Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.2% → 90.3%

Time: 12.6s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\left(\sqrt{\frac{A}{V}} \cdot {\ell}^{-0.5}\right) \cdot c0\_m\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-274}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\sqrt{\frac{\frac{c0\_m}{V} \cdot \left(A \cdot c0\_m\right)}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+237}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \]

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))
    (* (* (sqrt (/ A V)) (pow l -0.5)) c0_m)
    (if (<= (* V l) -1e-274)
      (* c0_m (/ (sqrt (- A)) (sqrt (* V (- l)))))
      (if (<= (* V l) 0.0)
        (sqrt (/ (* (/ c0_m V) (* A c0_m)) l))
        (if (<= (* V l) 4e+237)
          (* c0_m (/ (sqrt A) (sqrt (* V l))))
          (* 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) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = (sqrt((A / V)) * pow(l, -0.5)) * c0_m;
	} else if ((V * l) <= -1e-274) {
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = sqrt((((c0_m / V) * (A * c0_m)) / l));
	} else if ((V * l) <= 4e+237) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0_m * sqrt(((A / 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 = (Math.sqrt((A / V)) * Math.pow(l, -0.5)) * c0_m;
	} else if ((V * l) <= -1e-274) {
		tmp = c0_m * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = Math.sqrt((((c0_m / V) * (A * c0_m)) / l));
	} else if ((V * l) <= 4e+237) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0_m * Math.sqrt(((A / 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 = (math.sqrt((A / V)) * math.pow(l, -0.5)) * c0_m
	elif (V * l) <= -1e-274:
		tmp = c0_m * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = math.sqrt((((c0_m / V) * (A * c0_m)) / l))
	elif (V * l) <= 4e+237:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0_m * math.sqrt(((A / 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(Float64(sqrt(Float64(A / V)) * (l ^ -0.5)) * c0_m);
	elseif (Float64(V * l) <= -1e-274)
		tmp = Float64(c0_m * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(c0_m / V) * Float64(A * c0_m)) / l));
	elseif (Float64(V * l) <= 4e+237)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / 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 = (sqrt((A / V)) * (l ^ -0.5)) * c0_m;
	elseif ((V * l) <= -1e-274)
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = sqrt((((c0_m / V) * (A * c0_m)) / l));
	elseif ((V * l) <= 4e+237)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0_m * sqrt(((A / 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[(N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision] * c0$95$m), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-274], N[(c0$95$m * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Sqrt[N[(N[(N[(c0$95$m / V), $MachinePrecision] * N[(A * c0$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e+237], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0$95$m * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 38.4%

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

      1. clear-num 38.4%

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

      2. associate-/r/ 38.4%

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

      3. associate-/r* 38.4%

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

   4. Applied egg-rr 38.4%

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

   5. Taylor expanded in c0 around 0 38.4%

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

      1. associate-/l/ 63.3%

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

   7. Simplified 63.3%

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

      1. associate-/l/ 38.4%

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

      2. associate-/r* 63.3%

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

      3. frac-2neg 63.3%

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

      4. distribute-frac-neg 63.3%

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

      5. sqrt-undiv 30.4%

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

      6. div-inv 30.2%

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

      7. add-sqr-sqrt 30.2%

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

      8. sqrt-unprod 21.9%

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

      9. sqr-neg 21.9%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 8.7%

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

      12. clear-num 13.0%

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

      13. clear-num 8.7%

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

      14. pow1/2 8.7%

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

      15. pow-flip 8.7%

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

      16. add-sqr-sqrt 8.7%

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

      17. sqrt-unprod 37.2%

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

      18. sqr-neg 37.2%

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

      19. sqrt-unprod 61.0%

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

      20. add-sqr-sqrt 61.1%

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

      21. metadata-eval 61.1%

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

   9. Applied egg-rr 61.1%

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

   if -inf.0 < (*.f64 V l) < -9.99999999999999966e-275

   1. Initial program 84.3%

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

      1. frac-2neg 84.3%

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

      2. sqrt-div 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. distribute-rgt-neg-out 99.4%

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

      2. *-commutative 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

   6. Simplified 99.4%

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

   if -9.99999999999999966e-275 < (*.f64 V l) < -0.0

   1. Initial program 50.0%

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

      1. add-sqr-sqrt 41.1%

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

      2. sqrt-unprod 41.3%

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

      3. *-commutative 41.3%

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

      4. *-commutative 41.3%

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

      5. swap-sqr 40.9%

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

      6. add-sqr-sqrt 40.9%

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

      7. pow2 40.9%

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

   4. Applied egg-rr 40.9%

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

      1. associate-/r* 45.1%

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

   6. Simplified 45.1%

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

   7. Taylor expanded in A around 0 44.8%

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

      1. *-commutative 44.8%

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

      2. times-frac 49.0%

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

      3. associate-/r/ 45.1%

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

      4. associate-/r/ 45.2%

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

      5. associate-/r* 45.2%

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

   9. Simplified 45.2%

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

       1. unpow2 45.2%

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

       2. div-inv 45.2%

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

       3. times-frac 56.8%

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

   11. Applied egg-rr 56.8%

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

       1. div-inv 56.9%

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

       2. inv-pow 56.9%

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

       3. pow-flip 57.0%

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

       4. metadata-eval 57.0%

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

       5. pow1 57.0%

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

   13. Applied egg-rr 57.0%

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

   if -0.0 < (*.f64 V l) < 3.99999999999999976e237

   1. Initial program 86.1%

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

      1. sqrt-div 99.5%

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

      2. div-inv 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. associate-*r/ 99.5%

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

      2. *-rgt-identity 99.5%

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

   6. Simplified 99.5%

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

   if 3.99999999999999976e237 < (*.f64 V l)

