Henrywood and Agarwal, Equation (3)

Percentage Accurate: 74.5% → 89.6%

Time: 10.7s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;\ell \cdot V \leq -4 \cdot 10^{-284}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{0 - \mathsf{fma}\left(V, \ell, 0\right)}}{\sqrt{0 - A}}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) (- INFINITY))
   (/ c0 (* (sqrt l) (sqrt (/ V A))))
   (if (<= (* l V) -4e-284)
     (/ c0 (/ (sqrt (- 0.0 (fma V l 0.0))) (sqrt (- 0.0 A))))
     (if (<= (* l V) 0.0)
       (* (/ c0 (sqrt l)) (sqrt (/ A V)))
       (* c0 (/ (sqrt A) (sqrt (* l V))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	} else if ((l * V) <= -4e-284) {
		tmp = c0 / (sqrt((0.0 - fma(V, l, 0.0))) / sqrt((0.0 - A)));
	} else if ((l * V) <= 0.0) {
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	}
	return tmp;
}

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))));
	elseif (Float64(l * V) <= -4e-284)
		tmp = Float64(c0 / Float64(sqrt(Float64(0.0 - fma(V, l, 0.0))) / sqrt(Float64(0.0 - A))));
	elseif (Float64(l * V) <= 0.0)
		tmp = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(A / V)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	end
	return tmp
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -4e-284], N[(c0 / N[(N[Sqrt[N[(0.0 - N[(V * l + 0.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(0.0 - A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 41.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-lft-identity N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{0 + V \cdot \ell}} \cdot A} \]

      6. +-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{V \cdot \ell + 0}} \cdot A} \]

      7. accelerator-lowering-fma.f64 41.9

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}} \cdot A} \]

   4. Applied egg-rr 41.9%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{fma}\left(V, \ell, 0\right)} \cdot A}} \]
   5. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*l/ N/A

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

      3. sqrt-div N/A

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

      4. *-lft-identity N/A

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

      5. pow1/2 N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

      8. associate-/l/ N/A

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

      9. pow1/2 N/A

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

      10. sqrt-div N/A

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

      11. associate-*r/ N/A

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

      12. clear-num N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/r/ N/A

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

      16. sqrt-lowering-sqrt.f64 N/A

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

      17. associate-/r/ N/A

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

      18. associate-*l/ N/A

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

      19. *-commutative N/A

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

      20. /-lowering-/.f64 N/A

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

      21. +-rgt-identity N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell + 0}}{A}}} \]

      22. accelerator-lowering-fma.f64 41.9

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}{A}}} \]

   6. Applied egg-rr 41.9%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{\mathsf{fma}\left(V, \ell, 0\right)}{A}}}} \]
   7. Step-by-step derivation

      1. sqrt-div N/A

         \[\leadsto \frac{c0}{\color{blue}{\frac{\sqrt{V \cdot \ell + 0}}{\sqrt{A}}}} \]

      2. +-rgt-identity N/A

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

      3. sqrt-prod N/A

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

      4. associate-*l/ N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-undiv N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. sqrt-lowering-sqrt.f64 43.7

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

   8. Applied egg-rr 43.7%

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

   if -inf.0 < (*.f64 V l) < -4.00000000000000015e-284

   1. Initial program 85.7%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-lft-identity N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{0 + V \cdot \ell}} \cdot A} \]

      6. +-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{V \cdot \ell + 0}} \cdot A} \]

      7. accelerator-lowering-fma.f64 85.8

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}} \cdot A} \]

   4. Applied egg-rr 85.8%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{fma}\left(V, \ell, 0\right)} \cdot A}} \]
   5. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*l/ N/A

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

      3. sqrt-div N/A

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

      4. *-lft-identity N/A

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

      5. pow1/2 N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

      8. associate-/l/ N/A

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

      9. pow1/2 N/A

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

      10. sqrt-div N/A

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

      11. associate-*r/ N/A

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

      12. clear-num N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/r/ N/A

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

      16. sqrt-lowering-sqrt.f64 N/A

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

      17. associate-/r/ N/A

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

      18. associate-*l/ N/A

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

      19. *-commutative N/A

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

      20. /-lowering-/.f64 N/A

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

      21. +-rgt-identity N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell + 0}}{A}}} \]

      22. accelerator-lowering-fma.f64 86.6

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}{A}}} \]

   6. Applied egg-rr 86.6%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{\mathsf{fma}\left(V, \ell, 0\right)}{A}}}} \]
   7. Step-by-step derivation

      1. frac-2neg N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\mathsf{neg}\left(\left(V \cdot \ell + 0\right)\right)}{\mathsf{neg}\left(A\right)}}}} \]

      2. +-rgt-identity N/A

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

      3. sqrt-div N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. neg-sub0 N/A

         \[\leadsto \frac{c0}{\frac{\sqrt{\color{blue}{0 - V \cdot \ell}}}{\sqrt{\mathsf{neg}\left(A\right)}}} \]

      7. --lowering--.f64 N/A

         \[\leadsto \frac{c0}{\frac{\sqrt{\color{blue}{0 - V \cdot \ell}}}{\sqrt{\mathsf{neg}\left(A\right)}}} \]

      8. +-rgt-identity N/A

         \[\leadsto \frac{c0}{\frac{\sqrt{0 - \color{blue}{\left(V \cdot \ell + 0\right)}}}{\sqrt{\mathsf{neg}\left(A\right)}}} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{c0}{\frac{\sqrt{0 - \color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}}{\sqrt{\mathsf{neg}\left(A\right)}}} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{c0}{\frac{\sqrt{0 - \mathsf{fma}\left(V, \ell, 0\right)}}{\color{blue}{\sqrt{\mathsf{neg}\left(A\right)}}}} \]

      11. neg-sub0 N/A

          \[\leadsto \frac{c0}{\frac{\sqrt{0 - \mathsf{fma}\left(V, \ell, 0\right)}}{\sqrt{\color{blue}{0 - A}}}} \]

      12. --lowering--.f64 99.6

          \[\leadsto \frac{c0}{\frac{\sqrt{0 - \mathsf{fma}\left(V, \ell, 0\right)}}{\sqrt{\color{blue}{0 - A}}}} \]

