Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.6% → 91.0%

Time: 10.6s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-285}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-316}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\ell \cdot \frac{V}{A}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+262}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (* c0 (/ (sqrt (/ A (- l))) (sqrt (- V))))
   (if (<= (* V l) -2e-285)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 1e-316)
       (/ 1.0 (/ (sqrt (* l (/ V A))) c0))
       (if (<= (* V l) 5e+262)
         (* c0 (/ (sqrt A) (sqrt (* V l))))
         (/ c0 (sqrt (/ V (/ A l)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 * (sqrt((A / -l)) / sqrt(-V));
	} else if ((V * l) <= -2e-285) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 1e-316) {
		tmp = 1.0 / (sqrt((l * (V / A))) / c0);
	} else if ((V * l) <= 5e+262) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 / sqrt((V / (A / l)));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 * (Math.sqrt((A / -l)) / Math.sqrt(-V));
	} else if ((V * l) <= -2e-285) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 1e-316) {
		tmp = 1.0 / (Math.sqrt((l * (V / A))) / c0);
	} else if ((V * l) <= 5e+262) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 / Math.sqrt((V / (A / l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 * (math.sqrt((A / -l)) / math.sqrt(-V))
	elif (V * l) <= -2e-285:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 1e-316:
		tmp = 1.0 / (math.sqrt((l * (V / A))) / c0)
	elif (V * l) <= 5e+262:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 / math.sqrt((V / (A / l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 * Float64(sqrt(Float64(A / Float64(-l))) / sqrt(Float64(-V))));
	elseif (Float64(V * l) <= -2e-285)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 1e-316)
		tmp = Float64(1.0 / Float64(sqrt(Float64(l * Float64(V / A))) / c0));
	elseif (Float64(V * l) <= 5e+262)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 / sqrt(Float64(V / Float64(A / l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 * (sqrt((A / -l)) / sqrt(-V));
	elseif ((V * l) <= -2e-285)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 1e-316)
		tmp = 1.0 / (sqrt((l * (V / A))) / c0);
	elseif ((V * l) <= 5e+262)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 / sqrt((V / (A / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 * N[(N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-285], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-316], N[(1.0 / N[(N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+262], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 52.1%

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

      1. associate-/r* 68.3%

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

      2. div-inv 68.3%

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

      3. div-inv 68.3%

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

      4. associate-*l* 58.5%

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

   4. Applied egg-rr 58.5%

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

      1. *-commutative 58.5%

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

      2. associate-*r* 68.2%

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

      3. div-inv 68.3%

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

      4. div-inv 68.3%

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

      5. frac-2neg 68.3%

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

      6. sqrt-div 41.7%

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

      7. distribute-neg-frac2 41.7%

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

   6. Applied egg-rr 41.7%

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

      1. distribute-frac-neg2 41.7%

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

      2. distribute-frac-neg 41.7%

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

   8. Simplified 41.7%

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

   if -inf.0 < (*.f64 V l) < -2.00000000000000015e-285

   1. Initial program 88.5%

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

      1. frac-2neg 88.5%

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

      2. sqrt-div 99.4%

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

      3. *-commutative 99.4%

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

      4. distribute-rgt-neg-in 99.4%

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

   4. Applied egg-rr 99.4%

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

   if -2.00000000000000015e-285 < (*.f64 V l) < 9.999999837e-317

   1. Initial program 52.6%

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

      1. associate-/r* 72.5%

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

      2. div-inv 72.4%

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

      3. div-inv 72.5%

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

      4. associate-*l* 48.8%

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

   4. Applied egg-rr 48.8%

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

      1. add-sqr-sqrt 29.8%

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

      2. sqrt-unprod 26.7%

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

      3. swap-sqr 26.3%

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

      4. add-sqr-sqrt 26.3%

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

      5. frac-times 26.3%

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

      6. metadata-eval 26.3%

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

      7. div-inv 26.3%

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

      8. clear-num 26.3%

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

      9. associate-*r/ 30.4%

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

      10. metadata-eval 30.4%

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

      11. add-sqr-sqrt 30.4%

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

      12. frac-times 30.4%

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

      13. swap-sqr 33.8%

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

      14. sqrt-unprod 40.0%

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

   6. Applied egg-rr 72.6%

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

   if 9.999999837e-317 < (*.f64 V l) < 5.00000000000000008e262

   1. Initial program 79.2%

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

      1. sqrt-div 99.2%

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

      2. associate-*r/ 96.5%

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

   4. Applied egg-rr 96.5%

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

      1. associate-/l* 99.2%

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

   6. Simplified 99.2%

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

   if 5.00000000000000008e262 < (*.f64 V l)

