Henrywood and Agarwal, Equation (3)

Percentage Accurate: 74.4% → 90.0%

Time: 8.4s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{1}{\frac{\sqrt{-V}}{\sqrt{\frac{-A}{\ell}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-306}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-317}:\\ \;\;\;\;c0 \cdot \left({\left(\frac{-\ell}{A}\right)}^{-0.5} \cdot {\left(\frac{-1}{V}\right)}^{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (* c0 (/ 1.0 (/ (sqrt (- V)) (sqrt (/ (- A) l)))))
   (if (<= (* V l) -5e-306)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 1e-317)
       (* c0 (* (pow (/ (- l) A) -0.5) (pow (/ -1.0 V) 0.5)))
       (* c0 (* (sqrt A) (pow (* V l) -0.5)))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 * (1.0 / (sqrt(-V) / sqrt((-A / l))));
	} else if ((V * l) <= -5e-306) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 1e-317) {
		tmp = c0 * (pow((-l / A), -0.5) * pow((-1.0 / V), 0.5));
	} else {
		tmp = c0 * (sqrt(A) * pow((V * l), -0.5));
	}
	return tmp;
}

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 * (1.0 / (Math.sqrt(-V) / Math.sqrt((-A / l))));
	} else if ((V * l) <= -5e-306) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 1e-317) {
		tmp = c0 * (Math.pow((-l / A), -0.5) * Math.pow((-1.0 / V), 0.5));
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.pow((V * l), -0.5));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 * (1.0 / (math.sqrt(-V) / math.sqrt((-A / l))))
	elif (V * l) <= -5e-306:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 1e-317:
		tmp = c0 * (math.pow((-l / A), -0.5) * math.pow((-1.0 / V), 0.5))
	else:
		tmp = c0 * (math.sqrt(A) * math.pow((V * l), -0.5))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 * Float64(1.0 / Float64(sqrt(Float64(-V)) / sqrt(Float64(Float64(-A) / l)))));
	elseif (Float64(V * l) <= -5e-306)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 1e-317)
		tmp = Float64(c0 * Float64((Float64(Float64(-l) / A) ^ -0.5) * (Float64(-1.0 / V) ^ 0.5)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * (Float64(V * l) ^ -0.5)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 * (1.0 / (sqrt(-V) / sqrt((-A / l))));
	elseif ((V * l) <= -5e-306)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 1e-317)
		tmp = c0 * (((-l / A) ^ -0.5) * ((-1.0 / V) ^ 0.5));
	else
		tmp = c0 * (sqrt(A) * ((V * l) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 * N[(1.0 / N[(N[Sqrt[(-V)], $MachinePrecision] / N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-306], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-317], N[(c0 * N[(N[Power[N[((-l) / A), $MachinePrecision], -0.5], $MachinePrecision] * N[Power[N[(-1.0 / V), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 16.6%

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

      1. clear-num 16.6%

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

      2. sqrt-div 16.6%

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

      3. metadata-eval 16.6%

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

      4. associate-/l* 50.2%

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

   3. Applied egg-rr 50.2%

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

      1. associate-/l* 16.6%

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

   5. Simplified 16.6%

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

      1. associate-/l* 50.2%

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

      2. frac-2neg 50.2%

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

      3. sqrt-div 51.9%

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

   7. Applied egg-rr 51.9%

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

      1. distribute-neg-frac 51.9%

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

   9. Simplified 51.9%

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

   if -inf.0 < (*.f64 V l) < -4.99999999999999998e-306

   1. Initial program 84.8%

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

      1. frac-2neg 84.8%

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

      2. sqrt-div 99.5%

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

      3. *-commutative 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

   3. Applied egg-rr 99.5%

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

   if -4.99999999999999998e-306 < (*.f64 V l) < 1.00000023e-317

   1. Initial program 38.2%

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

      1. pow1/2 38.2%

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

      2. clear-num 38.2%

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

      3. inv-pow 38.2%

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

      4. pow-pow 38.2%

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

      5. associate-/l* 54.5%

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

      6. metadata-eval 54.5%

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

   3. Applied egg-rr 54.5%

      \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}} \]
   4. Step-by-step derivation

      1. associate-/l* 38.2%

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

   5. Simplified 38.2%

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

   6. Taylor expanded in V around -inf 34.1%

      \[\leadsto c0 \cdot \color{blue}{e^{-0.5 \cdot \left(\log \left(-1 \cdot \frac{\ell}{A}\right) + -1 \cdot \log \left(\frac{-1}{V}\right)\right)}} \]
   7. Step-by-step derivation