   8. Applied egg-rr 99.6%

      \[\leadsto \frac{c0}{\color{blue}{\frac{\sqrt{0 - \mathsf{fma}\left(V, \ell, 0\right)}}{\sqrt{0 - A}}}} \]

   if -4.00000000000000015e-284 < (*.f64 V l) < -0.0

   1. Initial program 34.7%

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

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. /-lowering-/.f64 50.1

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

   4. Applied egg-rr 50.1%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 81.5%

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

      1. sqrt-div N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. +-lft-identity N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{0 + V \cdot \ell}}} \]

      6. +-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell + 0}}} \]

      7. accelerator-lowering-fma.f64 92.0

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}} \]

   4. Applied egg-rr 92.0%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}}} \]
   5. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 92.0

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

   6. Applied egg-rr 92.0%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;\ell \cdot V \leq -4 \cdot 10^{-284}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{0 - \mathsf{fma}\left(V, \ell, 0\right)}}{\sqrt{0 - A}}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+263}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

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 (/ A (* l V))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 2e+263) (* c0 (sqrt 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 = A / (l * V);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 2e+263) {
		tmp = c0 * sqrt(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 = a / (l * v)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 2d+263) then
        tmp = c0 * sqrt(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 = A / (l * V);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 2e+263) {
		tmp = c0 * Math.sqrt(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 = A / (l * V)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 2e+263:
		tmp = c0 * math.sqrt(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(A / Float64(l * V))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 2e+263)
		tmp = Float64(c0 * sqrt(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 = A / (l * V);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 2e+263)
		tmp = c0 * sqrt(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[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+263], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 43.7%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 59.1

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

   4. Applied egg-rr 59.1%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 2.00000000000000003e263

   1. Initial program 98.3%

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

   if 2.00000000000000003e263 < (/.f64 A (*.f64 V l))

   1. Initial program 48.5%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 53.6

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

   4. Applied egg-rr 53.6%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{\ell \cdot V} \leq 2 \cdot 10^{+263}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+263}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* l V))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 2e+263) (* c0 (sqrt t_0)) (* c0 (sqrt (/ (/ A l) V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 2e+263) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (l * v)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 2d+263) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 * sqrt(((a / l) / v))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 2e+263) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (l * V)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 2e+263:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(l * V))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 2e+263)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (l * V);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 2e+263)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 * sqrt(((A / l) / V));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+263], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 43.7%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 59.1

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

   4. Applied egg-rr 59.1%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 2.00000000000000003e263

   1. Initial program 98.3%

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

   if 2.00000000000000003e263 < (/.f64 A (*.f64 V l))

   1. Initial program 48.5%

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

      1. associate-/l/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 52.0

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

   4. Applied egg-rr 52.0%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{\ell \cdot V} \leq 2 \cdot 10^{+263}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ t_1 := c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+303}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* l V))) (t_1 (* c0 (sqrt (/ (/ A V) l)))))
   (if (<= t_0 0.0) t_1 (if (<= t_0 5e+303) (* c0 (sqrt t_0)) t_1))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double t_1 = c0 * sqrt(((A / V) / l));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+303) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a / (l * v)
    t_1 = c0 * sqrt(((a / v) / l))
    if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 5d+303) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double t_1 = c0 * Math.sqrt(((A / V) / l));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+303) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (l * V)
	t_1 = c0 * math.sqrt(((A / V) / l))
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 5e+303:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = t_1
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(l * V))
	t_1 = Float64(c0 * sqrt(Float64(Float64(A / V) / l)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+303)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (l * V);
	t_1 = c0 * sqrt(((A / V) / l));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+303)
		tmp = c0 * sqrt(t_0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 5e+303], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 4.9999999999999997e303 < (/.f64 A (*.f64 V l))

   1. Initial program 41.0%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 55.1

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

   4. Applied egg-rr 55.1%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.9999999999999997e303

   1. Initial program 98.4%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{\ell \cdot V} \leq 5 \cdot 10^{+303}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;\ell \cdot V \leq -4 \cdot 10^{-284}:\\ \;\;\;\;\sqrt{0 - A} \cdot \frac{c0}{\sqrt{0 - \mathsf{fma}\left(V, \ell, 0\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) (- INFINITY))
   (/ c0 (* (sqrt l) (sqrt (/ V A))))
   (if (<= (* l V) -4e-284)
     (* (sqrt (- 0.0 A)) (/ c0 (sqrt (- 0.0 (fma V l 0.0)))))
     (if (<= (* l V) 0.0)
       (* (/ c0 (sqrt l)) (sqrt (/ A V)))
       (* c0 (/ (sqrt A) (sqrt (* l V))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	} else if ((l * V) <= -4e-284) {
		tmp = sqrt((0.0 - A)) * (c0 / sqrt((0.0 - fma(V, l, 0.0))));
	} else if ((l * V) <= 0.0) {
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	}
	return tmp;
}

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))));
	elseif (Float64(l * V) <= -4e-284)
		tmp = Float64(sqrt(Float64(0.0 - A)) * Float64(c0 / sqrt(Float64(0.0 - fma(V, l, 0.0)))));
	elseif (Float64(l * V) <= 0.0)
		tmp = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(A / V)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	end
	return tmp
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -4e-284], N[(N[Sqrt[N[(0.0 - A), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[Sqrt[N[(0.0 - N[(V * l + 0.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 41.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-lft-identity N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{0 + V \cdot \ell}} \cdot A} \]

      6. +-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{V \cdot \ell + 0}} \cdot A} \]

      7. accelerator-lowering-fma.f64 41.9

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}} \cdot A} \]

   4. Applied egg-rr 41.9%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{fma}\left(V, \ell, 0\right)} \cdot A}} \]
   5. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*l/ N/A

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

      3. sqrt-div N/A

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

      4. *-lft-identity N/A

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

      5. pow1/2 N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

      8. associate-/l/ N/A

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

      9. pow1/2 N/A

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

      10. sqrt-div N/A

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

      11. associate-*r/ N/A

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

      12. clear-num N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/r/ N/A

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

      16. sqrt-lowering-sqrt.f64 N/A

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

      17. associate-/r/ N/A

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

      18. associate-*l/ N/A

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

      19. *-commutative N/A

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

      20. /-lowering-/.f64 N/A

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

      21. +-rgt-identity N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell + 0}}{A}}} \]

      22. accelerator-lowering-fma.f64 41.9

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}{A}}} \]

