)}}^{\left(\frac{0.5}{2}\right)}\right) \cdot \sqrt{\sqrt{\frac{\frac{1}{\ell}}{V} \cdot A}} \]

      6. associate-*r/ 59.2%

         \[\leadsto \left(c0 \cdot {\color{blue}{\left(\frac{A \cdot \frac{1}{\ell}}{V}\right)}}^{\left(\frac{0.5}{2}\right)}\right) \cdot \sqrt{\sqrt{\frac{\frac{1}{\ell}}{V} \cdot A}} \]

      7. div-inv 59.2%

         \[\leadsto \left(c0 \cdot {\left(\frac{\color{blue}{\frac{A}{\ell}}}{V}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot \sqrt{\sqrt{\frac{\frac{1}{\ell}}{V} \cdot A}} \]

      8. clear-num 59.3%

         \[\leadsto \left(c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V}{\frac{A}{\ell}}}\right)}}^{\left(\frac{0.5}{2}\right)}\right) \cdot \sqrt{\sqrt{\frac{\frac{1}{\ell}}{V} \cdot A}} \]

      9. sqrt-pow1 59.3%

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

      10. pow1/2 59.3%

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

      11. pow1/2 59.3%

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

      12. sqrt-pow1 59.3%

          \[\leadsto \left(c0 \cdot \sqrt{\sqrt{\frac{1}{\frac{V}{\frac{A}{\ell}}}}}\right) \cdot \color{blue}{{\left(\frac{\frac{1}{\ell}}{V} \cdot A\right)}^{\left(\frac{0.5}{2}\right)}} \]

      13. *-commutative 59.3%

          \[\leadsto \left(c0 \cdot \sqrt{\sqrt{\frac{1}{\frac{V}{\frac{A}{\ell}}}}}\right) \cdot {\color{blue}{\left(A \cdot \frac{\frac{1}{\ell}}{V}\right)}}^{\left(\frac{0.5}{2}\right)} \]

      14. associate-*r/ 76.9%

          \[\leadsto \left(c0 \cdot \sqrt{\sqrt{\frac{1}{\frac{V}{\frac{A}{\ell}}}}}\right) \cdot {\color{blue}{\left(\frac{A \cdot \frac{1}{\ell}}{V}\right)}}^{\left(\frac{0.5}{2}\right)} \]

      15. div-inv 76.9%

          \[\leadsto \left(c0 \cdot \sqrt{\sqrt{\frac{1}{\frac{V}{\frac{A}{\ell}}}}}\right) \cdot {\left(\frac{\color{blue}{\frac{A}{\ell}}}{V}\right)}^{\left(\frac{0.5}{2}\right)} \]

      16. clear-num 76.8%

          \[\leadsto \left(c0 \cdot \sqrt{\sqrt{\frac{1}{\frac{V}{\frac{A}{\ell}}}}}\right) \cdot {\color{blue}{\left(\frac{1}{\frac{V}{\frac{A}{\ell}}}\right)}}^{\left(\frac{0.5}{2}\right)} \]

      17. sqrt-pow1 76.7%

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

   10. Applied egg-rr 77.3%

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

   if 4.0000049e-319 < (*.f64 V l) < 2.0000000000000001e281

   1. Initial program 87.4%

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

      1. sqrt-div 99.5%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.5%

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

      2. *-rgt-identity 99.5%

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

   6. Simplified 99.5%

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

   if 2.0000000000000001e281 < (*.f64 V l)

   1. Initial program 19.4%

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

      1. associate-/r* 67.4%

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

   3. Simplified 67.4%

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

Alternative 7: 87.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+143}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-164}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-319}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\ell \cdot \frac{V}{A}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+281}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -2e+143)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* V l) -2e-164)
     (* c0 (sqrt (/ A (* V l))))
     (if (<= (* V l) 4e-319)
       (/ 1.0 (/ (sqrt (* l (/ V A))) c0))
       (if (<= (* V l) 2e+281)
         (* c0 (/ (sqrt A) (sqrt (* V l))))
         (* c0 (sqrt (/ (/ A V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+143) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= -2e-164) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 4e-319) {
		tmp = 1.0 / (sqrt((l * (V / A))) / c0);
	} else if ((V * l) <= 2e+281) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-2d+143)) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= (-2d-164)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 4d-319) then
        tmp = 1.0d0 / (sqrt((l * (v / a))) / c0)
    else if ((v * l) <= 2d+281) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 * sqrt(((a / v) / l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+143) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= -2e-164) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 4e-319) {
		tmp = 1.0 / (Math.sqrt((l * (V / A))) / c0);
	} else if ((V * l) <= 2e+281) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -2e+143:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= -2e-164:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 4e-319:
		tmp = 1.0 / (math.sqrt((l * (V / A))) / c0)
	elif (V * l) <= 2e+281:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -2e+143)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= -2e-164)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 4e-319)
		tmp = Float64(1.0 / Float64(sqrt(Float64(l * Float64(V / A))) / c0));
	elseif (Float64(V * l) <= 2e+281)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -2e+143)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= -2e-164)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 4e-319)
		tmp = 1.0 / (sqrt((l * (V / A))) / c0);
	elseif ((V * l) <= 2e+281)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -2e+143], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-164], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e-319], N[(1.0 / N[(N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e+281], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -2e143