      1. distribute-lft-in 34.1%

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

      2. exp-sum 34.3%

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

      3. *-commutative 34.3%

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

      4. associate-*r/ 34.3%

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

      5. associate-/l* 34.3%

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

      6. metadata-eval 34.3%

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

      7. associate-/r* 34.3%

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

      8. neg-mul-1 34.3%

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

      9. exp-to-pow 34.3%

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

      10. neg-mul-1 34.3%

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

      11. associate-/r* 34.3%

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

      12. metadata-eval 34.3%

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

      13. associate-/l* 34.3%

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

      14. neg-mul-1 34.3%

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

      15. *-commutative 34.3%

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

      16. *-commutative 34.3%

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

      17. associate-*l* 34.3%

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

   8. Simplified 36.0%

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

   if 1.00000023e-317 < (*.f64 V l)

   1. Initial program 80.5%

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

      1. div-inv 80.5%

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

      2. sqrt-prod 91.6%

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

      3. pow1/2 91.6%

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

      4. inv-pow 91.6%

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

      5. pow-pow 92.2%

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

      6. metadata-eval 92.2%

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

   3. Applied egg-rr 92.2%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{1}{\frac{\sqrt{-V}}{\sqrt{\frac{-A}{\ell}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-306}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-317}:\\ \;\;\;\;c0 \cdot \left({\left(\frac{-\ell}{A}\right)}^{-0.5} \cdot {\left(\frac{-1}{V}\right)}^{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \end{array} \]

Alternative 2: 80.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ t_1 := \frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{-319}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+289}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{c0 \cdot A}{\ell \cdot \frac{V}{c0}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))) (t_1 (/ c0 (sqrt (/ (* V l) A)))))
   (if (<= t_0 -4e-319)
     t_1
     (if (<= t_0 0.0)
       (* c0 (sqrt (/ (/ A V) l)))
       (if (<= t_0 5e+289) t_1 (sqrt (/ (* c0 A) (* l (/ V c0)))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double t_1 = c0 / sqrt(((V * l) / A));
	double tmp;
	if (t_0 <= -4e-319) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 5e+289) {
		tmp = t_1;
	} else {
		tmp = sqrt(((c0 * A) / (l * (V / c0))));
	}
	return tmp;
}

Fortran

NOTE: 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 = c0 * sqrt((a / (v * l)))
    t_1 = c0 / sqrt(((v * l) / a))
    if (t_0 <= (-4d-319)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 5d+289) then
        tmp = t_1
    else
        tmp = sqrt(((c0 * a) / (l * (v / c0))))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double t_1 = c0 / Math.sqrt(((V * l) / A));
	double tmp;
	if (t_0 <= -4e-319) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 5e+289) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt(((c0 * A) / (l * (V / c0))));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	t_1 = c0 / math.sqrt(((V * l) / A))
	tmp = 0
	if t_0 <= -4e-319:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 5e+289:
		tmp = t_1
	else:
		tmp = math.sqrt(((c0 * A) / (l * (V / c0))))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	t_1 = Float64(c0 / sqrt(Float64(Float64(V * l) / A)))
	tmp = 0.0
	if (t_0 <= -4e-319)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 5e+289)
		tmp = t_1;
	else
		tmp = sqrt(Float64(Float64(c0 * A) / Float64(l * Float64(V / c0))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	t_1 = c0 / sqrt(((V * l) / A));
	tmp = 0.0;
	if (t_0 <= -4e-319)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 5e+289)
		tmp = t_1;
	else
		tmp = sqrt(((c0 * A) / (l * (V / c0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-319], t$95$1, 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, 5e+289], t$95$1, N[Sqrt[N[(N[(c0 * A), $MachinePrecision] / N[(l * N[(V / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < -4.0000049e-319 or -0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 5.00000000000000031e289

   1. Initial program 90.8%

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

      1. pow1/2 90.8%

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

      2. clear-num 90.7%

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

      3. inv-pow 90.7%

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

      4. pow-pow 91.6%

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

      5. associate-/l* 78.4%

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

      6. metadata-eval 78.4%

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

   3. Applied egg-rr 78.4%

      \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}} \]
   4. Step-by-step derivation

      1. associate-/l* 91.6%

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

   5. Simplified 91.6%

      \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}} \]
   6. Step-by-step derivation