   6. Applied egg-rr 41.9%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{\mathsf{fma}\left(V, \ell, 0\right)}{A}}}} \]
   7. Step-by-step derivation

      1. sqrt-div N/A

         \[\leadsto \frac{c0}{\color{blue}{\frac{\sqrt{V \cdot \ell + 0}}{\sqrt{A}}}} \]

      2. +-rgt-identity N/A

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

      3. sqrt-prod N/A

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

      4. associate-*l/ N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-undiv N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. sqrt-lowering-sqrt.f64 43.7

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

   8. Applied egg-rr 43.7%

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

   if -inf.0 < (*.f64 V l) < -4.00000000000000015e-284

   1. Initial program 85.7%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-lft-identity N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{0 + V \cdot \ell}} \cdot A} \]

      6. +-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{V \cdot \ell + 0}} \cdot A} \]

      7. accelerator-lowering-fma.f64 85.8

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}} \cdot A} \]

   4. Applied egg-rr 85.8%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{fma}\left(V, \ell, 0\right)} \cdot A}} \]
   5. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*l/ N/A

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

      3. sqrt-div N/A

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

      4. *-lft-identity N/A

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

      5. pow1/2 N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

      8. associate-/l/ N/A

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

      9. pow1/2 N/A

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

      10. sqrt-div N/A

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

      11. associate-*r/ N/A

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

      12. clear-num N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/r/ N/A

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

      16. sqrt-lowering-sqrt.f64 N/A

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

      17. associate-/r/ N/A

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

      18. associate-*l/ N/A

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

      19. *-commutative N/A

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

      20. /-lowering-/.f64 N/A

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

      21. +-rgt-identity N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell + 0}}{A}}} \]

      22. accelerator-lowering-fma.f64 86.6

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}{A}}} \]

   6. Applied egg-rr 86.6%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{\mathsf{fma}\left(V, \ell, 0\right)}{A}}}} \]
   7. Step-by-step derivation

      1. frac-2neg N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\mathsf{neg}\left(\left(V \cdot \ell + 0\right)\right)}{\mathsf{neg}\left(A\right)}}}} \]

      2. +-rgt-identity N/A

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

      3. sqrt-div N/A

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

      4. associate-/r/ N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{0 - V \cdot \ell}}} \cdot \sqrt{\mathsf{neg}\left(A\right)} \]

      9. --lowering--.f64 N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{0 - V \cdot \ell}}} \cdot \sqrt{\mathsf{neg}\left(A\right)} \]

      10. +-rgt-identity N/A

          \[\leadsto \frac{c0}{\sqrt{0 - \color{blue}{\left(V \cdot \ell + 0\right)}}} \cdot \sqrt{\mathsf{neg}\left(A\right)} \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{0 - \color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}} \cdot \sqrt{\mathsf{neg}\left(A\right)} \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{c0}{\sqrt{0 - \mathsf{fma}\left(V, \ell, 0\right)}} \cdot \color{blue}{\sqrt{\mathsf{neg}\left(A\right)}} \]

      13. neg-sub0 N/A

          \[\leadsto \frac{c0}{\sqrt{0 - \mathsf{fma}\left(V, \ell, 0\right)}} \cdot \sqrt{\color{blue}{0 - A}} \]

      14. --lowering--.f64 99.5

          \[\leadsto \frac{c0}{\sqrt{0 - \mathsf{fma}\left(V, \ell, 0\right)}} \cdot \sqrt{\color{blue}{0 - A}} \]

   8. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{0 - \mathsf{fma}\left(V, \ell, 0\right)}} \cdot \sqrt{0 - A}} \]

   if -4.00000000000000015e-284 < (*.f64 V l) < -0.0

   1. Initial program 34.7%

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

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. /-lowering-/.f64 50.1

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

   4. Applied egg-rr 50.1%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 81.5%

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

      1. sqrt-div N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. +-lft-identity N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{0 + V \cdot \ell}}} \]

      6. +-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell + 0}}} \]

      7. accelerator-lowering-fma.f64 92.0

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}} \]

   4. Applied egg-rr 92.0%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}}} \]
   5. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 92.0

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

   6. Applied egg-rr 92.0%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;\ell \cdot V \leq -4 \cdot 10^{-284}:\\ \;\;\;\;\sqrt{0 - A} \cdot \frac{c0}{\sqrt{0 - \mathsf{fma}\left(V, \ell, 0\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-244}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{0 - \mathsf{fma}\left(V, \ell, 0\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ c0 (* (sqrt l) (sqrt (/ V A))))))
   (if (<= (* l V) (- INFINITY))
     t_0
     (if (<= (* l V) -5e-244)
       (* c0 (/ (sqrt (- 0.0 A)) (sqrt (- 0.0 (fma V l 0.0)))))
       (if (<= (* l V) 0.0) t_0 (* c0 (/ (sqrt A) (sqrt (* l V)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 / (sqrt(l) * sqrt((V / A)));
	double tmp;
	if ((l * V) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((l * V) <= -5e-244) {
		tmp = c0 * (sqrt((0.0 - A)) / sqrt((0.0 - fma(V, l, 0.0))));
	} else if ((l * V) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	}
	return tmp;
}

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))))
	tmp = 0.0
	if (Float64(l * V) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(l * V) <= -5e-244)
		tmp = Float64(c0 * Float64(sqrt(Float64(0.0 - A)) / sqrt(Float64(0.0 - fma(V, l, 0.0)))));
	elseif (Float64(l * V) <= 0.0)
		tmp = t_0;
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	end
	return 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[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(l * V), $MachinePrecision], -5e-244], N[(c0 * N[(N[Sqrt[N[(0.0 - A), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(0.0 - N[(V * l + 0.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], t$95$0, N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -inf.0 or -4.99999999999999998e-244 < (*.f64 V l) < -0.0

   1. Initial program 37.4%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-lft-identity N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{0 + V \cdot \ell}} \cdot A} \]

      6. +-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{V \cdot \ell + 0}} \cdot A} \]

      7. accelerator-lowering-fma.f64 36.8

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}} \cdot A} \]

   4. Applied egg-rr 36.8%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{fma}\left(V, \ell, 0\right)} \cdot A}} \]
   5. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*l/ N/A

