Henrywood and Agarwal, Equation (3)

Percentage Accurate: 74.0% → 91.1%

Time: 10.5s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.0% 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: 91.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0\_m}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-307}:\\ \;\;\;\;\frac{c0\_m}{\frac{\sqrt{V \cdot \left(-\ell\right)}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-311}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0\_m}{\ell} \cdot \frac{c0\_m}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (*
  c0_s
  (if (<= (* V l) (- INFINITY))
    (/ c0_m (/ (sqrt l) (sqrt (/ A V))))
    (if (<= (* V l) -1e-307)
      (/ c0_m (/ (sqrt (* V (- l))) (sqrt (- A))))
      (if (<= (* V l) 1e-311)
        (sqrt (* A (* (/ c0_m l) (/ c0_m V))))
        (* c0_m (* (sqrt A) (sqrt (/ (/ 1.0 V) l)))))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0_m / (sqrt(l) / sqrt((A / V)));
	} else if ((V * l) <= -1e-307) {
		tmp = c0_m / (sqrt((V * -l)) / sqrt(-A));
	} else if ((V * l) <= 1e-311) {
		tmp = sqrt((A * ((c0_m / l) * (c0_m / V))));
	} else {
		tmp = c0_m * (sqrt(A) * sqrt(((1.0 / V) / l)));
	}
	return c0_s * tmp;
}

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0_m / (Math.sqrt(l) / Math.sqrt((A / V)));
	} else if ((V * l) <= -1e-307) {
		tmp = c0_m / (Math.sqrt((V * -l)) / Math.sqrt(-A));
	} else if ((V * l) <= 1e-311) {
		tmp = Math.sqrt((A * ((c0_m / l) * (c0_m / V))));
	} else {
		tmp = c0_m * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0_m / (math.sqrt(l) / math.sqrt((A / V)))
	elif (V * l) <= -1e-307:
		tmp = c0_m / (math.sqrt((V * -l)) / math.sqrt(-A))
	elif (V * l) <= 1e-311:
		tmp = math.sqrt((A * ((c0_m / l) * (c0_m / V))))
	else:
		tmp = c0_m * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0_m / Float64(sqrt(l) / sqrt(Float64(A / V))));
	elseif (Float64(V * l) <= -1e-307)
		tmp = Float64(c0_m / Float64(sqrt(Float64(V * Float64(-l))) / sqrt(Float64(-A))));
	elseif (Float64(V * l) <= 1e-311)
		tmp = sqrt(Float64(A * Float64(Float64(c0_m / l) * Float64(c0_m / V))));
	else
		tmp = Float64(c0_m * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0_m / (sqrt(l) / sqrt((A / V)));
	elseif ((V * l) <= -1e-307)
		tmp = c0_m / (sqrt((V * -l)) / sqrt(-A));
	elseif ((V * l) <= 1e-311)
		tmp = sqrt((A * ((c0_m / l) * (c0_m / V))));
	else
		tmp = c0_m * (sqrt(A) * sqrt(((1.0 / V) / l)));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0$95$m / N[(N[Sqrt[l], $MachinePrecision] / N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-307], N[(c0$95$m / N[(N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-311], N[Sqrt[N[(A * N[(N[(c0$95$m / l), $MachinePrecision] * N[(c0$95$m / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 42.3%