   1. Initial program 45.9%

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

      1. associate-/r* 79.7%

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

      2. div-inv 79.7%

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

      3. div-inv 79.8%

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

      4. associate-*l* 48.8%

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

   4. Applied egg-rr 48.8%

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

      1. add-sqr-sqrt 45.5%

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

      2. sqrt-unprod 40.9%

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

      3. swap-sqr 34.7%

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

      4. add-sqr-sqrt 34.7%

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

      5. frac-times 34.8%

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

      6. metadata-eval 34.8%

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

      7. div-inv 34.7%

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

      8. clear-num 34.7%

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

      9. associate-*r/ 34.9%

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

      10. metadata-eval 34.9%

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

      11. add-sqr-sqrt 34.9%

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

      12. frac-times 34.9%

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

      13. swap-sqr 40.8%

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

      14. sqrt-unprod 48.0%

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

   6. Applied egg-rr 79.6%

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

      1. *-commutative 79.6%

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

      2. associate-*l/ 45.9%

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

      3. associate-/l* 79.8%

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

   8. Simplified 79.8%

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

      1. clear-num 79.8%

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

      2. div-inv 79.8%

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

   10. Applied egg-rr 79.8%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-285}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-316}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\ell \cdot \frac{V}{A}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+262}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 4 \cdot 10^{+251}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (or (<= t_0 0.0) (not (<= t_0 4e+251)))
     (* c0 (sqrt (/ (/ A V) l)))
     t_0)))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 4e+251)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 4d+251))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 4e+251)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 4e+251):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 4e+251))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 4e+251)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 4e+251]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 0.0 or 4.0000000000000002e251 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 65.1%

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

      1. *-commutative 65.1%

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

      2. associate-/l/ 68.1%

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

   3. Simplified 68.1%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.0000000000000002e251

   1. Initial program 99.5%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0 \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 4 \cdot 10^{+251}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{\frac{\sqrt{-A}}{\sqrt{-V}} \cdot c0}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -1e-311)
   (/ (* (/ (sqrt (- A)) (sqrt (- V))) c0) (sqrt l))
   (* c0 (/ (sqrt A) (sqrt (* V l))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -1e-311) {
		tmp = ((sqrt(-A) / sqrt(-V)) * c0) / sqrt(l);
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (a <= (-1d-311)) then
        tmp = ((sqrt(-a) / sqrt(-v)) * c0) / sqrt(l)
    else
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -1e-311) {
		tmp = ((Math.sqrt(-A) / Math.sqrt(-V)) * c0) / Math.sqrt(l);
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -1e-311:
		tmp = ((math.sqrt(-A) / math.sqrt(-V)) * c0) / math.sqrt(l)
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -1e-311)
		tmp = Float64(Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(-V))) * c0) / sqrt(l));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (A <= -1e-311)
		tmp = ((sqrt(-A) / sqrt(-V)) * c0) / sqrt(l);
	else
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -1e-311], N[(N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[l], $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 < -9.99999999999948e-312