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

      3. sqrt-div N/A

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

      4. *-lft-identity N/A

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

      5. pow1/2 N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

      8. associate-/l/ N/A

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

      9. pow1/2 N/A

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

      10. sqrt-div N/A

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

      11. associate-*r/ N/A

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

      12. clear-num N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/r/ N/A

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

      16. sqrt-lowering-sqrt.f64 N/A

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

      17. associate-/r/ N/A

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

      18. associate-*l/ N/A

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

      19. *-commutative N/A

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

      20. /-lowering-/.f64 N/A

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

      21. +-rgt-identity N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell + 0}}{A}}} \]

      22. accelerator-lowering-fma.f64 37.4

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}{A}}} \]

   6. Applied egg-rr 37.4%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{\mathsf{fma}\left(V, \ell, 0\right)}{A}}}} \]
   7. Step-by-step derivation

      1. sqrt-div N/A

         \[\leadsto \frac{c0}{\color{blue}{\frac{\sqrt{V \cdot \ell + 0}}{\sqrt{A}}}} \]

      2. +-rgt-identity N/A

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

      3. sqrt-prod N/A

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

      4. associate-*l/ N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-undiv N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. sqrt-lowering-sqrt.f64 46.2

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

   8. Applied egg-rr 46.2%

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

   if -inf.0 < (*.f64 V l) < -4.99999999999999998e-244

   1. Initial program 86.6%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-lft-identity N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{0 + V \cdot \ell}} \cdot A} \]

      6. +-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{V \cdot \ell + 0}} \cdot A} \]

      7. accelerator-lowering-fma.f64 86.6

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}} \cdot A} \]

   4. Applied egg-rr 86.6%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{fma}\left(V, \ell, 0\right)} \cdot A}} \]
   5. Step-by-step derivation

      1. frac-2neg N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(V \cdot \ell + 0\right)\right)}} \cdot A} \]

      2. metadata-eval N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(V \cdot \ell + 0\right)\right)} \cdot A} \]

      3. associate-*l/ N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{-1 \cdot A}{\mathsf{neg}\left(\left(V \cdot \ell + 0\right)\right)}}} \]

      4. neg-mul-1 N/A

         \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\mathsf{neg}\left(A\right)}}{\mathsf{neg}\left(\left(V \cdot \ell + 0\right)\right)}} \]

      5. sqrt-div N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(A\right)}}{\sqrt{\mathsf{neg}\left(\left(V \cdot \ell + 0\right)\right)}}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(A\right)}}{\sqrt{\mathsf{neg}\left(\left(V \cdot \ell + 0\right)\right)}}} \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\mathsf{neg}\left(A\right)}}}{\sqrt{\mathsf{neg}\left(\left(V \cdot \ell + 0\right)\right)}} \]

      8. neg-sub0 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{0 - A}}}{\sqrt{\mathsf{neg}\left(\left(V \cdot \ell + 0\right)\right)}} \]

      9. --lowering--.f64 N/A

         \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{0 - A}}}{\sqrt{\mathsf{neg}\left(\left(V \cdot \ell + 0\right)\right)}} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{0 - A}}{\color{blue}{\sqrt{\mathsf{neg}\left(\left(V \cdot \ell + 0\right)\right)}}} \]

      11. +-rgt-identity N/A

          \[\leadsto c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{\mathsf{neg}\left(\color{blue}{V \cdot \ell}\right)}} \]

      12. neg-sub0 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{\color{blue}{0 - V \cdot \ell}}} \]

      13. --lowering--.f64 N/A

          \[\leadsto c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{\color{blue}{0 - V \cdot \ell}}} \]

      14. +-rgt-identity N/A

          \[\leadsto c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{0 - \color{blue}{\left(V \cdot \ell + 0\right)}}} \]

      15. accelerator-lowering-fma.f64 99.4

          \[\leadsto c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{0 - \color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}} \]

   6. Applied egg-rr 99.4%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{0 - A}}{\sqrt{0 - \mathsf{fma}\left(V, \ell, 0\right)}}} \]

   if -0.0 < (*.f64 V l)

   1. Initial program 81.5%

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

      1. sqrt-div N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. +-lft-identity N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{0 + V \cdot \ell}}} \]

      6. +-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell + 0}}} \]

      7. accelerator-lowering-fma.f64 92.0

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}} \]

   4. Applied egg-rr 92.0%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}}} \]
   5. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 92.0

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

   6. Applied egg-rr 92.0%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -\infty:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;\ell \cdot V \leq -5 \cdot 10^{-244}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{0 - A}}{\sqrt{0 - \mathsf{fma}\left(V, \ell, 0\right)}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -2 \cdot 10^{+164}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;\ell \cdot V \leq -4 \cdot 10^{-210}:\\ \;\;\;\;\frac{c0}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right) \cdot \frac{1}{A}}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) -2e+164)
   (/ c0 (* (sqrt l) (sqrt (/ V A))))
   (if (<= (* l V) -4e-210)
     (/ c0 (sqrt (* (fma V l 0.0) (/ 1.0 A))))
     (if (<= (* l V) 0.0)
       (* (/ c0 (sqrt l)) (sqrt (/ A V)))
       (* c0 (/ (sqrt A) (sqrt (* l V))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -2e+164) {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	} else if ((l * V) <= -4e-210) {
		tmp = c0 / sqrt((fma(V, l, 0.0) * (1.0 / A)));
	} else if ((l * V) <= 0.0) {
		tmp = (c0 / sqrt(l)) * sqrt((A / V));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	}
	return tmp;
}

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= -2e+164)
		tmp = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))));
	elseif (Float64(l * V) <= -4e-210)
		tmp = Float64(c0 / sqrt(Float64(fma(V, l, 0.0) * Float64(1.0 / A))));
	elseif (Float64(l * V) <= 0.0)
		tmp = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(A / V)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	end
	return tmp
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], -2e+164], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -4e-210], N[(c0 / N[Sqrt[N[(N[(V * l + 0.0), $MachinePrecision] * N[(1.0 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -2e164

   1. Initial program 58.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-lft-identity N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{0 + V \cdot \ell}} \cdot A} \]

      6. +-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{V \cdot \ell + 0}} \cdot A} \]

      7. accelerator-lowering-fma.f64 58.8

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}} \cdot A} \]

   4. Applied egg-rr 58.8%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{fma}\left(V, \ell, 0\right)} \cdot A}} \]
   5. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*l/ N/A