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

      1. *-un-lft-identity 42.3%

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

      2. times-frac 77.1%

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

   4. Applied egg-rr 77.1%

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

      1. frac-times 42.3%

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

      2. *-un-lft-identity 42.3%

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

      3. sqrt-undiv 0.0%

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

      4. clear-num 0.0%

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

      5. un-div-inv 0.0%

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

      6. sqrt-undiv 42.3%

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

   6. Applied egg-rr 42.3%

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

      1. associate-*r/ 71.3%

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

      2. *-commutative 71.3%

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

      3. associate-/r/ 71.3%

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

   8. Simplified 71.3%

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

      1. sqrt-div 54.8%

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

      2. div-inv 54.7%

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

   10. Applied egg-rr 54.7%

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

       1. associate-*r/ 54.8%

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

       2. *-rgt-identity 54.8%

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

   12. Simplified 54.8%

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

   if -inf.0 < (*.f64 V l) < -9.99999999999999909e-308

   1. Initial program 85.3%

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

      1. *-un-lft-identity 85.3%

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

      2. times-frac 79.0%

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

   4. Applied egg-rr 79.0%

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

      1. frac-times 85.3%

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

      2. *-un-lft-identity 85.3%

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

      3. sqrt-undiv 0.0%

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

      4. clear-num 0.0%

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

      5. un-div-inv 0.0%

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

      6. sqrt-undiv 86.8%

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

   6. Applied egg-rr 86.8%

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

      1. associate-*r/ 80.4%

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

      2. *-commutative 80.4%

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

      3. associate-/r/ 76.4%

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

   8. Simplified 76.4%

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

      1. associate-/r/ 80.4%

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

   10. Applied egg-rr 80.4%

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

       1. *-commutative 80.4%

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

       2. associate-/l* 86.8%

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

       3. frac-2neg 86.8%

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

       4. sqrt-div 99.5%

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

       5. *-commutative 99.5%

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

       6. distribute-rgt-neg-in 99.5%

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

   12. Applied egg-rr 99.5%

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

   if -9.99999999999999909e-308 < (*.f64 V l) < 9.99999999999948e-312

   1. Initial program 56.7%

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

      1. add-sqr-sqrt 21.7%

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

      2. sqrt-unprod 21.8%

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

      3. *-commutative 21.8%

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

      4. *-commutative 21.8%

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

      5. swap-sqr 21.5%

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

      6. add-sqr-sqrt 21.5%

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

      7. pow2 21.5%

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

   4. Applied egg-rr 21.5%

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

      1. associate-*l/ 21.3%

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

      2. *-commutative 21.3%

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

      3. times-frac 33.5%

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

   6. Simplified 33.5%

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

      1. div-inv 33.5%

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

      2. unpow2 33.5%

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

      3. associate-*l* 43.3%

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

   8. Applied egg-rr 43.3%

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

      1. pow1/2 43.3%

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

      2. associate-*r* 43.3%

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

      3. unpow-prod-down 40.2%

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

      4. pow1/2 31.6%

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

      5. un-div-inv 31.6%

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

   10. Applied egg-rr 31.6%

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

       1. unpow1/2 31.6%

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

       2. associate-*l/ 26.3%

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

       3. associate-/l* 31.6%

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

   12. Simplified 31.6%

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

       1. sqrt-unprod 43.4%

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

       2. associate-*l* 43.3%

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

   14. Applied egg-rr 43.3%

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

   if 9.99999999999948e-312 < (*.f64 V l)

   1. Initial program 75.9%

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

      1. pow1/2 75.9%

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

      2. div-inv 75.8%

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

      3. unpow-prod-down 92.9%

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

      4. pow1/2 92.9%

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

   4. Applied egg-rr 92.9%

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

      1. unpow1/2 92.9%

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

      2. associate-/r* 93.2%

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

   6. Simplified 93.2%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-307}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \left(-\ell\right)}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-311}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 1.3 \cdot 10^{+298}\right):\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (* c0_m (sqrt (/ A (* V l))))))
   (*
    c0_s
    (if (or (<= t_0 0.0) (not (<= t_0 1.3e+298)))
      (* c0_m (sqrt (/ (/ A V) l)))
      t_0))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 1.3e+298)) {
		tmp = c0_m * sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    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_m * sqrt((a / (v * l)))
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 1.3d+298))) then
        tmp = c0_m * sqrt(((a / v) / l))
    else
        tmp = t_0
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * Math.sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 1.3e+298)) {
		tmp = c0_m * Math.sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = c0_m * math.sqrt((A / (V * l)))
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 1.3e+298):
		tmp = c0_m * math.sqrt(((A / V) / l))
	else:
		tmp = t_0
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(c0_m * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 1.3e+298))
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = t_0;
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = c0_m * sqrt((A / (V * l)));
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 1.3e+298)))
		tmp = c0_m * sqrt(((A / V) / l));
	else
		tmp = t_0;
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 1.3e+298]], $MachinePrecision]], N[(c0$95$m * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 0.0 or 1.3e298 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 67.2%

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

      1. associate-/r* 70.2%

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

   3. Simplified 70.2%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.3e298

   1. Initial program 98.0%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0 \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 1.3 \cdot 10^{+298}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0\_m}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-307}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-311}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0\_m}{\ell} \cdot \frac{c0\_m}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (*
  c0_s
  (if (<= (* V l) (- INFINITY))
    (/ c0_m (/ (sqrt l) (sqrt (/ A V))))
    (if (<= (* V l) -1e-307)
      (* c0_m (/ (sqrt (- A)) (sqrt (* V (- l)))))
      (if (<= (* V l) 1e-311)
        (sqrt (* A (* (/ c0_m l) (/ c0_m V))))
        (* c0_m (* (sqrt A) (sqrt (/ (/ 1.0 V) l)))))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0_m / (sqrt(l) / sqrt((A / V)));
	} else if ((V * l) <= -1e-307) {
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 1e-311) {
		tmp = sqrt((A * ((c0_m / l) * (c0_m / V))));
	} else {
		tmp = c0_m * (sqrt(A) * sqrt(((1.0 / V) / l)));
	}
	return c0_s * tmp;
}