   1. Initial program 81.3%

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

      1. *-commutative 81.3%

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

      2. associate-/r* 74.0%

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

      3. sqrt-div 39.5%

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

      4. associate-*l/ 38.2%

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

   4. Applied egg-rr 38.2%

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

      1. frac-2neg 38.2%

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

      2. sqrt-div 46.2%

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

   6. Applied egg-rr 46.2%

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

   if -9.99999999999948e-312 < A

   1. Initial program 70.8%

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

      1. sqrt-div 85.6%

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

      2. associate-*r/ 83.6%

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

   4. Applied egg-rr 83.6%

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

      1. associate-/l* 85.6%

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

   6. Simplified 85.6%

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

Alternative 4: 87.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{-285}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-316}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\ell \cdot \frac{V}{A}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+262}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -2e-285)
   (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
   (if (<= (* V l) 1e-316)
     (/ 1.0 (/ (sqrt (* l (/ V A))) c0))
     (if (<= (* V l) 5e+262)
       (* c0 (/ (sqrt A) (sqrt (* V l))))
       (/ c0 (sqrt (/ V (/ A l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e-285) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 1e-316) {
		tmp = 1.0 / (sqrt((l * (V / A))) / c0);
	} else if ((V * l) <= 5e+262) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 / sqrt((V / (A / l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-2d-285)) then
        tmp = c0 * (sqrt(-a) / sqrt((v * -l)))
    else if ((v * l) <= 1d-316) then
        tmp = 1.0d0 / (sqrt((l * (v / a))) / c0)
    else if ((v * l) <= 5d+262) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 / sqrt((v / (a / l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e-285) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 1e-316) {
		tmp = 1.0 / (Math.sqrt((l * (V / A))) / c0);
	} else if ((V * l) <= 5e+262) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 / Math.sqrt((V / (A / l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -2e-285:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 1e-316:
		tmp = 1.0 / (math.sqrt((l * (V / A))) / c0)
	elif (V * l) <= 5e+262:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 / math.sqrt((V / (A / l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -2e-285)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 1e-316)
		tmp = Float64(1.0 / Float64(sqrt(Float64(l * Float64(V / A))) / c0));
	elseif (Float64(V * l) <= 5e+262)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 / sqrt(Float64(V / Float64(A / l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -2e-285)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 1e-316)
		tmp = 1.0 / (sqrt((l * (V / A))) / c0);
	elseif ((V * l) <= 5e+262)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 / sqrt((V / (A / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -2e-285], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-316], N[(1.0 / N[(N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+262], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -2.00000000000000015e-285

   1. Initial program 85.1%

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

      1. frac-2neg 85.1%

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

      2. sqrt-div 94.9%

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

      3. *-commutative 94.9%

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

      4. distribute-rgt-neg-in 94.9%

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

   4. Applied egg-rr 94.9%

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

   if -2.00000000000000015e-285 < (*.f64 V l) < 9.999999837e-317

   1. Initial program 52.6%

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

      1. associate-/r* 72.5%

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

      2. div-inv 72.4%

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

      3. div-inv 72.5%

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

      4. associate-*l* 48.8%

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

   4. Applied egg-rr 48.8%

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

      1. add-sqr-sqrt 29.8%

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

      2. sqrt-unprod 26.7%

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

      3. swap-sqr 26.3%

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

      4. add-sqr-sqrt 26.3%

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

      5. frac-times 26.3%

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

      6. metadata-eval 26.3%

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

      7. div-inv 26.3%

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

      8. clear-num 26.3%

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

      9. associate-*r/ 30.4%

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

      10. metadata-eval 30.4%

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

      11. add-sqr-sqrt 30.4%

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

      12. frac-times 30.4%

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

      13. swap-sqr 33.8%

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

      14. sqrt-unprod 40.0%

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

   6. Applied egg-rr 72.6%

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

   if 9.999999837e-317 < (*.f64 V l) < 5.00000000000000008e262

   1. Initial program 79.2%

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

      1. sqrt-div 99.2%

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

      2. associate-*r/ 96.5%

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

   4. Applied egg-rr 96.5%

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

      1. associate-/l* 99.2%

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

   6. Simplified 99.2%

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

   if 5.00000000000000008e262 < (*.f64 V l)