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

      3. sqrt-div N/A

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

      4. *-lft-identity N/A

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

      5. pow1/2 N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

      8. associate-/l/ N/A

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

      9. pow1/2 N/A

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

      10. sqrt-div N/A

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

      11. associate-*r/ N/A

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

      12. clear-num N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/r/ N/A

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

      16. sqrt-lowering-sqrt.f64 N/A

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

      17. associate-/r/ N/A

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

      18. associate-*l/ N/A

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

      19. *-commutative N/A

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

      20. /-lowering-/.f64 N/A

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

      21. +-rgt-identity N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell + 0}}{A}}} \]

      22. accelerator-lowering-fma.f64 58.8

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}{A}}} \]

   6. Applied egg-rr 58.8%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{\mathsf{fma}\left(V, \ell, 0\right)}{A}}}} \]
   7. Step-by-step derivation

      1. sqrt-div N/A

         \[\leadsto \frac{c0}{\color{blue}{\frac{\sqrt{V \cdot \ell + 0}}{\sqrt{A}}}} \]

      2. +-rgt-identity N/A

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

      3. sqrt-prod N/A

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

      4. associate-*l/ N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-undiv N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. sqrt-lowering-sqrt.f64 59.0

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

   8. Applied egg-rr 59.0%

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

   if -2e164 < (*.f64 V l) < -4.0000000000000002e-210

   1. Initial program 92.1%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-lft-identity N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{0 + V \cdot \ell}} \cdot A} \]

      6. +-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{V \cdot \ell + 0}} \cdot A} \]

      7. accelerator-lowering-fma.f64 92.1

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}} \cdot A} \]

   4. Applied egg-rr 92.1%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{fma}\left(V, \ell, 0\right)} \cdot A}} \]
   5. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*l/ N/A

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

      3. sqrt-div N/A

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

      4. *-lft-identity N/A

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

      5. pow1/2 N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

      8. associate-/l/ N/A

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

      9. pow1/2 N/A

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

      10. sqrt-div N/A

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

      11. associate-*r/ N/A

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

      12. clear-num N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/r/ N/A

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

      16. sqrt-lowering-sqrt.f64 N/A

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

      17. associate-/r/ N/A

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

      18. associate-*l/ N/A

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

      19. *-commutative N/A

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

      20. /-lowering-/.f64 N/A

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

      21. +-rgt-identity N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell + 0}}{A}}} \]

      22. accelerator-lowering-fma.f64 93.3

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}{A}}} \]

   6. Applied egg-rr 93.3%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{\mathsf{fma}\left(V, \ell, 0\right)}{A}}}} \]
   7. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\left(V \cdot \ell + 0\right) \cdot \frac{1}{A}}}} \]

      2. +-rgt-identity N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-rgt-identity N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\left(V \cdot \ell + 0\right)} \cdot \frac{1}{A}}} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)} \cdot \frac{1}{A}}} \]

      6. /-lowering-/.f64 93.4

         \[\leadsto \frac{c0}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right) \cdot \color{blue}{\frac{1}{A}}}} \]

   8. Applied egg-rr 93.4%

      \[\leadsto \frac{c0}{\sqrt{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right) \cdot \frac{1}{A}}}} \]

   if -4.0000000000000002e-210 < (*.f64 V l) < -0.0

   1. Initial program 40.5%

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

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. /-lowering-/.f64 42.9

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

   4. Applied egg-rr 42.9%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 81.5%

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

      1. sqrt-div N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. +-lft-identity N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{0 + V \cdot \ell}}} \]

      6. +-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell + 0}}} \]

      7. accelerator-lowering-fma.f64 92.0

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}} \]

   4. Applied egg-rr 92.0%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}}} \]
   5. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 92.0

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

   6. Applied egg-rr 92.0%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -2 \cdot 10^{+164}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;\ell \cdot V \leq -4 \cdot 10^{-210}:\\ \;\;\;\;\frac{c0}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right) \cdot \frac{1}{A}}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{if}\;\ell \cdot V \leq -2 \cdot 10^{+164}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq -4 \cdot 10^{-210}:\\ \;\;\;\;\frac{c0}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right) \cdot \frac{1}{A}}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* (/ c0 (sqrt l)) (sqrt (/ A V)))))
   (if (<= (* l V) -2e+164)
     t_0
     (if (<= (* l V) -4e-210)
       (/ c0 (sqrt (* (fma V l 0.0) (/ 1.0 A))))
       (if (<= (* l V) 0.0) t_0 (* c0 (/ (sqrt A) (sqrt (* l V)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = (c0 / sqrt(l)) * sqrt((A / V));
	double tmp;
	if ((l * V) <= -2e+164) {
		tmp = t_0;
	} else if ((l * V) <= -4e-210) {
		tmp = c0 / sqrt((fma(V, l, 0.0) * (1.0 / A)));
	} else if ((l * V) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	}
	return tmp;
}

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(Float64(c0 / sqrt(l)) * sqrt(Float64(A / V)))
	tmp = 0.0
	if (Float64(l * V) <= -2e+164)
		tmp = t_0;
	elseif (Float64(l * V) <= -4e-210)
		tmp = Float64(c0 / sqrt(Float64(fma(V, l, 0.0) * Float64(1.0 / A))));
	elseif (Float64(l * V) <= 0.0)
		tmp = t_0;
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	end
	return tmp
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], -2e+164], t$95$0, If[LessEqual[N[(l * V), $MachinePrecision], -4e-210], N[(c0 / N[Sqrt[N[(N[(V * l + 0.0), $MachinePrecision] * N[(1.0 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], t$95$0, N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -2e164 or -4.0000000000000002e-210 < (*.f64 V l) < -0.0

   1. Initial program 50.7%

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

      1. associate-/r* N/A

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

      2. sqrt-div N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. /-lowering-/.f64 51.8

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

   4. Applied egg-rr 51.8%

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

   if -2e164 < (*.f64 V l) < -4.0000000000000002e-210

   1. Initial program 92.1%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-lft-identity N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{0 + V \cdot \ell}} \cdot A} \]

      6. +-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{V \cdot \ell + 0}} \cdot A} \]

      7. accelerator-lowering-fma.f64 92.1

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}} \cdot A} \]

   4. Applied egg-rr 92.1%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{fma}\left(V, \ell, 0\right)} \cdot A}} \]
   5. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*l/ N/A