      1. sqr-pow 91.2%

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

      2. sqr-pow 91.6%

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

      3. metadata-eval 91.6%

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

      4. sqrt-pow2 91.5%

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

      5. inv-pow 91.5%

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

      6. un-div-inv 91.7%

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

      7. div-inv 91.6%

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

      8. associate-*l* 77.3%

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

      9. div-inv 77.3%

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

   7. Applied egg-rr 77.3%

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

   8. Taylor expanded in V around 0 91.7%

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

   if -4.0000049e-319 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < -0.0

   1. Initial program 38.5%

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

      1. add-sqr-sqrt 38.5%

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

      2. pow2 38.5%

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

      3. pow1/2 38.5%

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

      4. sqrt-pow1 38.5%

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

      5. metadata-eval 38.5%

         \[\leadsto c0 \cdot {\left({\left(\frac{A}{V \cdot \ell}\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   3. Applied egg-rr 38.5%

      \[\leadsto c0 \cdot \color{blue}{{\left({\left(\frac{A}{V \cdot \ell}\right)}^{0.25}\right)}^{2}} \]

   4. Taylor expanded in c0 around 0 38.5%

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

      1. associate-/r* 53.7%

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

   6. Simplified 53.7%

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

   if 5.00000000000000031e289 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 41.1%

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

      1. add-sqr-sqrt 41.1%

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

      2. pow2 41.1%

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

      3. pow1/2 41.1%

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

      4. sqrt-pow1 41.1%

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

      5. metadata-eval 41.1%

         \[\leadsto c0 \cdot {\left({\left(\frac{A}{V \cdot \ell}\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   3. Applied egg-rr 41.1%

      \[\leadsto c0 \cdot \color{blue}{{\left({\left(\frac{A}{V \cdot \ell}\right)}^{0.25}\right)}^{2}} \]
   4. Step-by-step derivation

      1. pow-pow 41.1%

         \[\leadsto c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{\left(0.25 \cdot 2\right)}} \]

      2. metadata-eval 41.1%

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

      3. pow1/2 41.1%

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

      4. clear-num 41.1%

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

      5. associate-*r/ 51.0%

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

      6. sqrt-div 51.0%

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

      7. metadata-eval 51.0%

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

      8. div-inv 51.0%

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

      9. add-sqr-sqrt 51.0%

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

      10. sqrt-unprod 51.0%

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

      11. frac-times 50.5%

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

      12. add-sqr-sqrt 50.5%

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

   5. Applied egg-rr 50.5%

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

      1. times-frac 64.4%

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

      2. associate-/r/ 77.6%

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

      3. associate-*l* 67.7%

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

      4. *-commutative 67.7%

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

      5. associate-*l/ 64.3%

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

      6. associate-/l* 67.7%

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

      7. associate-*r/ 72.7%

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

      8. associate-/r/ 72.6%

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

      9. *-commutative 72.6%

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

   7. Simplified 72.6%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq -4 \cdot 10^{-319}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 5 \cdot 10^{+289}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{c0 \cdot A}{\ell \cdot \frac{V}{c0}}}\\ \end{array} \]

Alternative 3: 90.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \frac{1}{\frac{\sqrt{-V}}{\sqrt{\frac{-A}{\ell}}}}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-306}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-317}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (/ 1.0 (/ (sqrt (- V)) (sqrt (/ (- A) l)))))))
   (if (<= (* V l) (- INFINITY))
     t_0
     (if (<= (* V l) -5e-306)
       (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
       (if (<= (* V l) 1e-317) t_0 (* c0 (* (sqrt A) (pow (* V l) -0.5))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (1.0 / (sqrt(-V) / sqrt((-A / l))));
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((V * l) <= -5e-306) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 1e-317) {
		tmp = t_0;
	} else {
		tmp = c0 * (sqrt(A) * pow((V * l), -0.5));
	}
	return tmp;
}

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (1.0 / (Math.sqrt(-V) / Math.sqrt((-A / l))));
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if ((V * l) <= -5e-306) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 1e-317) {
		tmp = t_0;
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.pow((V * l), -0.5));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = c0 * (1.0 / (math.sqrt(-V) / math.sqrt((-A / l))))
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = t_0
	elif (V * l) <= -5e-306:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 1e-317:
		tmp = t_0
	else:
		tmp = c0 * (math.sqrt(A) * math.pow((V * l), -0.5))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * Float64(1.0 / Float64(sqrt(Float64(-V)) / sqrt(Float64(Float64(-A) / l)))))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(V * l) <= -5e-306)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 1e-317)
		tmp = t_0;
	else
		tmp = Float64(c0 * Float64(sqrt(A) * (Float64(V * l) ^ -0.5)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * (1.0 / (sqrt(-V) / sqrt((-A / l))));
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = t_0;
	elseif ((V * l) <= -5e-306)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 1e-317)
		tmp = t_0;
	else
		tmp = c0 * (sqrt(A) * ((V * l) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: 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[(1.0 / N[(N[Sqrt[(-V)], $MachinePrecision] / N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -5e-306], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-317], t$95$0, N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -inf.0 or -4.99999999999999998e-306 < (*.f64 V l) < 1.00000023e-317