   1. Initial program 47.0%

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

      1. associate-/r* 70.8%

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

   3. Simplified 70.8%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\left(\sqrt{\frac{A}{V}} \cdot {\ell}^{-0.5}\right) \cdot c0\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-274}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\sqrt{\frac{\frac{c0}{V} \cdot \left(A \cdot c0\right)}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+237}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{-322}:\\ \;\;\;\;c0\_m \cdot \frac{1}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+300}:\\ \;\;\;\;c0\_m \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{c0\_m}{V} \cdot \left(A \cdot c0\_m\right)}{\ell}}\\ \end{array} \end{array} \end{array} \]

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 1e-322)
      (* c0_m (/ 1.0 (/ (sqrt l) (sqrt (/ A V)))))
      (if (<= t_0 4e+300)
        (* c0_m (pow (/ (* V l) A) -0.5))
        (sqrt (/ (* (/ c0_m V) (* A c0_m)) 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 t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 1e-322) {
		tmp = c0_m * (1.0 / (sqrt(l) / sqrt((A / V))));
	} else if (t_0 <= 4e+300) {
		tmp = c0_m * pow(((V * l) / A), -0.5);
	} else {
		tmp = sqrt((((c0_m / V) * (A * c0_m)) / l));
	}
	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 <= 1d-322) then
        tmp = c0_m * (1.0d0 / (sqrt(l) / sqrt((a / v))))
    else if (t_0 <= 4d+300) then
        tmp = c0_m * (((v * l) / a) ** (-0.5d0))
    else
        tmp = sqrt((((c0_m / v) * (a * c0_m)) / l))
    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 <= 1e-322) {
		tmp = c0_m * (1.0 / (Math.sqrt(l) / Math.sqrt((A / V))));
	} else if (t_0 <= 4e+300) {
		tmp = c0_m * Math.pow(((V * l) / A), -0.5);
	} else {
		tmp = Math.sqrt((((c0_m / V) * (A * c0_m)) / 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):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 1e-322:
		tmp = c0_m * (1.0 / (math.sqrt(l) / math.sqrt((A / V))))
	elif t_0 <= 4e+300:
		tmp = c0_m * math.pow(((V * l) / A), -0.5)
	else:
		tmp = math.sqrt((((c0_m / V) * (A * c0_m)) / 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)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 1e-322)
		tmp = Float64(c0_m * Float64(1.0 / Float64(sqrt(l) / sqrt(Float64(A / V)))));
	elseif (t_0 <= 4e+300)
		tmp = Float64(c0_m * (Float64(Float64(V * l) / A) ^ -0.5));
	else
		tmp = sqrt(Float64(Float64(Float64(c0_m / V) * Float64(A * c0_m)) / 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)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 1e-322)
		tmp = c0_m * (1.0 / (sqrt(l) / sqrt((A / V))));
	elseif (t_0 <= 4e+300)
		tmp = c0_m * (((V * l) / A) ^ -0.5);
	else
		tmp = sqrt((((c0_m / V) * (A * c0_m)) / 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_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[t$95$0, 1e-322], N[(c0$95$m * N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] / N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+300], N[(c0$95$m * N[Power[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[(c0$95$m / V), $MachinePrecision] * N[(A * c0$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 9.88131e-323

   1. Initial program 35.5%

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

      1. associate-/r* 55.0%

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

      2. sqrt-div 47.5%

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

      3. clear-num 47.6%

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

   4. Applied egg-rr 47.6%

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

   if 9.88131e-323 < (/.f64 A (*.f64 V l)) < 4.0000000000000002e300

   1. Initial program 99.6%

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

      1. frac-2neg 99.6%

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

      2. sqrt-div 46.3%

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

      3. distribute-rgt-neg-in 46.3%

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

   4. Applied egg-rr 46.3%

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

      1. distribute-rgt-neg-out 46.3%

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

      2. *-commutative 46.3%

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

      3. distribute-rgt-neg-in 46.3%

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

   6. Simplified 46.3%

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

      1. clear-num 46.2%

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

      2. sqrt-undiv 99.4%

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

      3. distribute-rgt-neg-out 99.4%

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

      4. frac-2neg 99.4%

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

      5. sqrt-div 53.1%

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

      6. add-sqr-sqrt 22.8%

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

      7. sqrt-unprod 12.7%

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

      8. sqr-neg 12.7%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 0.0%

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

      11. *-commutative 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 12.7%

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

      14. sqr-neg 12.7%

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

      15. sqrt-unprod 22.8%

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

      16. add-sqr-sqrt 53.1%

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

      17. sqrt-div 99.4%

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

      18. associate-*r/ 84.5%

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

   8. Applied egg-rr 99.7%

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

   if 4.0000000000000002e300 < (/.f64 A (*.f64 V l))

   1. Initial program 46.8%

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

      1. add-sqr-sqrt 28.0%

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

      2. sqrt-unprod 28.1%

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

      3. *-commutative 28.1%

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

      4. *-commutative 28.1%

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

      5. swap-sqr 27.7%

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

      6. add-sqr-sqrt 27.7%

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

      7. pow2 27.7%

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

   4. Applied egg-rr 27.7%

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

      1. associate-/r* 29.7%

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

   6. Simplified 29.7%

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

   7. Taylor expanded in A around 0 32.6%

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

      1. *-commutative 32.6%

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

      2. times-frac 32.6%

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

      3. associate-/r/ 29.7%

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

      4. associate-/r/ 29.8%

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

      5. associate-/r* 31.4%

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

   9. Simplified 31.4%

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

       1. unpow2 31.4%

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

       2. div-inv 31.4%

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

       3. times-frac 43.0%

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

   11. Applied egg-rr 43.0%

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

       1. div-inv 43.0%

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

       2. inv-pow 43.0%

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

       3. pow-flip 43.0%

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

       4. metadata-eval 43.0%

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

       5. pow1 43.0%

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

   13. Applied egg-rr 43.0%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 10^{-322}:\\ \;\;\;\;c0 \cdot \frac{1}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 4 \cdot 10^{+300}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{c0}{V} \cdot \left(A \cdot c0\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{-322}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+300}:\\ \;\;\;\;c0\_m \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{c0\_m}{V} \cdot \left(A \cdot c0\_m\right)}{\ell}}\\ \end{array} \end{array} \end{array} \]