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

      3. sqrt-div N/A

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

      4. *-lft-identity N/A

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

      5. pow1/2 N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

      8. associate-/l/ N/A

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

      9. pow1/2 N/A

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

      10. sqrt-div N/A

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

      11. associate-*r/ N/A

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

      12. clear-num N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/r/ N/A

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

      16. sqrt-lowering-sqrt.f64 N/A

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

      17. associate-/r/ N/A

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

      18. associate-*l/ N/A

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

      19. *-commutative N/A

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

      20. /-lowering-/.f64 N/A

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

      21. +-rgt-identity N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell + 0}}{A}}} \]

      22. accelerator-lowering-fma.f64 93.3

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}{A}}} \]

   6. Applied egg-rr 93.3%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{\mathsf{fma}\left(V, \ell, 0\right)}{A}}}} \]
   7. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\left(V \cdot \ell + 0\right) \cdot \frac{1}{A}}}} \]

      2. +-rgt-identity N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-rgt-identity N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\left(V \cdot \ell + 0\right)} \cdot \frac{1}{A}}} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)} \cdot \frac{1}{A}}} \]

      6. /-lowering-/.f64 93.4

         \[\leadsto \frac{c0}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right) \cdot \color{blue}{\frac{1}{A}}}} \]

   8. Applied egg-rr 93.4%

      \[\leadsto \frac{c0}{\sqrt{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right) \cdot \frac{1}{A}}}} \]

   if -0.0 < (*.f64 V l)

   1. Initial program 81.5%

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

      1. sqrt-div N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. +-lft-identity N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{0 + V \cdot \ell}}} \]

      6. +-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell + 0}}} \]

      7. accelerator-lowering-fma.f64 92.0

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}} \]

   4. Applied egg-rr 92.0%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}}} \]
   5. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 92.0

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

   6. Applied egg-rr 92.0%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -2 \cdot 10^{+164}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{elif}\;\ell \cdot V \leq -4 \cdot 10^{-210}:\\ \;\;\;\;\frac{c0}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right) \cdot \frac{1}{A}}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\frac{A}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -1 \cdot 10^{+191}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell} \cdot \frac{1}{V}}\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-189}:\\ \;\;\;\;\frac{c0}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right) \cdot \frac{1}{A}}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{-309}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) -1e+191)
   (* c0 (sqrt (* (/ A l) (/ 1.0 V))))
   (if (<= (* l V) -2e-189)
     (/ c0 (sqrt (* (fma V l 0.0) (/ 1.0 A))))
     (if (<= (* l V) 5e-309)
       (* c0 (sqrt (/ (/ A l) V)))
       (* c0 (/ (sqrt A) (sqrt (* l V))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -1e+191) {
		tmp = c0 * sqrt(((A / l) * (1.0 / V)));
	} else if ((l * V) <= -2e-189) {
		tmp = c0 / sqrt((fma(V, l, 0.0) * (1.0 / A)));
	} else if ((l * V) <= 5e-309) {
		tmp = c0 * sqrt(((A / l) / V));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	}
	return tmp;
}

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= -1e+191)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) * Float64(1.0 / V))));
	elseif (Float64(l * V) <= -2e-189)
		tmp = Float64(c0 / sqrt(Float64(fma(V, l, 0.0) * Float64(1.0 / A))));
	elseif (Float64(l * V) <= 5e-309)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	end
	return tmp
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], -1e+191], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] * N[(1.0 / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -2e-189], N[(c0 / N[Sqrt[N[(N[(V * l + 0.0), $MachinePrecision] * N[(1.0 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 5e-309], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -1.00000000000000007e191

   1. Initial program 59.8%

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

      1. associate-/l/ N/A

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

      2. div-inv N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 73.5

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

   4. Applied egg-rr 73.5%

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

   if -1.00000000000000007e191 < (*.f64 V l) < -2.00000000000000014e-189

   1. Initial program 88.5%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-lft-identity N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{0 + V \cdot \ell}} \cdot A} \]

      6. +-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{V \cdot \ell + 0}} \cdot A} \]

      7. accelerator-lowering-fma.f64 88.5

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}} \cdot A} \]

   4. Applied egg-rr 88.5%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{fma}\left(V, \ell, 0\right)} \cdot A}} \]
   5. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*l/ N/A

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

      3. sqrt-div N/A

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

      4. *-lft-identity N/A

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

      5. pow1/2 N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

      8. associate-/l/ N/A

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

      9. pow1/2 N/A

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

      10. sqrt-div N/A

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

      11. associate-*r/ N/A

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

      12. clear-num N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/r/ N/A

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

      16. sqrt-lowering-sqrt.f64 N/A

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

      17. associate-/r/ N/A

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

      18. associate-*l/ N/A

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

      19. *-commutative N/A

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

      20. /-lowering-/.f64 N/A

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

      21. +-rgt-identity N/A

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell + 0}}{A}}} \]

      22. accelerator-lowering-fma.f64 89.6

          \[\leadsto \frac{c0}{\sqrt{\frac{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}{A}}} \]

   6. Applied egg-rr 89.6%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{\frac{\mathsf{fma}\left(V, \ell, 0\right)}{A}}}} \]
   7. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\left(V \cdot \ell + 0\right) \cdot \frac{1}{A}}}} \]

      2. +-rgt-identity N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-rgt-identity N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\left(V \cdot \ell + 0\right)} \cdot \frac{1}{A}}} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{c0}{\sqrt{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)} \cdot \frac{1}{A}}} \]

      6. /-lowering-/.f64 89.7

         \[\leadsto \frac{c0}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right) \cdot \color{blue}{\frac{1}{A}}}} \]

   8. Applied egg-rr 89.7%

      \[\leadsto \frac{c0}{\sqrt{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right) \cdot \frac{1}{A}}}} \]

   if -2.00000000000000014e-189 < (*.f64 V l) < 4.9999999999999995e-309

   1. Initial program 47.9%

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

      1. associate-/l/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 61.7

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

   4. Applied egg-rr 61.7%

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

   if 4.9999999999999995e-309 < (*.f64 V l)

   1. Initial program 81.2%

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

      1. sqrt-div N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. +-lft-identity N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{0 + V \cdot \ell}}} \]

      6. +-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell + 0}}} \]