   1. Initial program 45.9%

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

      1. associate-/r* 79.7%

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

      2. div-inv 79.7%

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

      3. div-inv 79.8%

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

      4. associate-*l* 48.8%

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

   4. Applied egg-rr 48.8%

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

      1. add-sqr-sqrt 45.5%

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

      2. sqrt-unprod 40.9%

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

      3. swap-sqr 34.7%

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

      4. add-sqr-sqrt 34.7%

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

      5. frac-times 34.8%

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

      6. metadata-eval 34.8%

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

      7. div-inv 34.7%

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

      8. clear-num 34.7%

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

      9. associate-*r/ 34.9%

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

      10. metadata-eval 34.9%

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

      11. add-sqr-sqrt 34.9%

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

      12. frac-times 34.9%

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

      13. swap-sqr 40.8%

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

      14. sqrt-unprod 48.0%

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

   6. Applied egg-rr 79.6%

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

      1. *-commutative 79.6%

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

      2. associate-*l/ 45.9%

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

      3. associate-/l* 79.8%

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

   8. Simplified 79.8%

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

      1. clear-num 79.8%

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

      2. div-inv 79.8%

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

   10. Applied egg-rr 79.8%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{-285}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-316}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\ell \cdot \frac{V}{A}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+262}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{-122}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \left(\frac{1}{V} \cdot \frac{1}{\ell}\right)}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+262}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -5e-122)
   (* c0 (sqrt (* A (* (/ 1.0 V) (/ 1.0 l)))))
   (if (<= (* V l) 0.0)
     (/ c0 (* (sqrt l) (sqrt (/ V A))))
     (if (<= (* V l) 5e+262)
       (* c0 (/ (sqrt A) (sqrt (* V l))))
       (/ c0 (sqrt (/ V (/ A l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e-122) {
		tmp = c0 * sqrt((A * ((1.0 / V) * (1.0 / l))));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	} else if ((V * l) <= 5e+262) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 / sqrt((V / (A / l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-5d-122)) then
        tmp = c0 * sqrt((a * ((1.0d0 / v) * (1.0d0 / l))))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 / (sqrt(l) * sqrt((v / a)))
    else if ((v * l) <= 5d+262) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 / sqrt((v / (a / l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e-122) {
		tmp = c0 * Math.sqrt((A * ((1.0 / V) * (1.0 / l))));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (Math.sqrt(l) * Math.sqrt((V / A)));
	} else if ((V * l) <= 5e+262) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 / Math.sqrt((V / (A / l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -5e-122:
		tmp = c0 * math.sqrt((A * ((1.0 / V) * (1.0 / l))))
	elif (V * l) <= 0.0:
		tmp = c0 / (math.sqrt(l) * math.sqrt((V / A)))
	elif (V * l) <= 5e+262:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 / math.sqrt((V / (A / l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -5e-122)
		tmp = Float64(c0 * sqrt(Float64(A * Float64(Float64(1.0 / V) * Float64(1.0 / l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))));
	elseif (Float64(V * l) <= 5e+262)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 / sqrt(Float64(V / Float64(A / l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -5e-122)
		tmp = c0 * sqrt((A * ((1.0 / V) * (1.0 / l))));
	elseif ((V * l) <= 0.0)
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	elseif ((V * l) <= 5e+262)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 / sqrt((V / (A / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -5e-122], N[(c0 * N[Sqrt[N[(A * N[(N[(1.0 / V), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+262], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -4.9999999999999999e-122

   1. Initial program 87.5%

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

      1. associate-/r* 75.2%

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

      2. div-inv 75.2%

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

      3. div-inv 75.2%

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

      4. associate-*l* 88.3%

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

   4. Applied egg-rr 88.3%

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

   if -4.9999999999999999e-122 < (*.f64 V l) < 0.0

   1. Initial program 65.1%

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

      1. associate-/r* 72.6%

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

      2. div-inv 72.6%

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

      3. div-inv 72.6%

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

      4. associate-*l* 64.0%

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

   4. Applied egg-rr 64.0%

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

      1. frac-times 64.1%

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

      2. metadata-eval 64.1%

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

      3. div-inv 65.1%

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

      4. clear-num 65.0%

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

      5. associate-*r/ 70.9%

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

      6. metadata-eval 70.9%

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

      7. add-sqr-sqrt 70.9%

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

      8. frac-times 70.9%

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

      9. sqrt-unprod 70.9%

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

      10. add-sqr-sqrt 70.9%

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

      11. inv-pow 70.9%

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

      12. sqrt-pow2 71.0%

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

      13. associate-*r/ 65.0%

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

      14. *-commutative 65.0%

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

      15. *-un-lft-identity 65.0%

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

      16. times-frac 72.8%

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

      17. /-rgt-identity 72.8%

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

      18. metadata-eval 72.8%

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

   6. Applied egg-rr 72.8%

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

      1. metadata-eval 72.8%

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

      2. pow-flip 72.6%

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

      3. pow1/2 72.6%

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

      4. div-inv 72.8%

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

      5. sqrt-prod 41.3%

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

      6. associate-/r* 39.8%

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

   8. Applied egg-rr 39.8%

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

      1. associate-/r* 41.3%

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

   10. Simplified 41.3%

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

   if 0.0 < (*.f64 V l) < 5.00000000000000008e262

   1. Initial program 78.9%

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

      1. sqrt-div 98.7%

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

      2. associate-*r/ 96.1%

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

   4. Applied egg-rr 96.1%

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

      1. associate-/l* 98.7%

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

   6. Simplified 98.7%

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

   if 5.00000000000000008e262 < (*.f64 V l)