   1. Initial program 62.9%

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

      1. associate-/r* 73.1%

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

      2. sqrt-div 53.6%

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

      3. div-inv 53.6%

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

   4. Applied egg-rr 53.6%

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

      1. associate-*r/ 53.6%

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

      2. *-rgt-identity 53.6%

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

   6. Simplified 53.6%

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

   if -2e143 < (*.f64 V l) < -1.99999999999999992e-164

   1. Initial program 97.7%

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

   if -1.99999999999999992e-164 < (*.f64 V l) < 4.0000049e-319

   1. Initial program 61.4%

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

      1. clear-num 61.5%

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

      2. associate-/r/ 59.5%

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

      3. associate-/r* 59.5%

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

   4. Applied egg-rr 59.5%

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

      1. clear-num 59.5%

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

      2. associate-/r/ 59.5%

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

   6. Applied egg-rr 59.5%

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

      1. un-div-inv 59.5%

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

   8. Applied egg-rr 59.5%

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

      1. add-sqr-sqrt 59.5%

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

      2. associate-*r* 59.5%

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

      3. pow1/2 59.5%

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

      4. sqrt-pow1 59.5%

         \[\leadsto \left(c0 \cdot \color{blue}{{\left(\frac{\frac{1}{\ell}}{V} \cdot A\right)}^{\left(\frac{0.5}{2}\right)}}\right) \cdot \sqrt{\sqrt{\frac{\frac{1}{\ell}}{V} \cdot A}} \]

      5. *-commutative 59.5%

         \[\leadsto \left(c0 \cdot {\color{blue}{\left(A \cdot \frac{\frac{1}{\ell}}{V}\right)}}^{\left(\frac{0.5}{2}\right)}\right) \cdot \sqrt{\sqrt{\frac{\frac{1}{\ell}}{V} \cdot A}} \]

      6. associate-*r/ 59.2%

         \[\leadsto \left(c0 \cdot {\color{blue}{\left(\frac{A \cdot \frac{1}{\ell}}{V}\right)}}^{\left(\frac{0.5}{2}\right)}\right) \cdot \sqrt{\sqrt{\frac{\frac{1}{\ell}}{V} \cdot A}} \]

      7. div-inv 59.2%

         \[\leadsto \left(c0 \cdot {\left(\frac{\color{blue}{\frac{A}{\ell}}}{V}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot \sqrt{\sqrt{\frac{\frac{1}{\ell}}{V} \cdot A}} \]

      8. clear-num 59.3%

         \[\leadsto \left(c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V}{\frac{A}{\ell}}}\right)}}^{\left(\frac{0.5}{2}\right)}\right) \cdot \sqrt{\sqrt{\frac{\frac{1}{\ell}}{V} \cdot A}} \]

      9. sqrt-pow1 59.3%

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

      10. pow1/2 59.3%

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

      11. pow1/2 59.3%

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

      12. sqrt-pow1 59.3%

          \[\leadsto \left(c0 \cdot \sqrt{\sqrt{\frac{1}{\frac{V}{\frac{A}{\ell}}}}}\right) \cdot \color{blue}{{\left(\frac{\frac{1}{\ell}}{V} \cdot A\right)}^{\left(\frac{0.5}{2}\right)}} \]

      13. *-commutative 59.3%

          \[\leadsto \left(c0 \cdot \sqrt{\sqrt{\frac{1}{\frac{V}{\frac{A}{\ell}}}}}\right) \cdot {\color{blue}{\left(A \cdot \frac{\frac{1}{\ell}}{V}\right)}}^{\left(\frac{0.5}{2}\right)} \]

      14. associate-*r/ 76.9%

          \[\leadsto \left(c0 \cdot \sqrt{\sqrt{\frac{1}{\frac{V}{\frac{A}{\ell}}}}}\right) \cdot {\color{blue}{\left(\frac{A \cdot \frac{1}{\ell}}{V}\right)}}^{\left(\frac{0.5}{2}\right)} \]