      7. accelerator-lowering-fma.f64 91.9

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}} \]

   4. Applied egg-rr 91.9%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}}} \]
   5. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 91.9

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

   6. Applied egg-rr 91.9%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -1 \cdot 10^{+191}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell} \cdot \frac{1}{V}}\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-189}:\\ \;\;\;\;\frac{c0}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right) \cdot \frac{1}{A}}}\\ \mathbf{elif}\;\ell \cdot V \leq 5 \cdot 10^{-309}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -2 \cdot 10^{+228}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-91}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{1}{\ell \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 10^{-272}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* l V) -2e+228)
   (* c0 (sqrt (/ (/ A V) l)))
   (if (<= (* l V) -2e-91)
     (* c0 (sqrt (* A (/ 1.0 (* l V)))))
     (if (<= (* l V) 1e-272)
       (/ c0 (sqrt (* V (/ l A))))
       (* c0 (/ (sqrt A) (sqrt (* l V))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -2e+228) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if ((l * V) <= -2e-91) {
		tmp = c0 * sqrt((A * (1.0 / (l * V))));
	} else if ((l * V) <= 1e-272) {
		tmp = c0 / sqrt((V * (l / A)));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((l * v) <= (-2d+228)) then
        tmp = c0 * sqrt(((a / v) / l))
    else if ((l * v) <= (-2d-91)) then
        tmp = c0 * sqrt((a * (1.0d0 / (l * v))))
    else if ((l * v) <= 1d-272) then
        tmp = c0 / sqrt((v * (l / a)))
    else
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((l * V) <= -2e+228) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if ((l * V) <= -2e-91) {
		tmp = c0 * Math.sqrt((A * (1.0 / (l * V))));
	} else if ((l * V) <= 1e-272) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (l * V) <= -2e+228:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif (l * V) <= -2e-91:
		tmp = c0 * math.sqrt((A * (1.0 / (l * V))))
	elif (l * V) <= 1e-272:
		tmp = c0 / math.sqrt((V * (l / A)))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(l * V) <= -2e+228)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (Float64(l * V) <= -2e-91)
		tmp = Float64(c0 * sqrt(Float64(A * Float64(1.0 / Float64(l * V)))));
	elseif (Float64(l * V) <= 1e-272)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(l * V))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((l * V) <= -2e+228)
		tmp = c0 * sqrt(((A / V) / l));
	elseif ((l * V) <= -2e-91)
		tmp = c0 * sqrt((A * (1.0 / (l * V))));
	elseif ((l * V) <= 1e-272)
		tmp = c0 / sqrt((V * (l / A)));
	else
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(l * V), $MachinePrecision], -2e+228], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], -2e-91], N[(c0 * N[Sqrt[N[(A * N[(1.0 / N[(l * V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 1e-272], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -1.9999999999999998e228

   1. Initial program 47.3%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 68.6

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

   4. Applied egg-rr 68.6%

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

   if -1.9999999999999998e228 < (*.f64 V l) < -2.00000000000000004e-91

   1. Initial program 91.7%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-lft-identity N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{0 + V \cdot \ell}} \cdot A} \]

      6. +-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{V \cdot \ell + 0}} \cdot A} \]

      7. accelerator-lowering-fma.f64 91.8

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}} \cdot A} \]

   4. Applied egg-rr 91.8%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{fma}\left(V, \ell, 0\right)} \cdot A}} \]

   5. Taylor expanded in V around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 91.8

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

   7. Simplified 91.8%

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

   if -2.00000000000000004e-91 < (*.f64 V l) < 9.9999999999999993e-273

   1. Initial program 58.9%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 67.7

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

   4. Applied egg-rr 67.7%

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

   if 9.9999999999999993e-273 < (*.f64 V l)

   1. Initial program 81.1%

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

      1. sqrt-div N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. +-lft-identity N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{0 + V \cdot \ell}}} \]

      6. +-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell + 0}}} \]

      7. accelerator-lowering-fma.f64 91.8

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}} \]

   4. Applied egg-rr 91.8%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}}} \]
   5. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 91.8

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

   6. Applied egg-rr 91.8%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -2 \cdot 10^{+228}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq -2 \cdot 10^{-91}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{1}{\ell \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 10^{-272}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 88.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}} \cdot \sqrt{0 - A}}{\sqrt{0 - V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -5e-310)
   (/ (* (/ c0 (sqrt l)) (sqrt (- 0.0 A))) (sqrt (- 0.0 V)))
   (* c0 (/ (sqrt A) (sqrt (* l V))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = ((c0 / sqrt(l)) * sqrt((0.0 - A))) / sqrt((0.0 - V));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (a <= (-5d-310)) then
        tmp = ((c0 / sqrt(l)) * sqrt((0.0d0 - a))) / sqrt((0.0d0 - v))
    else
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = ((c0 / Math.sqrt(l)) * Math.sqrt((0.0 - A))) / Math.sqrt((0.0 - V));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -5e-310:
		tmp = ((c0 / math.sqrt(l)) * math.sqrt((0.0 - A))) / math.sqrt((0.0 - V))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	return tmp

Julia

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

MATLAB

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

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -5e-310], N[(N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.0 - A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(0.0 - V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -4.999999999999985e-310

   1. Initial program 72.0%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-lft-identity N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{0 + V \cdot \ell}} \cdot A} \]

      6. +-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{V \cdot \ell + 0}} \cdot A} \]

      7. accelerator-lowering-fma.f64 71.8

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}} \cdot A} \]

   4. Applied egg-rr 71.8%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{fma}\left(V, \ell, 0\right)} \cdot A}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\color{blue}{A \cdot \frac{1}{V \cdot \ell + 0}}} \]

      2. +-rgt-identity N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. sqrt-div N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. frac-2neg N/A

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

      9. sqrt-div N/A

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

      10. associate-*r/ N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. sqrt-lowering-sqrt.f64 N/A

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

      15. sqrt-lowering-sqrt.f64 N/A

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

      16. neg-sub0 N/A

          \[\leadsto \frac{\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\color{blue}{0 - A}}}{\sqrt{\mathsf{neg}\left(V\right)}} \]

      17. --lowering--.f64 N/A

          \[\leadsto \frac{\frac{c0}{\sqrt{\ell}} \cdot \sqrt{\color{blue}{0 - A}}}{\sqrt{\mathsf{neg}\left(V\right)}} \]