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 1e-322)
      (* c0_m (/ (sqrt (/ A V)) (sqrt l)))
      (if (<= t_0 4e+300)
        (* c0_m (pow (/ (* V l) A) -0.5))
        (sqrt (/ (* (/ c0_m V) (* A c0_m)) 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 t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 1e-322) {
		tmp = c0_m * (sqrt((A / V)) / sqrt(l));
	} else if (t_0 <= 4e+300) {
		tmp = c0_m * pow(((V * l) / A), -0.5);
	} else {
		tmp = sqrt((((c0_m / V) * (A * c0_m)) / l));
	}
	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 <= 1d-322) then
        tmp = c0_m * (sqrt((a / v)) / sqrt(l))
    else if (t_0 <= 4d+300) then
        tmp = c0_m * (((v * l) / a) ** (-0.5d0))
    else
        tmp = sqrt((((c0_m / v) * (a * c0_m)) / l))
    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 <= 1e-322) {
		tmp = c0_m * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if (t_0 <= 4e+300) {
		tmp = c0_m * Math.pow(((V * l) / A), -0.5);
	} else {
		tmp = Math.sqrt((((c0_m / V) * (A * c0_m)) / 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):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 1e-322:
		tmp = c0_m * (math.sqrt((A / V)) / math.sqrt(l))
	elif t_0 <= 4e+300:
		tmp = c0_m * math.pow(((V * l) / A), -0.5)
	else:
		tmp = math.sqrt((((c0_m / V) * (A * c0_m)) / 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)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 1e-322)
		tmp = Float64(c0_m * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (t_0 <= 4e+300)
		tmp = Float64(c0_m * (Float64(Float64(V * l) / A) ^ -0.5));
	else
		tmp = sqrt(Float64(Float64(Float64(c0_m / V) * Float64(A * c0_m)) / 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)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 1e-322)
		tmp = c0_m * (sqrt((A / V)) / sqrt(l));
	elseif (t_0 <= 4e+300)
		tmp = c0_m * (((V * l) / A) ^ -0.5);
	else
		tmp = sqrt((((c0_m / V) * (A * c0_m)) / 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_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[t$95$0, 1e-322], N[(c0$95$m * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+300], N[(c0$95$m * N[Power[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[(c0$95$m / V), $MachinePrecision] * N[(A * c0$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 9.88131e-323

   1. Initial program 35.5%

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

      1. associate-/r* 55.0%

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

      2. sqrt-div 47.5%

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

      3. div-inv 47.4%

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

   4. Applied egg-rr 47.4%

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

      1. associate-*r/ 47.5%

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

      2. *-rgt-identity 47.5%

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

   6. Simplified 47.5%

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

   if 9.88131e-323 < (/.f64 A (*.f64 V l)) < 4.0000000000000002e300

   1. Initial program 99.6%

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

      1. frac-2neg 99.6%

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

      2. sqrt-div 46.3%

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

      3. distribute-rgt-neg-in 46.3%

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

   4. Applied egg-rr 46.3%

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

      1. distribute-rgt-neg-out 46.3%

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

      2. *-commutative 46.3%

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

      3. distribute-rgt-neg-in 46.3%

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

   6. Simplified 46.3%

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

      1. clear-num 46.2%

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

      2. sqrt-undiv 99.4%

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

      3. distribute-rgt-neg-out 99.4%

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

      4. frac-2neg 99.4%

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

      5. sqrt-div 53.1%

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

      6. add-sqr-sqrt 22.8%

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

      7. sqrt-unprod 12.7%

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

      8. sqr-neg 12.7%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 0.0%

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

      11. *-commutative 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 12.7%

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

      14. sqr-neg 12.7%

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

      15. sqrt-unprod 22.8%

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

      16. add-sqr-sqrt 53.1%

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

      17. sqrt-div 99.4%

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

      18. associate-*r/ 84.5%

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

   8. Applied egg-rr 99.7%

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

   if 4.0000000000000002e300 < (/.f64 A (*.f64 V l))

   1. Initial program 46.8%

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

      1. add-sqr-sqrt 28.0%

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

      2. sqrt-unprod 28.1%

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

      3. *-commutative 28.1%

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

      4. *-commutative 28.1%

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

      5. swap-sqr 27.7%

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

      6. add-sqr-sqrt 27.7%

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

      7. pow2 27.7%

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

   4. Applied egg-rr 27.7%

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

      1. associate-/r* 29.7%

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

   6. Simplified 29.7%

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

   7. Taylor expanded in A around 0 32.6%

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

      1. *-commutative 32.6%

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

      2. times-frac 32.6%

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

      3. associate-/r/ 29.7%

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

      4. associate-/r/ 29.8%

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

      5. associate-/r* 31.4%

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

   9. Simplified 31.4%

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

       1. unpow2 31.4%

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

       2. div-inv 31.4%

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

       3. times-frac 43.0%

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

   11. Applied egg-rr 43.0%

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

       1. div-inv 43.0%

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

       2. inv-pow 43.0%

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

       3. pow-flip 43.0%

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

       4. metadata-eval 43.0%

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

       5. pow1 43.0%

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

   13. Applied egg-rr 43.0%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 10^{-322}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 4 \cdot 10^{+300}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{c0}{V} \cdot \left(A \cdot c0\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{-322} \lor \neg \left(t\_0 \leq 4 \cdot 10^{+300}\right):\\ \;\;\;\;\sqrt{\frac{c0\_m}{V} \cdot \frac{A \cdot c0\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \end{array} \end{array} \end{array} \]