   1. Initial program 27.8%

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

      1. clear-num 27.8%

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

      2. sqrt-div 27.8%

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

      3. metadata-eval 27.8%

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

      4. associate-/l* 52.4%

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

   3. Applied egg-rr 52.4%

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

      1. associate-/l* 27.8%

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

   5. Simplified 27.8%

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

      1. associate-/l* 52.4%

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

      2. frac-2neg 52.4%

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

      3. sqrt-div 43.6%

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

   7. Applied egg-rr 43.6%

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

      1. distribute-neg-frac 43.6%

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

   9. Simplified 43.6%

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

   if -inf.0 < (*.f64 V l) < -4.99999999999999998e-306

   1. Initial program 84.8%

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

      1. frac-2neg 84.8%

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

      2. sqrt-div 99.5%

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

      3. *-commutative 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

   3. Applied egg-rr 99.5%

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

   if 1.00000023e-317 < (*.f64 V l)

   1. Initial program 80.5%

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

      1. div-inv 80.5%

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

      2. sqrt-prod 91.6%

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

      3. pow1/2 91.6%

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

      4. inv-pow 91.6%

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

      5. pow-pow 92.2%

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

      6. metadata-eval 92.2%

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

   3. Applied egg-rr 92.2%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{1}{\frac{\sqrt{-V}}{\sqrt{\frac{-A}{\ell}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-306}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-317}:\\ \;\;\;\;c0 \cdot \frac{1}{\frac{\sqrt{-V}}{\sqrt{\frac{-A}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \end{array} \]

Alternative 4: 90.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-306}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{-1}{\frac{-A}{\ell}}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* V l) -5e-306)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 0.0)
       (* c0 (pow (* V (/ -1.0 (/ (- A) l))) -0.5))
       (* c0 (* (sqrt A) (pow (* V l) -0.5)))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= -5e-306) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * pow((V * (-1.0 / (-A / l))), -0.5);
	} else {
		tmp = c0 * (sqrt(A) * pow((V * l), -0.5));
	}
	return tmp;
}

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= -5e-306) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * Math.pow((V * (-1.0 / (-A / l))), -0.5);
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.pow((V * l), -0.5));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= -5e-306:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = c0 * math.pow((V * (-1.0 / (-A / l))), -0.5)
	else:
		tmp = c0 * (math.sqrt(A) * math.pow((V * l), -0.5))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= -5e-306)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * (Float64(V * Float64(-1.0 / Float64(Float64(-A) / l))) ^ -0.5));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * (Float64(V * l) ^ -0.5)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= -5e-306)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = c0 * ((V * (-1.0 / (-A / l))) ^ -0.5);
	else
		tmp = c0 * (sqrt(A) * ((V * l) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-306], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[Power[N[(V * N[(-1.0 / N[((-A) / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 16.6%

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

      1. associate-/r* 50.4%

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

      2. sqrt-div 52.2%

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

   3. Applied egg-rr 52.2%

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

   if -inf.0 < (*.f64 V l) < -4.99999999999999998e-306

   1. Initial program 84.8%

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

      1. frac-2neg 84.8%

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

      2. sqrt-div 99.5%

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

      3. *-commutative 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

   3. Applied egg-rr 99.5%

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

   if -4.99999999999999998e-306 < (*.f64 V l) < -0.0

   1. Initial program 39.6%

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

      1. pow1/2 39.6%

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

      2. clear-num 39.6%

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

      3. inv-pow 39.6%

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

      4. pow-pow 39.6%

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

      5. associate-/l* 56.6%

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

      6. metadata-eval 56.6%

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

   3. Applied egg-rr 56.6%

      \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}} \]
   4. Step-by-step derivation

      1. associate-/l* 39.6%

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

   5. Simplified 39.6%

      \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}} \]
   6. Step-by-step derivation