      15. div-inv 76.9%

          \[\leadsto \left(c0 \cdot \sqrt{\sqrt{\frac{1}{\frac{V}{\frac{A}{\ell}}}}}\right) \cdot {\left(\frac{\color{blue}{\frac{A}{\ell}}}{V}\right)}^{\left(\frac{0.5}{2}\right)} \]

      16. clear-num 76.8%

          \[\leadsto \left(c0 \cdot \sqrt{\sqrt{\frac{1}{\frac{V}{\frac{A}{\ell}}}}}\right) \cdot {\color{blue}{\left(\frac{1}{\frac{V}{\frac{A}{\ell}}}\right)}}^{\left(\frac{0.5}{2}\right)} \]

      17. sqrt-pow1 76.7%

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

   10. Applied egg-rr 77.3%

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

   if 4.0000049e-319 < (*.f64 V l) < 2.0000000000000001e281

   1. Initial program 87.4%

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

      1. sqrt-div 99.5%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.5%

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

      2. *-rgt-identity 99.5%

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

   6. Simplified 99.5%

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

   if 2.0000000000000001e281 < (*.f64 V l)

   1. Initial program 19.4%

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

      1. associate-/r* 67.4%

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

   3. Simplified 67.4%

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

Alternative 8: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-164}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-319}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\ell \cdot \frac{V}{A}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+281}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (/ c0 (sqrt (* V (/ l A))))
   (if (<= (* V l) -2e-164)
     (* c0 (sqrt (/ A (* V l))))
     (if (<= (* V l) 4e-319)
       (/ 1.0 (/ (sqrt (* l (/ V A))) c0))
       (if (<= (* V l) 2e+281)
         (* c0 (/ (sqrt A) (sqrt (* V l))))
         (* c0 (sqrt (/ (/ A V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 / sqrt((V * (l / A)));
	} else if ((V * l) <= -2e-164) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 4e-319) {
		tmp = 1.0 / (sqrt((l * (V / A))) / c0);
	} else if ((V * l) <= 2e+281) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else if ((V * l) <= -2e-164) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 4e-319) {
		tmp = 1.0 / (Math.sqrt((l * (V / A))) / c0);
	} else if ((V * l) <= 2e+281) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((A / V) / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 / math.sqrt((V * (l / A)))
	elif (V * l) <= -2e-164:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 4e-319:
		tmp = 1.0 / (math.sqrt((l * (V / A))) / c0)
	elif (V * l) <= 2e+281:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 * math.sqrt(((A / V) / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	elseif (Float64(V * l) <= -2e-164)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 4e-319)
		tmp = Float64(1.0 / Float64(sqrt(Float64(l * Float64(V / A))) / c0));
	elseif (Float64(V * l) <= 2e+281)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 / sqrt((V * (l / A)));
	elseif ((V * l) <= -2e-164)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 4e-319)
		tmp = 1.0 / (sqrt((l * (V / A))) / c0);
	elseif ((V * l) <= 2e+281)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 * sqrt(((A / V) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-164], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e-319], N[(1.0 / N[(N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e+281], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 36.3%