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 1e-322) (not (<= t_0 4e+300)))
      (sqrt (* (/ c0_m V) (/ (* A c0_m) l)))
      (* c0_m (pow (/ (* V l) A) -0.5))))))

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 <= 1e-322) || !(t_0 <= 4e+300)) {
		tmp = sqrt(((c0_m / V) * ((A * c0_m) / l)));
	} else {
		tmp = c0_m * pow(((V * l) / A), -0.5);
	}
	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 <= 1d-322) .or. (.not. (t_0 <= 4d+300))) then
        tmp = sqrt(((c0_m / v) * ((a * c0_m) / l)))
    else
        tmp = c0_m * (((v * l) / a) ** (-0.5d0))
    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 <= 1e-322) || !(t_0 <= 4e+300)) {
		tmp = Math.sqrt(((c0_m / V) * ((A * c0_m) / l)));
	} else {
		tmp = c0_m * Math.pow(((V * l) / A), -0.5);
	}
	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 <= 1e-322) or not (t_0 <= 4e+300):
		tmp = math.sqrt(((c0_m / V) * ((A * c0_m) / l)))
	else:
		tmp = c0_m * math.pow(((V * l) / A), -0.5)
	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 <= 1e-322) || !(t_0 <= 4e+300))
		tmp = sqrt(Float64(Float64(c0_m / V) * Float64(Float64(A * c0_m) / l)));
	else
		tmp = Float64(c0_m * (Float64(Float64(V * l) / A) ^ -0.5));
	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 <= 1e-322) || ~((t_0 <= 4e+300)))
		tmp = sqrt(((c0_m / V) * ((A * c0_m) / l)));
	else
		tmp = c0_m * (((V * l) / A) ^ -0.5);
	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, 1e-322], N[Not[LessEqual[t$95$0, 4e+300]], $MachinePrecision]], N[Sqrt[N[(N[(c0$95$m / V), $MachinePrecision] * N[(N[(A * c0$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(c0$95$m * N[Power[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 9.88131e-323 or 4.0000000000000002e300 < (/.f64 A (*.f64 V l))

   1. Initial program 40.8%

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

      1. add-sqr-sqrt 31.8%

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

      2. sqrt-unprod 31.8%

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

      3. *-commutative 31.8%

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

      4. *-commutative 31.8%

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

      5. swap-sqr 31.3%

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

      6. add-sqr-sqrt 31.3%

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

      7. pow2 31.3%

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

   4. Applied egg-rr 31.3%

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

      1. associate-/r* 33.6%

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

   6. Simplified 33.6%

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

   7. Taylor expanded in A around 0 34.4%

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

      1. *-commutative 34.4%

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

      2. times-frac 36.5%

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

      3. associate-/r/ 32.9%

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

      4. associate-/r/ 32.9%

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

      5. associate-/r* 35.9%

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

   9. Simplified 35.9%

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

       1. unpow2 35.9%

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

       2. div-inv 35.9%

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

       3. times-frac 45.0%

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

   11. Applied egg-rr 45.0%

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

       1. associate-/l* 44.1%

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

       2. div-inv 44.1%

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

       3. inv-pow 44.1%

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

       4. pow-flip 44.2%

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

       5. metadata-eval 44.2%

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

       6. pow1 44.2%

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

   13. Applied egg-rr 44.2%

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

   if 9.88131e-323 < (/.f64 A (*.f64 V l)) < 4.0000000000000002e300

   1. Initial program 99.6%

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

      1. frac-2neg 99.6%

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

      2. sqrt-div 46.3%

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

      3. distribute-rgt-neg-in 46.3%

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

   4. Applied egg-rr 46.3%

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

      1. distribute-rgt-neg-out 46.3%

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

      2. *-commutative 46.3%

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

      3. distribute-rgt-neg-in 46.3%

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

   6. Simplified 46.3%

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

      1. clear-num 46.2%

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

      2. sqrt-undiv 99.4%

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

      3. distribute-rgt-neg-out 99.4%

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

      4. frac-2neg 99.4%

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

      5. sqrt-div 53.1%

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

      6. add-sqr-sqrt 22.8%

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

      7. sqrt-unprod 12.7%

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

      8. sqr-neg 12.7%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 0.0%

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

      11. *-commutative 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 12.7%

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

      14. sqr-neg 12.7%

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

      15. sqrt-unprod 22.8%

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

      16. add-sqr-sqrt 53.1%

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

      17. sqrt-div 99.4%

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

      18. associate-*r/ 84.5%

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

   8. Applied egg-rr 99.7%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 10^{-322} \lor \neg \left(\frac{A}{V \cdot \ell} \leq 4 \cdot 10^{+300}\right):\\ \;\;\;\;\sqrt{\frac{c0}{V} \cdot \frac{A \cdot c0}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 4 \cdot 10^{+300}\right):\\ \;\;\;\;\sqrt{\frac{\frac{c0\_m}{V} \cdot \left(A \cdot c0\_m\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \end{array} \end{array} \end{array} \]