      1. associate-/l* 56.6%

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

      2. frac-2neg 56.6%

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

      3. div-inv 56.7%

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

   7. Applied egg-rr 56.7%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 79.8%

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

      1. div-inv 79.8%

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

      2. sqrt-prod 90.8%

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

      3. pow1/2 90.8%

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

      4. inv-pow 90.8%

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

      5. pow-pow 91.9%

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

      6. metadata-eval 91.9%

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

   3. Applied egg-rr 91.9%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-306}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{-1}{\frac{-A}{\ell}}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \end{array} \]

Alternative 5: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: 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 (/ A V)) (sqrt l)))
   (* c0 (* (sqrt A) (pow (* V l) -0.5)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else {
		tmp = c0 * (sqrt(A) * pow((V * l), -0.5));
	}
	return tmp;
}

Fortran

NOTE: 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((a / v)) / sqrt(l))
    else
        tmp = c0 * (sqrt(a) * ((v * l) ** (-0.5d0)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.pow((V * l), -0.5));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -5e-310:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	else:
		tmp = c0 * (math.sqrt(A) * math.pow((V * l), -0.5))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -5e-310)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * (Float64(V * l) ^ -0.5)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (A <= -5e-310)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	else
		tmp = c0 * (sqrt(A) * ((V * l) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -5e-310], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -4.999999999999985e-310

   1. Initial program 68.3%

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

      1. associate-/r* 71.5%

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

      2. sqrt-div 40.0%

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

   3. Applied egg-rr 40.0%

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

   if -4.999999999999985e-310 < A

   1. Initial program 76.6%

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

      1. div-inv 76.6%

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

      2. sqrt-prod 86.4%

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

      3. pow1/2 86.4%

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

      4. inv-pow 86.4%

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

      5. pow-pow 87.4%

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

      6. metadata-eval 87.4%

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

   3. Applied egg-rr 87.4%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \end{array} \]

Alternative 6: 81.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: 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 <= (-5d-310)) then
        tmp = c0 * sqrt((a / (v * l)))
    else
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 73.3%

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

   if -4.999999999999985e-310 < l

   1. Initial program 71.2%

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

      1. associate-/r* 74.4%

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

      2. sqrt-div 87.0%

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

   3. Applied egg-rr 87.0%

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

4. Final simplification 79.7%

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

Alternative 7: 83.1% accurate, 0.5× speedup

Math

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

FPCore

NOTE: 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 (/ A V)) (sqrt l)))
   (* (sqrt A) (/ c0 (sqrt (* V l))))))

C

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

Fortran

NOTE: 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((a / v)) / sqrt(l))
    else
        tmp = sqrt(a) * (c0 / sqrt((v * l)))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if A < -4.999999999999985e-310

   1. Initial program 68.3%

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

      1. associate-/r* 71.5%

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

      2. sqrt-div 40.0%

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

   3. Applied egg-rr 40.0%

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

   if -4.999999999999985e-310 < A

   1. Initial program 76.6%

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

      1. sqrt-div 87.3%

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

      2. associate-*r/ 83.6%

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

   3. Applied egg-rr 83.6%

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

      1. associate-*l/ 83.6%

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

   5. Simplified 83.6%

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

4. Final simplification 61.0%

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

Alternative 8: 84.4% accurate, 0.5× speedup

Math

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

FPCore

NOTE: 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 (/ A V)) (sqrt l)))
   (* c0 (/ (sqrt A) (sqrt (* V l))))))

C

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

Fortran

NOTE: 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((a / v)) / sqrt(l))
    else
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if A < -4.999999999999985e-310

   1. Initial program 68.3%

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

      1. associate-/r* 71.5%

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

      2. sqrt-div 40.0%

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

   3. Applied egg-rr 40.0%

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

   if -4.999999999999985e-310 < A

   1. Initial program 76.6%

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

      1. sqrt-div 87.3%

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

      2. associate-*r/ 83.6%

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

   3. Applied egg-rr 83.6%

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

      1. *-commutative 83.6%

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

      2. associate-*l/ 87.3%

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

   5. Simplified 87.3%

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

4. Final simplification 62.7%

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

Alternative 9: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+297}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 5e+297) (* c0 (sqrt t_0)) (/ c0 (sqrt (* V (/ l A))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 5e+297) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: 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 / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 5d+297) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 5e+297) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 5e+297:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 5e+297)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 5e+297)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, 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, 5e+297], 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 32.0%