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0_m / (Math.sqrt(l) / Math.sqrt((A / V)));
	} else if ((V * l) <= -1e-307) {
		tmp = c0_m * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 1e-311) {
		tmp = Math.sqrt((A * ((c0_m / l) * (c0_m / V))));
	} else {
		tmp = c0_m * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0_m / (math.sqrt(l) / math.sqrt((A / V)))
	elif (V * l) <= -1e-307:
		tmp = c0_m * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 1e-311:
		tmp = math.sqrt((A * ((c0_m / l) * (c0_m / V))))
	else:
		tmp = c0_m * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0_m / Float64(sqrt(l) / sqrt(Float64(A / V))));
	elseif (Float64(V * l) <= -1e-307)
		tmp = Float64(c0_m * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 1e-311)
		tmp = sqrt(Float64(A * Float64(Float64(c0_m / l) * Float64(c0_m / V))));
	else
		tmp = Float64(c0_m * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0_m / (sqrt(l) / sqrt((A / V)));
	elseif ((V * l) <= -1e-307)
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 1e-311)
		tmp = sqrt((A * ((c0_m / l) * (c0_m / V))));
	else
		tmp = c0_m * (sqrt(A) * sqrt(((1.0 / V) / l)));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0$95$m / N[(N[Sqrt[l], $MachinePrecision] / N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-307], N[(c0$95$m * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-311], N[Sqrt[N[(A * N[(N[(c0$95$m / l), $MachinePrecision] * N[(c0$95$m / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 42.3%

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

      1. *-un-lft-identity 42.3%

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

      2. times-frac 77.1%

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

   4. Applied egg-rr 77.1%

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

      1. frac-times 42.3%

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

      2. *-un-lft-identity 42.3%

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

      3. sqrt-undiv 0.0%

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

      4. clear-num 0.0%

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

      5. un-div-inv 0.0%

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

      6. sqrt-undiv 42.3%

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

   6. Applied egg-rr 42.3%

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

      1. associate-*r/ 71.3%

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

      2. *-commutative 71.3%

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

      3. associate-/r/ 71.3%

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

   8. Simplified 71.3%

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

      1. sqrt-div 54.8%

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

      2. div-inv 54.7%

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

   10. Applied egg-rr 54.7%

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

       1. associate-*r/ 54.8%

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

       2. *-rgt-identity 54.8%

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

   12. Simplified 54.8%

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

   if -inf.0 < (*.f64 V l) < -9.99999999999999909e-308

   1. Initial program 85.3%

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

      1. frac-2neg 85.3%

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

      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 -9.99999999999999909e-308 < (*.f64 V l) < 9.99999999999948e-312

   1. Initial program 56.7%

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

      1. add-sqr-sqrt 21.7%

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

      2. sqrt-unprod 21.8%

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

      3. *-commutative 21.8%

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

      4. *-commutative 21.8%

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

      5. swap-sqr 21.5%

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

      6. add-sqr-sqrt 21.5%

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

      7. pow2 21.5%

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

   4. Applied egg-rr 21.5%

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

      1. associate-*l/ 21.3%

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

      2. *-commutative 21.3%

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

      3. times-frac 33.5%

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

   6. Simplified 33.5%

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

      1. div-inv 33.5%

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

      2. unpow2 33.5%

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

      3. associate-*l* 43.3%

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

   8. Applied egg-rr 43.3%

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

      1. pow1/2 43.3%

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

      2. associate-*r* 43.3%

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

      3. unpow-prod-down 40.2%

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

      4. pow1/2 31.6%

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

      5. un-div-inv 31.6%

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

   10. Applied egg-rr 31.6%

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

       1. unpow1/2 31.6%

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

       2. associate-*l/ 26.3%

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

       3. associate-/l* 31.6%

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

   12. Simplified 31.6%

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

       1. sqrt-unprod 43.4%

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

       2. associate-*l* 43.3%

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

   14. Applied egg-rr 43.3%

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

   if 9.99999999999948e-312 < (*.f64 V l)