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 4e+300)))
      (sqrt (/ (* (/ c0_m V) (* A c0_m)) l))
      (* 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 <= 4e+300)) {
		tmp = sqrt((((c0_m / V) * (A * c0_m)) / l));
	} 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 <= 4d+300))) then
        tmp = sqrt((((c0_m / v) * (a * c0_m)) / l))
    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 <= 4e+300)) {
		tmp = Math.sqrt((((c0_m / V) * (A * c0_m)) / l));
	} 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 <= 4e+300):
		tmp = math.sqrt((((c0_m / V) * (A * c0_m)) / l))
	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 <= 4e+300))
		tmp = sqrt(Float64(Float64(Float64(c0_m / V) * Float64(A * c0_m)) / l));
	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 <= 4e+300)))
		tmp = sqrt((((c0_m / V) * (A * c0_m)) / l));
	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, 4e+300]], $MachinePrecision]], N[Sqrt[N[(N[(N[(c0$95$m / V), $MachinePrecision] * N[(A * c0$95$m), $MachinePrecision]), $MachinePrecision] / l), $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 4.0000000000000002e300 < (/.f64 A (*.f64 V l))

   1. Initial program 40.9%

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

      1. add-sqr-sqrt 32.1%

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

      2. sqrt-unprod 32.1%

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

      3. *-commutative 32.1%

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

      4. *-commutative 32.1%

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

      5. swap-sqr 31.6%

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

      6. add-sqr-sqrt 31.6%

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

      7. pow2 31.6%

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

   4. Applied egg-rr 31.6%

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

      1. associate-/r* 33.9%

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

   6. Simplified 33.9%

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

   7. Taylor expanded in A around 0 34.7%

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

      1. *-commutative 34.7%

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

      2. times-frac 36.8%

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

      3. associate-/r/ 33.2%

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

      4. associate-/r/ 33.2%

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

      5. associate-/r* 36.2%

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

   9. Simplified 36.2%

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

       1. unpow2 36.2%

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

       2. div-inv 36.2%

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

       3. times-frac 45.4%

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

   11. Applied egg-rr 45.4%

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

       1. div-inv 45.4%

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

       2. inv-pow 45.4%

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

       3. pow-flip 45.4%

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

       4. metadata-eval 45.4%

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

       5. pow1 45.4%

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

   13. Applied egg-rr 45.4%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.0000000000000002e300

   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 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 4 \cdot 10^{+300}\right):\\ \;\;\;\;\sqrt{\frac{\frac{c0}{V} \cdot \left(A \cdot c0\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{\frac{c0\_m}{V} \cdot \frac{c0\_m}{\frac{1}{A}}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+300}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{c0\_m}{V} \cdot \left(A \cdot c0\_m\right)}{\ell}}\\ \end{array} \end{array} \end{array} \]

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 (/ (* (/ c0_m V) (/ c0_m (/ 1.0 A))) l))
      (if (<= t_0 4e+300)
        (* c0_m (sqrt t_0))
        (sqrt (/ (* (/ c0_m V) (* A c0_m)) 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 t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = sqrt((((c0_m / V) * (c0_m / (1.0 / A))) / l));
	} else if (t_0 <= 4e+300) {
		tmp = c0_m * sqrt(t_0);
	} else {
		tmp = sqrt((((c0_m / V) * (A * c0_m)) / l));
	}
	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((((c0_m / v) * (c0_m / (1.0d0 / a))) / l))
    else if (t_0 <= 4d+300) then
        tmp = c0_m * sqrt(t_0)
    else
        tmp = sqrt((((c0_m / v) * (a * c0_m)) / l))
    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((((c0_m / V) * (c0_m / (1.0 / A))) / l));
	} else if (t_0 <= 4e+300) {
		tmp = c0_m * Math.sqrt(t_0);
	} else {
		tmp = Math.sqrt((((c0_m / V) * (A * c0_m)) / 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):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.sqrt((((c0_m / V) * (c0_m / (1.0 / A))) / l))
	elif t_0 <= 4e+300:
		tmp = c0_m * math.sqrt(t_0)
	else:
		tmp = math.sqrt((((c0_m / V) * (A * c0_m)) / 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)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(c0_m / V) * Float64(c0_m / Float64(1.0 / A))) / l));
	elseif (t_0 <= 4e+300)
		tmp = Float64(c0_m * sqrt(t_0));
	else
		tmp = sqrt(Float64(Float64(Float64(c0_m / V) * Float64(A * c0_m)) / 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)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = sqrt((((c0_m / V) * (c0_m / (1.0 / A))) / l));
	elseif (t_0 <= 4e+300)
		tmp = c0_m * sqrt(t_0);
	else
		tmp = sqrt((((c0_m / V) * (A * c0_m)) / 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_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[t$95$0, 0.0], N[Sqrt[N[(N[(N[(c0$95$m / V), $MachinePrecision] * N[(c0$95$m / N[(1.0 / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 4e+300], N[(c0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[(c0$95$m / V), $MachinePrecision] * N[(A * c0$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 35.7%

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

      1. add-sqr-sqrt 35.7%

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

      2. sqrt-unprod 35.7%

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

      3. *-commutative 35.7%

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

      4. *-commutative 35.7%

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

      5. swap-sqr 35.0%

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

      6. add-sqr-sqrt 35.0%

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

      7. pow2 35.0%

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

   4. Applied egg-rr 35.0%

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

      1. associate-/r* 37.5%

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

   6. Simplified 37.5%

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

   7. Taylor expanded in A around 0 36.6%

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

      1. *-commutative 36.6%

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

      2. times-frac 40.6%

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

      3. associate-/r/ 36.2%

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

      4. associate-/r/ 36.2%

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

      5. associate-/r* 40.4%

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

   9. Simplified 40.4%

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

       1. unpow2 40.4%

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

       2. div-inv 40.4%

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

       3. times-frac 47.5%

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

   11. Applied egg-rr 47.5%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.0000000000000002e300

   1. Initial program 99.1%

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

   if 4.0000000000000002e300 < (/.f64 A (*.f64 V l))