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

      1. add-sqr-sqrt 32.0%

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

      2. pow2 32.0%

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

      3. pow1/2 32.0%

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

      4. sqrt-pow1 32.0%

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

      5. metadata-eval 32.0%

         \[\leadsto c0 \cdot {\left({\left(\frac{A}{V \cdot \ell}\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   3. Applied egg-rr 32.0%

      \[\leadsto c0 \cdot \color{blue}{{\left({\left(\frac{A}{V \cdot \ell}\right)}^{0.25}\right)}^{2}} \]

   4. Taylor expanded in c0 around 0 32.0%

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

      1. associate-/r* 48.8%

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

   6. Simplified 48.8%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.9999999999999998e297

   1. Initial program 98.9%

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

   if 4.9999999999999998e297 < (/.f64 A (*.f64 V l))

   1. Initial program 32.5%

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

      1. pow1/2 32.5%

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

      2. clear-num 32.5%

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

      3. inv-pow 32.5%

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

      4. pow-pow 35.3%

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

      5. associate-/l* 41.7%

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

      6. metadata-eval 41.7%

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

   3. Applied egg-rr 41.7%

      \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}} \]
   4. Step-by-step derivation

      1. associate-/l* 35.3%

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

   5. Simplified 35.3%

      \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}} \]
   6. Step-by-step derivation

      1. sqr-pow 35.3%

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

      2. sqr-pow 35.3%

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

      3. metadata-eval 35.3%

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

      4. sqrt-pow2 35.3%

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

      5. inv-pow 35.3%

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

      6. un-div-inv 35.3%

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

      7. div-inv 35.3%

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

      8. associate-*l* 41.7%

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

      9. div-inv 41.7%

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

   7. Applied egg-rr 41.7%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+297}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]

Alternative 10: 78.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: 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 / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = c0 * sqrt(t_0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 32.0%

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

      1. add-sqr-sqrt 32.0%

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

      2. pow2 32.0%

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

      3. pow1/2 32.0%

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

      4. sqrt-pow1 32.0%

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

      5. metadata-eval 32.0%

         \[\leadsto c0 \cdot {\left({\left(\frac{A}{V \cdot \ell}\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   3. Applied egg-rr 32.0%

      \[\leadsto c0 \cdot \color{blue}{{\left({\left(\frac{A}{V \cdot \ell}\right)}^{0.25}\right)}^{2}} \]

   4. Taylor expanded in c0 around 0 32.0%

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

      1. associate-/r* 48.8%

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

   6. Simplified 48.8%

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

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

   1. Initial program 83.9%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]

Alternative 11: 78.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \end{array} \end{array} \]

FPCore

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

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((A / (V * l)) <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = c0 / sqrt(((V * l) / A));
	}
	return tmp;
}

Fortran

NOTE: 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 / (v * l)) <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = c0 / sqrt(((v * l) / a))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 32.0%

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

      1. add-sqr-sqrt 32.0%

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

      2. pow2 32.0%

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

      3. pow1/2 32.0%

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

      4. sqrt-pow1 32.0%

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

      5. metadata-eval 32.0%

         \[\leadsto c0 \cdot {\left({\left(\frac{A}{V \cdot \ell}\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   3. Applied egg-rr 32.0%

      \[\leadsto c0 \cdot \color{blue}{{\left({\left(\frac{A}{V \cdot \ell}\right)}^{0.25}\right)}^{2}} \]

   4. Taylor expanded in c0 around 0 32.0%

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

      1. associate-/r* 48.8%

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

   6. Simplified 48.8%

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

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

   1. Initial program 83.9%

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

      1. pow1/2 83.9%

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

      2. clear-num 83.8%

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

      3. inv-pow 83.8%

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

      4. pow-pow 84.5%

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

      5. associate-/l* 75.0%

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

      6. metadata-eval 75.0%

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

   3. Applied egg-rr 75.0%

      \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}} \]
   4. Step-by-step derivation

      1. associate-/l* 84.5%

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

   5. Simplified 84.5%

      \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}} \]
   6. Step-by-step derivation

      1. sqr-pow 84.2%

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

      2. sqr-pow 84.5%

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

      3. metadata-eval 84.5%

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

      4. sqrt-pow2 84.4%

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

      5. inv-pow 84.4%

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

      6. un-div-inv 84.5%

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

      7. div-inv 84.5%

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

      8. associate-*l* 74.2%

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

      9. div-inv 74.2%

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

   7. Applied egg-rr 74.2%

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

   8. Taylor expanded in V around 0 84.5%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \end{array} \]

Alternative 12: 74.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.3%

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

2. Final simplification 72.3%

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

Reproduce

herbie shell --seed 2023261 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))