   1. Initial program 45.9%

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

      1. associate-/r* 79.7%

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

      2. div-inv 79.7%

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

      3. div-inv 79.8%

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

      4. associate-*l* 48.8%

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

   4. Applied egg-rr 48.8%

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

      1. add-sqr-sqrt 45.5%

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

      2. sqrt-unprod 40.9%

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

      3. swap-sqr 34.7%

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

      4. add-sqr-sqrt 34.7%

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

      5. frac-times 34.8%

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

      6. metadata-eval 34.8%

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

      7. div-inv 34.7%

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

      8. clear-num 34.7%

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

      9. associate-*r/ 34.9%

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

      10. metadata-eval 34.9%

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

      11. add-sqr-sqrt 34.9%

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

      12. frac-times 34.9%

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

      13. swap-sqr 40.8%

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

      14. sqrt-unprod 48.0%

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

   6. Applied egg-rr 79.6%

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

      1. *-commutative 79.6%

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

      2. associate-*l/ 45.9%

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

      3. associate-/l* 79.8%

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

   8. Simplified 79.8%

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

      1. clear-num 79.8%

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

      2. div-inv 79.8%

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

   10. Applied egg-rr 79.8%

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

Alternative 6: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{-124}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \left(\frac{1}{V} \cdot \frac{1}{\ell}\right)}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+262}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e-124)
   (* c0 (sqrt (* A (* (/ 1.0 V) (/ 1.0 l)))))
   (if (<= (* V l) 0.0)
     (* c0 (/ (sqrt (/ A V)) (sqrt l)))
     (if (<= (* V l) 5e+262)
       (* c0 (/ (sqrt A) (sqrt (* V l))))
       (/ c0 (sqrt (/ V (/ A l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e-124) {
		tmp = c0 * sqrt((A * ((1.0 / V) * (1.0 / l))));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= 5e+262) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 / sqrt((V / (A / l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-1d-124)) then
        tmp = c0 * sqrt((a * ((1.0d0 / v) * (1.0d0 / l))))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= 5d+262) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 / sqrt((v / (a / l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e-124) {
		tmp = c0 * Math.sqrt((A * ((1.0 / V) * (1.0 / l))));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= 5e+262) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 / Math.sqrt((V / (A / l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e-124:
		tmp = c0 * math.sqrt((A * ((1.0 / V) * (1.0 / l))))
	elif (V * l) <= 0.0:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= 5e+262:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 / math.sqrt((V / (A / l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e-124)
		tmp = Float64(c0 * sqrt(Float64(A * Float64(Float64(1.0 / V) * Float64(1.0 / l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= 5e+262)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 / sqrt(Float64(V / Float64(A / l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e-124)
		tmp = c0 * sqrt((A * ((1.0 / V) * (1.0 / l))));
	elseif ((V * l) <= 0.0)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= 5e+262)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 / sqrt((V / (A / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -1e-124], N[(c0 * N[Sqrt[N[(A * N[(N[(1.0 / V), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+262], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -9.99999999999999933e-125

   1. Initial program 87.6%

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

      1. associate-/r* 75.5%

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

      2. div-inv 75.4%

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

      3. div-inv 75.4%

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

      4. associate-*l* 88.4%

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

   4. Applied egg-rr 88.4%

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

   if -9.99999999999999933e-125 < (*.f64 V l) < 0.0

   1. Initial program 64.5%

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

      1. associate-/r* 72.2%

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

      2. sqrt-div 42.2%

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

      3. associate-*r/ 42.2%

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

   4. Applied egg-rr 42.2%

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

      1. associate-/l* 42.2%

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

   6. Simplified 42.2%

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

   if 0.0 < (*.f64 V l) < 5.00000000000000008e262

   1. Initial program 78.9%

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

      1. sqrt-div 98.7%

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

      2. associate-*r/ 96.1%

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

   4. Applied egg-rr 96.1%

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

      1. associate-/l* 98.7%

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

   6. Simplified 98.7%

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

   if 5.00000000000000008e262 < (*.f64 V l)