   1. Initial program 46.8%

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

      1. add-sqr-sqrt 28.0%

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

      2. sqrt-unprod 28.1%

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

      3. *-commutative 28.1%

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

      4. *-commutative 28.1%

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

      5. swap-sqr 27.7%

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

      6. add-sqr-sqrt 27.7%

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

      7. pow2 27.7%

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

   4. Applied egg-rr 27.7%

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

      1. associate-/r* 29.7%

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

   6. Simplified 29.7%

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

   7. Taylor expanded in A around 0 32.6%

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

      1. *-commutative 32.6%

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

      2. times-frac 32.6%

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

      3. associate-/r/ 29.7%

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

      4. associate-/r/ 29.8%

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

      5. associate-/r* 31.4%

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

   9. Simplified 31.4%

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

       1. unpow2 31.4%

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

       2. div-inv 31.4%

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

       3. times-frac 43.0%

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

   11. Applied egg-rr 43.0%

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

       1. div-inv 43.0%

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

       2. inv-pow 43.0%

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

       3. pow-flip 43.0%

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

       4. metadata-eval 43.0%

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

       5. pow1 43.0%

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

   13. Applied egg-rr 43.0%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;\sqrt{\frac{\frac{c0}{V} \cdot \frac{c0}{\frac{1}{A}}}{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 4 \cdot 10^{+300}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{c0}{V} \cdot \left(A \cdot c0\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-254}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+300}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \end{array} \end{array} \]

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 2e-254)
      (* c0_m (sqrt (/ (/ A l) V)))
      (if (<= t_0 4e+300)
        (* c0_m (sqrt t_0))
        (* c0_m (pow (* V (/ l A)) -0.5)))))))

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 <= 2e-254) {
		tmp = c0_m * sqrt(((A / l) / V));
	} else if (t_0 <= 4e+300) {
		tmp = c0_m * sqrt(t_0);
	} else {
		tmp = c0_m * pow((V * (l / A)), -0.5);
	}
	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 <= 2d-254) then
        tmp = c0_m * sqrt(((a / l) / v))
    else if (t_0 <= 4d+300) then
        tmp = c0_m * sqrt(t_0)
    else
        tmp = c0_m * ((v * (l / a)) ** (-0.5d0))
    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 <= 2e-254) {
		tmp = c0_m * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 4e+300) {
		tmp = c0_m * Math.sqrt(t_0);
	} else {
		tmp = c0_m * Math.pow((V * (l / A)), -0.5);
	}
	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 <= 2e-254:
		tmp = c0_m * math.sqrt(((A / l) / V))
	elif t_0 <= 4e+300:
		tmp = c0_m * math.sqrt(t_0)
	else:
		tmp = c0_m * math.pow((V * (l / A)), -0.5)
	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 <= 2e-254)
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 4e+300)
		tmp = Float64(c0_m * sqrt(t_0));
	else
		tmp = Float64(c0_m * (Float64(V * Float64(l / A)) ^ -0.5));
	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 <= 2e-254)
		tmp = c0_m * sqrt(((A / l) / V));
	elseif (t_0 <= 4e+300)
		tmp = c0_m * sqrt(t_0);
	else
		tmp = c0_m * ((V * (l / A)) ^ -0.5);
	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, 2e-254], N[(c0$95$m * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+300], N[(c0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0$95$m * N[Power[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 1.9999999999999998e-254

   1. Initial program 45.5%

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

      1. clear-num 45.1%

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

      2. associate-/r/ 45.5%

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

      3. associate-/r* 45.9%

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

   4. Applied egg-rr 45.9%

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

   5. Taylor expanded in c0 around 0 45.5%

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

      1. associate-/l/ 60.6%

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

   7. Simplified 60.6%

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

   if 1.9999999999999998e-254 < (/.f64 A (*.f64 V l)) < 4.0000000000000002e300

   1. Initial program 99.6%

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

   if 4.0000000000000002e300 < (/.f64 A (*.f64 V l))

   1. Initial program 46.8%

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

      1. frac-2neg 46.8%

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

      2. sqrt-div 31.1%

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

      3. distribute-rgt-neg-in 31.1%

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

   4. Applied egg-rr 31.1%

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

      1. distribute-rgt-neg-out 31.1%

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

      2. *-commutative 31.1%

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

      3. distribute-rgt-neg-in 31.1%

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

   6. Simplified 31.1%

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

      1. clear-num 31.1%

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

      2. sqrt-undiv 51.5%

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

      3. distribute-rgt-neg-out 51.5%

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

      4. frac-2neg 51.5%

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

      5. sqrt-div 46.6%

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

      6. add-sqr-sqrt 27.0%

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

      7. sqrt-unprod 23.5%

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

      8. sqr-neg 23.5%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 0.1%

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

      11. *-commutative 0.1%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 23.5%

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

      14. sqr-neg 23.5%

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

      15. sqrt-unprod 27.0%

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

      16. add-sqr-sqrt 46.6%

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

      17. sqrt-div 51.5%

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

      18. associate-*r/ 59.9%

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

   8. Applied egg-rr 51.5%

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

      1. associate-/l* 59.9%

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

   10. Simplified 59.9%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{-254}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 4 \cdot 10^{+300}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{-322}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+300}:\\ \;\;\;\;c0\_m \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \end{array} \end{array} \]

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 1e-322)
      (* c0_m (sqrt (/ (/ A V) l)))
      (if (<= t_0 4e+300)
        (* c0_m (pow (/ (* V l) A) -0.5))
        (* c0_m (pow (* V (/ l A)) -0.5)))))))