   1. Initial program 75.9%

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

      1. pow1/2 75.9%

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

      2. div-inv 75.8%

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

      3. unpow-prod-down 92.9%

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

      4. pow1/2 92.9%

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

   4. Applied egg-rr 92.9%

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

      1. unpow1/2 92.9%

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

      2. associate-/r* 93.2%

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

   6. Simplified 93.2%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-307}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-311}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0\_m}{\sqrt{\ell}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+294}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0\_m}{\ell} \cdot \frac{c0\_m}{V}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (*
    c0_s
    (if (<= t_0 0.0)
      (* (sqrt (/ A V)) (/ c0_m (sqrt l)))
      (if (<= t_0 5e+294)
        (* c0_m (sqrt t_0))
        (sqrt (* A (* (/ c0_m l) (/ c0_m V)))))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = sqrt((A / V)) * (c0_m / sqrt(l));
	} else if (t_0 <= 5e+294) {
		tmp = c0_m * sqrt(t_0);
	} else {
		tmp = sqrt((A * ((c0_m / l) * (c0_m / V))));
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    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 = sqrt((a / v)) * (c0_m / sqrt(l))
    else if (t_0 <= 5d+294) then
        tmp = c0_m * sqrt(t_0)
    else
        tmp = sqrt((a * ((c0_m / l) * (c0_m / v))))
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.sqrt((A / V)) * (c0_m / Math.sqrt(l));
	} else if (t_0 <= 5e+294) {
		tmp = c0_m * Math.sqrt(t_0);
	} else {
		tmp = Math.sqrt((A * ((c0_m / l) * (c0_m / V))));
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.sqrt((A / V)) * (c0_m / math.sqrt(l))
	elif t_0 <= 5e+294:
		tmp = c0_m * math.sqrt(t_0)
	else:
		tmp = math.sqrt((A * ((c0_m / l) * (c0_m / V))))
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(sqrt(Float64(A / V)) * Float64(c0_m / sqrt(l)));
	elseif (t_0 <= 5e+294)
		tmp = Float64(c0_m * sqrt(t_0));
	else
		tmp = sqrt(Float64(A * Float64(Float64(c0_m / l) * Float64(c0_m / V))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = sqrt((A / V)) * (c0_m / sqrt(l));
	elseif (t_0 <= 5e+294)
		tmp = c0_m * sqrt(t_0);
	else
		tmp = sqrt((A * ((c0_m / l) * (c0_m / V))));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[t$95$0, 0.0], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0$95$m / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+294], N[(c0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(A * N[(N[(c0$95$m / l), $MachinePrecision] * N[(c0$95$m / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 35.6%

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

      1. associate-/r* 50.9%

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

      2. sqrt-div 51.7%

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

      3. associate-*r/ 50.2%

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

   4. Applied egg-rr 50.2%

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

      1. *-commutative 50.2%

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

      2. associate-/l* 51.8%

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

   6. Simplified 51.8%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.9999999999999999e294

   1. Initial program 98.9%

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

   if 4.9999999999999999e294 < (/.f64 A (*.f64 V l))