   1. Initial program 45.9%

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

      1. associate-/r* 79.7%

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

      2. div-inv 79.7%

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

      3. div-inv 79.8%

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

      4. associate-*l* 48.8%

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

   4. Applied egg-rr 48.8%

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

      1. add-sqr-sqrt 45.5%

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

      2. sqrt-unprod 40.9%

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

      3. swap-sqr 34.7%

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

      4. add-sqr-sqrt 34.7%

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

      5. frac-times 34.8%

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

      6. metadata-eval 34.8%

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

      7. div-inv 34.7%

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

      8. clear-num 34.7%

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

      9. associate-*r/ 34.9%

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

      10. metadata-eval 34.9%

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

      11. add-sqr-sqrt 34.9%

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

      12. frac-times 34.9%

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

      13. swap-sqr 40.8%

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

      14. sqrt-unprod 48.0%

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

   6. Applied egg-rr 79.6%

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

      1. *-commutative 79.6%

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

      2. associate-*l/ 45.9%

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

      3. associate-/l* 79.8%

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

   8. Simplified 79.8%

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

      1. clear-num 79.8%

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

      2. div-inv 79.8%

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

   10. Applied egg-rr 79.8%

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

Alternative 7: 83.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{-214}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \left(\frac{1}{V} \cdot \frac{1}{\ell}\right)}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-316}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\ell \cdot \frac{V}{A}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+262}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -2e-214)
   (* c0 (sqrt (* A (* (/ 1.0 V) (/ 1.0 l)))))
   (if (<= (* V l) 1e-316)
     (/ 1.0 (/ (sqrt (* l (/ V A))) c0))
     (if (<= (* V l) 5e+262)
       (* c0 (/ (sqrt A) (sqrt (* V l))))
       (/ c0 (sqrt (/ V (/ A l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e-214) {
		tmp = c0 * sqrt((A * ((1.0 / V) * (1.0 / l))));
	} else if ((V * l) <= 1e-316) {
		tmp = 1.0 / (sqrt((l * (V / A))) / c0);
	} else if ((V * l) <= 5e+262) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 / sqrt((V / (A / l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-2d-214)) then
        tmp = c0 * sqrt((a * ((1.0d0 / v) * (1.0d0 / l))))
    else if ((v * l) <= 1d-316) then
        tmp = 1.0d0 / (sqrt((l * (v / a))) / c0)
    else if ((v * l) <= 5d+262) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0 / sqrt((v / (a / l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e-214) {
		tmp = c0 * Math.sqrt((A * ((1.0 / V) * (1.0 / l))));
	} else if ((V * l) <= 1e-316) {
		tmp = 1.0 / (Math.sqrt((l * (V / A))) / c0);
	} else if ((V * l) <= 5e+262) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 / Math.sqrt((V / (A / l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -2e-214:
		tmp = c0 * math.sqrt((A * ((1.0 / V) * (1.0 / l))))
	elif (V * l) <= 1e-316:
		tmp = 1.0 / (math.sqrt((l * (V / A))) / c0)
	elif (V * l) <= 5e+262:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0 / math.sqrt((V / (A / l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -2e-214)
		tmp = Float64(c0 * sqrt(Float64(A * Float64(Float64(1.0 / V) * Float64(1.0 / l)))));
	elseif (Float64(V * l) <= 1e-316)
		tmp = Float64(1.0 / Float64(sqrt(Float64(l * Float64(V / A))) / c0));
	elseif (Float64(V * l) <= 5e+262)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 / sqrt(Float64(V / Float64(A / l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -2e-214)
		tmp = c0 * sqrt((A * ((1.0 / V) * (1.0 / l))));
	elseif ((V * l) <= 1e-316)
		tmp = 1.0 / (sqrt((l * (V / A))) / c0);
	elseif ((V * l) <= 5e+262)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0 / sqrt((V / (A / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -2e-214], N[(c0 * N[Sqrt[N[(A * N[(N[(1.0 / V), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-316], N[(1.0 / N[(N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+262], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -1.99999999999999983e-214

   1. Initial program 86.6%

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

      1. associate-/r* 75.2%

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

      2. div-inv 75.1%

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

      3. div-inv 75.1%

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

      4. associate-*l* 87.3%

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

   4. Applied egg-rr 87.3%

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

   if -1.99999999999999983e-214 < (*.f64 V l) < 9.999999837e-317

   1. Initial program 59.3%

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

      1. associate-/r* 72.5%

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

      2. div-inv 72.4%

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

      3. div-inv 72.5%

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

      4. associate-*l* 56.6%

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

   4. Applied egg-rr 56.6%

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

      1. add-sqr-sqrt 24.9%

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

      2. sqrt-unprod 21.0%

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

      3. swap-sqr 20.8%

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

      4. add-sqr-sqrt 20.8%

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

      5. frac-times 20.8%

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

      6. metadata-eval 20.8%

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

      7. div-inv 20.8%

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

      8. clear-num 20.8%

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

      9. associate-*r/ 23.5%

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

      10. metadata-eval 23.5%

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

      11. add-sqr-sqrt 23.5%

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

      12. frac-times 23.4%

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

      13. swap-sqr 25.8%

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

      14. sqrt-unprod 31.8%

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

   6. Applied egg-rr 72.7%

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

   if 9.999999837e-317 < (*.f64 V l) < 5.00000000000000008e262

   1. Initial program 79.2%

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

      1. sqrt-div 99.2%

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

      2. associate-*r/ 96.5%

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

   4. Applied egg-rr 96.5%

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

      1. associate-/l* 99.2%

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

   6. Simplified 99.2%

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

   if 5.00000000000000008e262 < (*.f64 V l)