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 <= 1e-322) {
		tmp = c0_m * sqrt(((A / V) / l));
	} else if (t_0 <= 4e+300) {
		tmp = c0_m * pow(((V * l) / A), -0.5);
	} else {
		tmp = c0_m * pow((V * (l / A)), -0.5);
	}
	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 <= 1d-322) then
        tmp = c0_m * sqrt(((a / v) / l))
    else if (t_0 <= 4d+300) then
        tmp = c0_m * (((v * l) / a) ** (-0.5d0))
    else
        tmp = c0_m * ((v * (l / a)) ** (-0.5d0))
    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 <= 1e-322) {
		tmp = c0_m * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 4e+300) {
		tmp = c0_m * Math.pow(((V * l) / A), -0.5);
	} else {
		tmp = c0_m * Math.pow((V * (l / A)), -0.5);
	}
	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 <= 1e-322:
		tmp = c0_m * math.sqrt(((A / V) / l))
	elif t_0 <= 4e+300:
		tmp = c0_m * math.pow(((V * l) / A), -0.5)
	else:
		tmp = c0_m * math.pow((V * (l / A)), -0.5)
	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 <= 1e-322)
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 4e+300)
		tmp = Float64(c0_m * (Float64(Float64(V * l) / A) ^ -0.5));
	else
		tmp = Float64(c0_m * (Float64(V * Float64(l / A)) ^ -0.5));
	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 <= 1e-322)
		tmp = c0_m * sqrt(((A / V) / l));
	elseif (t_0 <= 4e+300)
		tmp = c0_m * (((V * l) / A) ^ -0.5);
	else
		tmp = c0_m * ((V * (l / A)) ^ -0.5);
	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, 1e-322], N[(c0$95$m * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+300], N[(c0$95$m * N[Power[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(c0$95$m * N[Power[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 9.88131e-323

   1. Initial program 35.5%

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

      1. associate-/r* 55.0%

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

   3. Simplified 55.0%

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

   if 9.88131e-323 < (/.f64 A (*.f64 V l)) < 4.0000000000000002e300

   1. Initial program 99.6%

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

      1. frac-2neg 99.6%

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

      2. sqrt-div 46.3%

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

      3. distribute-rgt-neg-in 46.3%

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

   4. Applied egg-rr 46.3%

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

      1. distribute-rgt-neg-out 46.3%

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

      2. *-commutative 46.3%

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

      3. distribute-rgt-neg-in 46.3%

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

   6. Simplified 46.3%

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

      1. clear-num 46.2%

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

      2. sqrt-undiv 99.4%

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

      3. distribute-rgt-neg-out 99.4%

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

      4. frac-2neg 99.4%

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

      5. sqrt-div 53.1%

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

      6. add-sqr-sqrt 22.8%

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

      7. sqrt-unprod 12.7%

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

      8. sqr-neg 12.7%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 0.0%

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

      11. *-commutative 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 12.7%

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

      14. sqr-neg 12.7%

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

      15. sqrt-unprod 22.8%

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

      16. add-sqr-sqrt 53.1%

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

      17. sqrt-div 99.4%

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

      18. associate-*r/ 84.5%

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

   8. Applied egg-rr 99.7%

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

   if 4.0000000000000002e300 < (/.f64 A (*.f64 V l))

   1. Initial program 46.8%

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

      1. frac-2neg 46.8%

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

      2. sqrt-div 31.1%

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

      3. distribute-rgt-neg-in 31.1%

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

   4. Applied egg-rr 31.1%

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

      1. distribute-rgt-neg-out 31.1%

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

      2. *-commutative 31.1%

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

      3. distribute-rgt-neg-in 31.1%

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

   6. Simplified 31.1%

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

      1. clear-num 31.1%

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

      2. sqrt-undiv 51.5%

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

      3. distribute-rgt-neg-out 51.5%

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

      4. frac-2neg 51.5%

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

      5. sqrt-div 46.6%

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

      6. add-sqr-sqrt 27.0%

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

      7. sqrt-unprod 23.5%

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

      8. sqr-neg 23.5%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 0.1%

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

      11. *-commutative 0.1%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 23.5%

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

      14. sqr-neg 23.5%

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

      15. sqrt-unprod 27.0%

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

      16. add-sqr-sqrt 46.6%

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

      17. sqrt-div 51.5%

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

      18. associate-*r/ 59.9%

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

   8. Applied egg-rr 51.5%

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

      1. associate-/l* 59.9%

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

   10. Simplified 59.9%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 10^{-322}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 4 \cdot 10^{+300}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-308} \lor \neg \left(t\_0 \leq 4 \cdot 10^{+300}\right):\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \end{array} \end{array} \end{array} \]

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 4e-308) (not (<= t_0 4e+300)))
      (* c0_m (sqrt (/ (/ A V) l)))
      (* 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 <= 4e-308) || !(t_0 <= 4e+300)) {
		tmp = c0_m * sqrt(((A / V) / l));
	} 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 <= 4d-308) .or. (.not. (t_0 <= 4d+300))) then
        tmp = c0_m * sqrt(((a / v) / l))
    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 <= 4e-308) || !(t_0 <= 4e+300)) {
		tmp = c0_m * Math.sqrt(((A / V) / l));
	} 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 <= 4e-308) or not (t_0 <= 4e+300):
		tmp = c0_m * math.sqrt(((A / V) / l))
	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 <= 4e-308) || !(t_0 <= 4e+300))
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / V) / l)));
	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 <= 4e-308) || ~((t_0 <= 4e+300)))
		tmp = c0_m * sqrt(((A / V) / l));
	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, 4e-308], N[Not[LessEqual[t$95$0, 4e+300]], $MachinePrecision]], N[(c0$95$m * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $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)) < 4.00000000000000013e-308 or 4.0000000000000002e300 < (/.f64 A (*.f64 V l))

   1. Initial program 41.3%

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

      1. associate-/r* 55.9%

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

   3. Simplified 55.9%

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

   if 4.00000000000000013e-308 < (/.f64 A (*.f64 V l)) < 4.0000000000000002e300

   1. Initial program 99.6%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 4 \cdot 10^{-308} \lor \neg \left(\frac{A}{V \cdot \ell} \leq 4 \cdot 10^{+300}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-254}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+300}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \end{array} \end{array} \]