   1. Initial program 51.1%

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

      1. add-sqr-sqrt 18.9%

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

      2. sqrt-unprod 19.1%

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

      3. *-commutative 19.1%

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

      4. *-commutative 19.1%

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

      5. swap-sqr 18.7%

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

      6. add-sqr-sqrt 18.7%

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

      7. pow2 18.7%

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

   4. Applied egg-rr 18.7%

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

      1. associate-*l/ 19.1%

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

      2. *-commutative 19.1%

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

      3. times-frac 25.3%

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

   6. Simplified 25.3%

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

      1. div-inv 25.3%

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

      2. unpow2 25.3%

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

      3. associate-*l* 31.5%

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

   8. Applied egg-rr 31.5%

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

      1. pow1/2 31.5%

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

      2. associate-*r* 33.2%

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

      3. unpow-prod-down 29.3%

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

      4. pow1/2 20.2%

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

      5. un-div-inv 20.2%

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

   10. Applied egg-rr 20.2%

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

       1. unpow1/2 20.2%

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

       2. associate-*l/ 16.8%

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

       3. associate-/l* 20.2%

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

   12. Simplified 20.2%

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

       1. sqrt-unprod 35.0%

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

       2. associate-*l* 35.0%

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

   14. Applied egg-rr 35.0%

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

Alternative 5: 86.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 5 \cdot 10^{+294}\right):\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0\_m}{\ell} \cdot \frac{c0\_m}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (*
    c0_s
    (if (or (<= t_0 0.0) (not (<= t_0 5e+294)))
      (sqrt (* A (* (/ c0_m l) (/ c0_m V))))
      (* c0_m (sqrt t_0))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+294)) {
		tmp = sqrt((A * ((c0_m / l) * (c0_m / V))));
	} else {
		tmp = c0_m * sqrt(t_0);
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    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) .or. (.not. (t_0 <= 5d+294))) then
        tmp = sqrt((a * ((c0_m / l) * (c0_m / v))))
    else
        tmp = c0_m * sqrt(t_0)
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+294)) {
		tmp = Math.sqrt((A * ((c0_m / l) * (c0_m / V))));
	} else {
		tmp = c0_m * Math.sqrt(t_0);
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 5e+294):
		tmp = math.sqrt((A * ((c0_m / l) * (c0_m / V))))
	else:
		tmp = c0_m * math.sqrt(t_0)
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 5e+294))
		tmp = sqrt(Float64(A * Float64(Float64(c0_m / l) * Float64(c0_m / V))));
	else
		tmp = Float64(c0_m * sqrt(t_0));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 5e+294)))
		tmp = sqrt((A * ((c0_m / l) * (c0_m / V))));
	else
		tmp = c0_m * sqrt(t_0);
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 5e+294]], $MachinePrecision]], N[Sqrt[N[(A * N[(N[(c0$95$m / l), $MachinePrecision] * N[(c0$95$m / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(c0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 4.9999999999999999e294 < (/.f64 A (*.f64 V l))

   1. Initial program 43.2%

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

      1. add-sqr-sqrt 27.5%

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

      2. sqrt-unprod 27.6%

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

      3. *-commutative 27.6%

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

      4. *-commutative 27.6%

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

      5. swap-sqr 26.9%

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

      6. add-sqr-sqrt 26.9%

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

      7. pow2 26.9%

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

   4. Applied egg-rr 26.9%

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

      1. associate-*l/ 27.9%

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

      2. *-commutative 27.9%

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

      3. times-frac 35.4%

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

   6. Simplified 35.4%

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

      1. div-inv 35.3%

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

      2. unpow2 35.3%

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

      3. associate-*l* 41.1%

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

   8. Applied egg-rr 41.1%

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

      1. pow1/2 41.1%

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

      2. associate-*r* 42.1%

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

      3. unpow-prod-down 31.5%

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

      4. pow1/2 27.1%

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

      5. un-div-inv 27.1%

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

   10. Applied egg-rr 27.1%

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

       1. unpow1/2 27.1%

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

       2. associate-*l/ 27.1%

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

       3. associate-/l* 29.6%

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

   12. Simplified 29.6%

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

       1. sqrt-unprod 47.1%

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

       2. associate-*l* 43.7%

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

   14. Applied egg-rr 43.7%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.9999999999999999e294

   1. Initial program 98.9%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+294}\right):\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \left(c0\_m \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\right) \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (* c0_s (* c0_m (sqrt (/ (/ A l) V)))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	return c0_s * (c0_m * sqrt(((A / l) / V)));
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    code = c0_s * (c0_m * sqrt(((a / l) / v)))
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	return c0_s * (c0_m * Math.sqrt(((A / l) / V)));
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	return c0_s * (c0_m * math.sqrt(((A / l) / V)))

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	return Float64(c0_s * Float64(c0_m * sqrt(Float64(Float64(A / l) / V))))
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp = code(c0_s, c0_m, A, V, l)
	tmp = c0_s * (c0_m * sqrt(((A / l) / V)));
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := N[(c0$95$s * N[(c0$95$m * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.3%

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

3. Taylor expanded in c0 around 0 74.3%

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

   1. *-commutative 74.3%

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

   2. associate-/r* 75.7%

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

5. Simplified 75.7%

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

6. Final simplification 75.7%

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

Alternative 7: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \left(c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\right) \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (* c0_s (* c0_m (sqrt (/ A (* V l))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	return c0_s * (c0_m * sqrt((A / (V * l))));
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    code = c0_s * (c0_m * sqrt((a / (v * l))))
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	return c0_s * (c0_m * Math.sqrt((A / (V * l))));
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	return c0_s * (c0_m * math.sqrt((A / (V * l))))

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	return Float64(c0_s * Float64(c0_m * sqrt(Float64(A / Float64(V * l)))))
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp = code(c0_s, c0_m, A, V, l)
	tmp = c0_s * (c0_m * sqrt((A / (V * l))));
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := N[(c0$95$s * N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.3%

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

Reproduce

herbie shell --seed 2024132 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))