   1. Initial program 45.9%

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

      1. associate-/r* 79.7%

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

      2. div-inv 79.7%

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

      3. div-inv 79.8%

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

      4. associate-*l* 48.8%

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

   4. Applied egg-rr 48.8%

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

      1. add-sqr-sqrt 45.5%

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

      2. sqrt-unprod 40.9%

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

      3. swap-sqr 34.7%

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

      4. add-sqr-sqrt 34.7%

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

      5. frac-times 34.8%

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

      6. metadata-eval 34.8%

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

      7. div-inv 34.7%

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

      8. clear-num 34.7%

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

      9. associate-*r/ 34.9%

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

      10. metadata-eval 34.9%

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

      11. add-sqr-sqrt 34.9%

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

      12. frac-times 34.9%

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

      13. swap-sqr 40.8%

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

      14. sqrt-unprod 48.0%

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

   6. Applied egg-rr 79.6%

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

      1. *-commutative 79.6%

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

      2. associate-*l/ 45.9%

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

      3. associate-/l* 79.8%

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

   8. Simplified 79.8%

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

      1. clear-num 79.8%

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

      2. div-inv 79.8%

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

   10. Applied egg-rr 79.8%

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

Alternative 8: 73.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-312} \lor \neg \left(t\_0 \leq 10^{+297}\right):\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (or (<= t_0 2e-312) (not (<= t_0 1e+297)))
     (sqrt (* (/ A l) (* c0 (/ c0 V))))
     (* c0 (sqrt t_0)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 2e-312) || !(t_0 <= 1e+297)) {
		tmp = sqrt(((A / l) * (c0 * (c0 / V))));
	} else {
		tmp = c0 * sqrt(t_0);
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if ((t_0 <= 2d-312) .or. (.not. (t_0 <= 1d+297))) then
        tmp = sqrt(((a / l) * (c0 * (c0 / v))))
    else
        tmp = c0 * sqrt(t_0)
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 2e-312) || !(t_0 <= 1e+297)) {
		tmp = Math.sqrt(((A / l) * (c0 * (c0 / V))));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 2e-312) or not (t_0 <= 1e+297):
		tmp = math.sqrt(((A / l) * (c0 * (c0 / V))))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 2e-312) || !(t_0 <= 1e+297))
		tmp = sqrt(Float64(Float64(A / l) * Float64(c0 * Float64(c0 / V))));
	else
		tmp = Float64(c0 * sqrt(t_0));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 2e-312) || ~((t_0 <= 1e+297)))
		tmp = sqrt(((A / l) * (c0 * (c0 / V))));
	else
		tmp = c0 * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 2e-312], N[Not[LessEqual[t$95$0, 1e+297]], $MachinePrecision]], N[Sqrt[N[(N[(A / l), $MachinePrecision] * N[(c0 * N[(c0 / V), $MachinePrecision]), $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)) < 2.0000000000019e-312 or 1e297 < (/.f64 A (*.f64 V l))

   1. Initial program 36.0%

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

      1. add-sqr-sqrt 23.8%

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

      2. sqrt-unprod 23.8%

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

      3. *-commutative 23.8%

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

      4. *-commutative 23.8%

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

      5. swap-sqr 23.0%

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

      6. add-sqr-sqrt 23.0%

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

      7. pow2 23.0%

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

   4. Applied egg-rr 23.0%

      \[\leadsto \color{blue}{\sqrt{\frac{A}{V \cdot \ell} \cdot {c0}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*l/ 29.5%

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

      2. *-commutative 29.5%

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

      3. times-frac 30.8%

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

   6. Simplified 30.8%

      \[\leadsto \color{blue}{\sqrt{\frac{A}{\ell} \cdot \frac{{c0}^{2}}{V}}} \]
   7. Step-by-step derivation

      1. unpow2 30.8%

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

      2. *-un-lft-identity 30.8%

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

      3. times-frac 38.4%

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

      4. /-rgt-identity 38.4%

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

   8. Applied egg-rr 38.4%

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

   if 2.0000000000019e-312 < (/.f64 A (*.f64 V l)) < 1e297

   1. Initial program 99.1%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{-312} \lor \neg \left(\frac{A}{V \cdot \ell} \leq 10^{+297}\right):\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+282}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

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 (* V l))))
   (if (<= t_0 0.0)
     (* c0 (pow (/ V (/ A l)) -0.5))
     (if (<= t_0 4e+282) (* c0 (sqrt t_0)) (/ c0 (sqrt (* l (/ V A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * pow((V / (A / l)), -0.5);
	} else if (t_0 <= 4e+282) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((l * (V / 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 / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * ((v / (a / l)) ** (-0.5d0))
    else if (t_0 <= 4d+282) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((l * (v / 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 / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.pow((V / (A / l)), -0.5);
	} else if (t_0 <= 4e+282) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((l * (V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.pow((V / (A / l)), -0.5)
	elif t_0 <= 4e+282:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((l * (V / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * (Float64(V / Float64(A / l)) ^ -0.5));
	elseif (t_0 <= 4e+282)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / 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 / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * ((V / (A / l)) ^ -0.5);
	elseif (t_0 <= 4e+282)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((l * (V / 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[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Power[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+282], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 30.5%