      18. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\frac{c0}{\sqrt{\ell}} \cdot \sqrt{0 - A}}{\color{blue}{\sqrt{\mathsf{neg}\left(V\right)}}} \]

      19. neg-sub0 N/A

          \[\leadsto \frac{\frac{c0}{\sqrt{\ell}} \cdot \sqrt{0 - A}}{\sqrt{\color{blue}{0 - V}}} \]

      20. --lowering--.f64 44.6

          \[\leadsto \frac{\frac{c0}{\sqrt{\ell}} \cdot \sqrt{0 - A}}{\sqrt{\color{blue}{0 - V}}} \]

   6. Applied egg-rr 44.6%

      \[\leadsto \color{blue}{\frac{\frac{c0}{\sqrt{\ell}} \cdot \sqrt{0 - A}}{\sqrt{0 - V}}} \]

   if -4.999999999999985e-310 < A

   1. Initial program 78.1%

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

      1. sqrt-div N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. +-lft-identity N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{0 + V \cdot \ell}}} \]

      6. +-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell + 0}}} \]

      7. accelerator-lowering-fma.f64 87.7

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}} \]

   4. Applied egg-rr 87.7%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}}} \]
   5. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 87.7

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

   6. Applied egg-rr 87.7%

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

Alternative 12: 88.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{0 - A}}{\sqrt{0 - V} \cdot \frac{\sqrt{\ell}}{c0}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -5e-310)
   (/ (sqrt (- 0.0 A)) (* (sqrt (- 0.0 V)) (/ (sqrt l) c0)))
   (* c0 (/ (sqrt A) (sqrt (* l V))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = sqrt((0.0 - A)) / (sqrt((0.0 - V)) * (sqrt(l) / c0));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((l * V)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (a <= (-5d-310)) then
        tmp = sqrt((0.0d0 - a)) / (sqrt((0.0d0 - v)) * (sqrt(l) / c0))
    else
        tmp = c0 * (sqrt(a) / sqrt((l * v)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = Math.sqrt((0.0 - A)) / (Math.sqrt((0.0 - V)) * (Math.sqrt(l) / c0));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((l * V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -5e-310:
		tmp = math.sqrt((0.0 - A)) / (math.sqrt((0.0 - V)) * (math.sqrt(l) / c0))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((l * V)))
	return tmp

Julia

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

MATLAB

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

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -5e-310], N[(N[Sqrt[N[(0.0 - A), $MachinePrecision]], $MachinePrecision] / N[(N[Sqrt[N[(0.0 - V), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[l], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -4.999999999999985e-310

   1. Initial program 72.0%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-lft-identity N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{0 + V \cdot \ell}} \cdot A} \]

      6. +-commutative N/A

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{V \cdot \ell + 0}} \cdot A} \]

      7. accelerator-lowering-fma.f64 71.8

         \[\leadsto c0 \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}} \cdot A} \]

   4. Applied egg-rr 71.8%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{fma}\left(V, \ell, 0\right)} \cdot A}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{V \cdot \ell + 0} \cdot A} \cdot c0} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{A \cdot \frac{1}{V \cdot \ell + 0}}} \cdot c0 \]

      3. +-rgt-identity N/A

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

      4. div-inv N/A

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

      5. associate-/r* N/A

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

      6. sqrt-div N/A

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

      7. associate-/r/ N/A

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

      8. frac-2neg N/A

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

      9. sqrt-div N/A

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

      10. associate-/l/ N/A

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

      11. /-lowering-/.f64 N/A

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

      12. sqrt-lowering-sqrt.f64 N/A

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

      13. neg-sub0 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{0 - A}}}{\frac{\sqrt{\ell}}{c0} \cdot \sqrt{\mathsf{neg}\left(V\right)}} \]

      14. --lowering--.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{0 - A}}}{\frac{\sqrt{\ell}}{c0} \cdot \sqrt{\mathsf{neg}\left(V\right)}} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \frac{\sqrt{0 - A}}{\color{blue}{\frac{\sqrt{\ell}}{c0} \cdot \sqrt{\mathsf{neg}\left(V\right)}}} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \frac{\sqrt{0 - A}}{\color{blue}{\frac{\sqrt{\ell}}{c0}} \cdot \sqrt{\mathsf{neg}\left(V\right)}} \]

      17. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{0 - A}}{\frac{\color{blue}{\sqrt{\ell}}}{c0} \cdot \sqrt{\mathsf{neg}\left(V\right)}} \]

      18. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{0 - A}}{\frac{\sqrt{\ell}}{c0} \cdot \color{blue}{\sqrt{\mathsf{neg}\left(V\right)}}} \]

      19. neg-sub0 N/A

          \[\leadsto \frac{\sqrt{0 - A}}{\frac{\sqrt{\ell}}{c0} \cdot \sqrt{\color{blue}{0 - V}}} \]

      20. --lowering--.f64 45.1

          \[\leadsto \frac{\sqrt{0 - A}}{\frac{\sqrt{\ell}}{c0} \cdot \sqrt{\color{blue}{0 - V}}} \]

   6. Applied egg-rr 45.1%

      \[\leadsto \color{blue}{\frac{\sqrt{0 - A}}{\frac{\sqrt{\ell}}{c0} \cdot \sqrt{0 - V}}} \]

   if -4.999999999999985e-310 < A

   1. Initial program 78.1%

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

      1. sqrt-div N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. +-lft-identity N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{0 + V \cdot \ell}}} \]

      6. +-commutative N/A

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{V \cdot \ell + 0}}} \]

      7. accelerator-lowering-fma.f64 87.7

         \[\leadsto c0 \cdot \frac{\sqrt{A}}{\sqrt{\color{blue}{\mathsf{fma}\left(V, \ell, 0\right)}}} \]

   4. Applied egg-rr 87.7%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\mathsf{fma}\left(V, \ell, 0\right)}}} \]
   5. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 87.7

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

   6. Applied egg-rr 87.7%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{0 - A}}{\sqrt{0 - V} \cdot \frac{\sqrt{\ell}}{c0}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{\ell \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.5% accurate, 1.0× speedup

Math

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

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 (* l V)))))

C

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

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 / (l * v)))
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 / (l * V)));
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 74.8%

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

3. Final simplification 74.8%

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

Reproduce

herbie shell --seed 2024196 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))