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 2e-254)
      (* c0_m (sqrt (/ (/ A l) V)))
      (if (<= t_0 4e+300)
        (* c0_m (sqrt t_0))
        (* 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) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 2e-254) {
		tmp = c0_m * sqrt(((A / l) / V));
	} else if (t_0 <= 4e+300) {
		tmp = c0_m * sqrt(t_0);
	} else {
		tmp = c0_m * sqrt(((A / V) / l));
	}
	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 <= 2d-254) then
        tmp = c0_m * sqrt(((a / l) / v))
    else if (t_0 <= 4d+300) then
        tmp = c0_m * sqrt(t_0)
    else
        tmp = c0_m * sqrt(((a / v) / l))
    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 <= 2e-254) {
		tmp = c0_m * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 4e+300) {
		tmp = c0_m * Math.sqrt(t_0);
	} else {
		tmp = c0_m * Math.sqrt(((A / 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):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 2e-254:
		tmp = c0_m * math.sqrt(((A / l) / V))
	elif t_0 <= 4e+300:
		tmp = c0_m * math.sqrt(t_0)
	else:
		tmp = c0_m * math.sqrt(((A / 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)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 2e-254)
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 4e+300)
		tmp = Float64(c0_m * sqrt(t_0));
	else
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / 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)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 2e-254)
		tmp = c0_m * sqrt(((A / l) / V));
	elseif (t_0 <= 4e+300)
		tmp = c0_m * sqrt(t_0);
	else
		tmp = c0_m * sqrt(((A / 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_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[t$95$0, 2e-254], N[(c0$95$m * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+300], N[(c0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0$95$m * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 1.9999999999999998e-254

   1. Initial program 45.5%

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

      1. clear-num 45.1%

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

      2. associate-/r/ 45.5%

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

      3. associate-/r* 45.9%

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

   4. Applied egg-rr 45.9%

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

   5. Taylor expanded in c0 around 0 45.5%

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

      1. associate-/l/ 60.6%

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

   7. Simplified 60.6%

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

   if 1.9999999999999998e-254 < (/.f64 A (*.f64 V l)) < 4.0000000000000002e300

   1. Initial program 99.6%

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

   if 4.0000000000000002e300 < (/.f64 A (*.f64 V l))

   1. Initial program 46.8%

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

      1. associate-/r* 56.1%

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

   3. Simplified 56.1%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{-254}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 4 \cdot 10^{+300}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-254}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+288}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \end{array} \]

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 2e-254)
      (* c0_m (sqrt (/ (/ A l) V)))
      (if (<= t_0 5e+288)
        (* c0_m (sqrt t_0))
        (/ c0_m (sqrt (* l (/ V A)))))))))

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 <= 2e-254) {
		tmp = c0_m * sqrt(((A / l) / V));
	} else if (t_0 <= 5e+288) {
		tmp = c0_m * sqrt(t_0);
	} else {
		tmp = c0_m / sqrt((l * (V / A)));
	}
	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 <= 2d-254) then
        tmp = c0_m * sqrt(((a / l) / v))
    else if (t_0 <= 5d+288) then
        tmp = c0_m * sqrt(t_0)
    else
        tmp = c0_m / sqrt((l * (v / a)))
    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 <= 2e-254) {
		tmp = c0_m * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 5e+288) {
		tmp = c0_m * Math.sqrt(t_0);
	} else {
		tmp = c0_m / Math.sqrt((l * (V / A)));
	}
	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 <= 2e-254:
		tmp = c0_m * math.sqrt(((A / l) / V))
	elif t_0 <= 5e+288:
		tmp = c0_m * math.sqrt(t_0)
	else:
		tmp = c0_m / math.sqrt((l * (V / A)))
	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 <= 2e-254)
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 5e+288)
		tmp = Float64(c0_m * sqrt(t_0));
	else
		tmp = Float64(c0_m / sqrt(Float64(l * Float64(V / A))));
	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 <= 2e-254)
		tmp = c0_m * sqrt(((A / l) / V));
	elseif (t_0 <= 5e+288)
		tmp = c0_m * sqrt(t_0);
	else
		tmp = c0_m / sqrt((l * (V / A)));
	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, 2e-254], N[(c0$95$m * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+288], N[(c0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0$95$m / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 1.9999999999999998e-254

   1. Initial program 45.5%

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

      1. clear-num 45.1%

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

      2. associate-/r/ 45.5%

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

      3. associate-/r* 45.9%

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

   4. Applied egg-rr 45.9%

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

   5. Taylor expanded in c0 around 0 45.5%

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

      1. associate-/l/ 60.6%

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

   7. Simplified 60.6%

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

   if 1.9999999999999998e-254 < (/.f64 A (*.f64 V l)) < 5.0000000000000003e288

   1. Initial program 99.6%

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

   if 5.0000000000000003e288 < (/.f64 A (*.f64 V l))

   1. Initial program 49.7%

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

      1. clear-num 49.7%

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

      2. associate-/r/ 49.6%

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

      3. associate-/r* 49.6%

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

   4. Applied egg-rr 49.6%

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

      1. *-commutative 49.6%

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

      2. associate-/r* 49.6%

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

      3. div-inv 49.7%

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

      4. sqrt-undiv 47.6%

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

      5. clear-num 47.6%

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

      6. un-div-inv 47.7%

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

      7. sqrt-undiv 54.1%

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

      8. clear-num 49.7%

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

      9. associate-/r* 58.4%

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

      10. clear-num 62.5%

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

      11. div-inv 62.6%

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

      12. clear-num 62.6%

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

   6. Applied egg-rr 62.6%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{-254}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+288}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.2% 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 73.9%

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

3. Final simplification 73.9%

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

Reproduce

herbie shell --seed 2024059 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))