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

      1. associate-/r* 51.0%

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

      2. div-inv 51.0%

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

      3. div-inv 51.0%

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

      4. associate-*l* 33.7%

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

   4. Applied egg-rr 33.7%

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

      1. frac-times 30.5%

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

      2. metadata-eval 30.5%

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

      3. div-inv 30.5%

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

      4. clear-num 30.5%

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

      5. associate-*r/ 51.0%

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

      6. metadata-eval 51.0%

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

      7. add-sqr-sqrt 50.9%

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

      8. frac-times 51.0%

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

      9. sqrt-unprod 50.9%

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

      10. add-sqr-sqrt 51.0%

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

      11. inv-pow 51.0%

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

      12. sqrt-pow2 51.0%

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

      13. associate-*r/ 30.5%

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

      14. *-commutative 30.5%

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

      15. *-un-lft-identity 30.5%

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

      16. times-frac 51.0%

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

      17. /-rgt-identity 51.0%

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

      18. metadata-eval 51.0%

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

   6. Applied egg-rr 51.0%

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

      1. associate-*r/ 30.5%

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

      2. *-commutative 30.5%

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

      3. associate-*r/ 51.0%

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

      4. clear-num 51.0%

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

      5. un-div-inv 51.0%

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

   8. Applied egg-rr 51.0%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.00000000000000013e282

   1. Initial program 99.2%

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

   if 4.00000000000000013e282 < (/.f64 A (*.f64 V l))

   1. Initial program 43.4%

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

      1. associate-/r* 51.4%

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

      2. div-inv 51.3%

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

      3. div-inv 51.4%

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

      4. associate-*l* 42.4%

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

   4. Applied egg-rr 42.4%

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

      1. add-sqr-sqrt 19.0%

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

      2. sqrt-unprod 19.2%

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

      3. swap-sqr 17.0%

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

      4. add-sqr-sqrt 17.0%

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

      5. frac-times 17.0%

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

      6. metadata-eval 17.0%

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

      7. div-inv 17.0%

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

      8. clear-num 17.0%

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

      9. associate-*r/ 19.0%

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

      10. metadata-eval 19.0%

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

      11. add-sqr-sqrt 19.0%

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

      12. frac-times 19.0%

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

      13. swap-sqr 21.1%

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

      14. sqrt-unprod 22.6%

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

   6. Applied egg-rr 51.4%

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

Alternative 10: 80.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+282}:\\ \;\;\;\;c0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

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 (* V l))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 4e+282) (* c0 (sqrt t_0)) (/ c0 (sqrt (* l (/ V A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 4e+282) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((l * (V / 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 / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 4d+282) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((l * (v / 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 / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 4e+282) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((l * (V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, 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 <= 4e+282:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((l * (V / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 4e+282)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / 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 / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 4e+282)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((l * (V / 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[(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, 4e+282], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 30.5%

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

      1. *-commutative 30.5%

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

      2. associate-/l/ 51.0%

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

   3. Simplified 51.0%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.00000000000000013e282

   1. Initial program 99.2%

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

   if 4.00000000000000013e282 < (/.f64 A (*.f64 V l))

   1. Initial program 43.4%

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

      1. associate-/r* 51.4%

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

      2. div-inv 51.3%

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

      3. div-inv 51.4%

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

      4. associate-*l* 42.4%

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

   4. Applied egg-rr 42.4%

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

      1. add-sqr-sqrt 19.0%

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

      2. sqrt-unprod 19.2%

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

      3. swap-sqr 17.0%

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

      4. add-sqr-sqrt 17.0%

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

      5. frac-times 17.0%

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

      6. metadata-eval 17.0%

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

      7. div-inv 17.0%

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

      8. clear-num 17.0%

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

      9. associate-*r/ 19.0%

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

      10. metadata-eval 19.0%

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

      11. add-sqr-sqrt 19.0%

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

      12. frac-times 19.0%

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

      13. swap-sqr 21.1%

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

      14. sqrt-unprod 22.6%

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

   6. Applied egg-rr 51.4%

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

Alternative 11: 73.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

Reproduce

herbie shell --seed 2024165 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))