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

      1. clear-num 36.3%

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

      2. associate-/r/ 36.3%

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

      3. associate-/r* 36.3%

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

   4. Applied egg-rr 36.3%

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

      1. *-commutative 36.3%

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

      2. associate-/l/ 36.3%

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

      3. div-inv 36.3%

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

      4. frac-2neg 36.3%

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

      5. distribute-lft-neg-out 36.3%

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

      6. *-commutative 36.3%

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

      7. sqrt-undiv 36.3%

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

      8. clear-num 36.3%

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

      9. un-div-inv 36.3%

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

      10. sqrt-undiv 36.3%

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

      11. add-sqr-sqrt 11.1%

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

      12. sqrt-unprod 11.1%

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

      13. sqr-neg 11.1%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 0.0%

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

      16. add-sqr-sqrt 0.0%

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

      17. sqrt-unprod 0.0%

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

      18. sqr-neg 0.0%

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

      19. sqrt-unprod 0.0%

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

      20. add-sqr-sqrt 36.3%

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

   6. Applied egg-rr 36.3%

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

      1. associate-/l* 68.2%

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

   8. Simplified 68.2%

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

   if -inf.0 < (*.f64 V l) < -1.99999999999999992e-164

   1. Initial program 91.3%

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

   if -1.99999999999999992e-164 < (*.f64 V l) < 4.0000049e-319

   1. Initial program 61.4%

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

      1. clear-num 61.5%

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

      2. associate-/r/ 59.5%

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

      3. associate-/r* 59.5%

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

   4. Applied egg-rr 59.5%

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

      1. clear-num 59.5%

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

      2. associate-/r/ 59.5%

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

   6. Applied egg-rr 59.5%

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

      1. un-div-inv 59.5%

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

   8. Applied egg-rr 59.5%

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

      1. add-sqr-sqrt 59.5%

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

      2. associate-*r* 59.5%

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

      3. pow1/2 59.5%

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

      4. sqrt-pow1 59.5%

         \[\leadsto \left(c0 \cdot \color{blue}{{\left(\frac{\frac{1}{\ell}}{V} \cdot A\right)}^{\left(\frac{0.5}{2}\right)}}\right) \cdot \sqrt{\sqrt{\frac{\frac{1}{\ell}}{V} \cdot A}} \]

      5. *-commutative 59.5%

         \[\leadsto \left(c0 \cdot {\color{blue}{\left(A \cdot \frac{\frac{1}{\ell}}{V}\right)}}^{\left(\frac{0.5}{2}\right)}\right) \cdot \sqrt{\sqrt{\frac{\frac{1}{\ell}}{V} \cdot A}} \]

      6. associate-*r/ 59.2%

         \[\leadsto \left(c0 \cdot {\color{blue}{\left(\frac{A \cdot \frac{1}{\ell}}{V}\right)}}^{\left(\frac{0.5}{2}\right)}\right) \cdot \sqrt{\sqrt{\frac{\frac{1}{\ell}}{V} \cdot A}} \]

      7. div-inv 59.2%

         \[\leadsto \left(c0 \cdot {\left(\frac{\color{blue}{\frac{A}{\ell}}}{V}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot \sqrt{\sqrt{\frac{\frac{1}{\ell}}{V} \cdot A}} \]

      8. clear-num 59.3%

         \[\leadsto \left(c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V}{\frac{A}{\ell}}}\right)}}^{\left(\frac{0.5}{2}\right)}\right) \cdot \sqrt{\sqrt{\frac{\frac{1}{\ell}}{V} \cdot A}} \]

      9. sqrt-pow1 59.3%

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

      10. pow1/2 59.3%

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

      11. pow1/2 59.3%

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

      12. sqrt-pow1 59.3%

          \[\leadsto \left(c0 \cdot \sqrt{\sqrt{\frac{1}{\frac{V}{\frac{A}{\ell}}}}}\right) \cdot \color{blue}{{\left(\frac{\frac{1}{\ell}}{V} \cdot A\right)}^{\left(\frac{0.5}{2}\right)}} \]

      13. *-commutative 59.3%

          \[\leadsto \left(c0 \cdot \sqrt{\sqrt{\frac{1}{\frac{V}{\frac{A}{\ell}}}}}\right) \cdot {\color{blue}{\left(A \cdot \frac{\frac{1}{\ell}}{V}\right)}}^{\left(\frac{0.5}{2}\right)} \]

      14. associate-*r/ 76.9%

          \[\leadsto \left(c0 \cdot \sqrt{\sqrt{\frac{1}{\frac{V}{\frac{A}{\ell}}}}}\right) \cdot {\color{blue}{\left(\frac{A \cdot \frac{1}{\ell}}{V}\right)}}^{\left(\frac{0.5}{2}\right)} \]

      15. div-inv 76.9%

          \[\leadsto \left(c0 \cdot \sqrt{\sqrt{\frac{1}{\frac{V}{\frac{A}{\ell}}}}}\right) \cdot {\left(\frac{\color{blue}{\frac{A}{\ell}}}{V}\right)}^{\left(\frac{0.5}{2}\right)} \]

      16. clear-num 76.8%

          \[\leadsto \left(c0 \cdot \sqrt{\sqrt{\frac{1}{\frac{V}{\frac{A}{\ell}}}}}\right) \cdot {\color{blue}{\left(\frac{1}{\frac{V}{\frac{A}{\ell}}}\right)}}^{\left(\frac{0.5}{2}\right)} \]

      17. sqrt-pow1 76.7%

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

   10. Applied egg-rr 77.3%

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

   if 4.0000049e-319 < (*.f64 V l) < 2.0000000000000001e281

   1. Initial program 87.4%

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

      1. sqrt-div 99.5%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.5%

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

      2. *-rgt-identity 99.5%

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

   6. Simplified 99.5%

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

   if 2.0000000000000001e281 < (*.f64 V l)

   1. Initial program 19.4%

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

      1. associate-/r* 67.4%

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

   3. Simplified 67.4%

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

Alternative 9: 73.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

Reproduce

herbie shell --seed 2024137 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))