Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.7% → 92.0%

Time: 10.3s

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: 73.7% 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: 92.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+274}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-317}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+304}:\\ \;\;\;\;c0\_m \cdot \left(\sqrt{A} \cdot \frac{1}{\sqrt{V \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{c0\_m \cdot \frac{A}{\ell}}{\frac{V}{c0\_m}}}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (* c0_m (/ (sqrt (/ A (- l))) (sqrt (- V))))))
   (*
    c0_s
    (if (<= (* V l) -1e+274)
      t_0
      (if (<= (* V l) -1e-317)
        (* c0_m (/ (sqrt (- A)) (sqrt (* l (- V)))))
        (if (<= (* V l) 0.0)
          t_0
          (if (<= (* V l) 2e+304)
            (* c0_m (* (sqrt A) (/ 1.0 (sqrt (* V l)))))
            (sqrt (/ (* c0_m (/ A l)) (/ V c0_m))))))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * (sqrt((A / -l)) / sqrt(-V));
	double tmp;
	if ((V * l) <= -1e+274) {
		tmp = t_0;
	} else if ((V * l) <= -1e-317) {
		tmp = c0_m * (sqrt(-A) / sqrt((l * -V)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 2e+304) {
		tmp = c0_m * (sqrt(A) * (1.0 / sqrt((V * l))));
	} else {
		tmp = sqrt(((c0_m * (A / l)) / (V / c0_m)));
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0_m * (sqrt((a / -l)) / sqrt(-v))
    if ((v * l) <= (-1d+274)) then
        tmp = t_0
    else if ((v * l) <= (-1d-317)) then
        tmp = c0_m * (sqrt(-a) / sqrt((l * -v)))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else if ((v * l) <= 2d+304) then
        tmp = c0_m * (sqrt(a) * (1.0d0 / sqrt((v * l))))
    else
        tmp = sqrt(((c0_m * (a / l)) / (v / c0_m)))
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * (Math.sqrt((A / -l)) / Math.sqrt(-V));
	double tmp;
	if ((V * l) <= -1e+274) {
		tmp = t_0;
	} else if ((V * l) <= -1e-317) {
		tmp = c0_m * (Math.sqrt(-A) / Math.sqrt((l * -V)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 2e+304) {
		tmp = c0_m * (Math.sqrt(A) * (1.0 / Math.sqrt((V * l))));
	} else {
		tmp = Math.sqrt(((c0_m * (A / l)) / (V / c0_m)));
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = c0_m * (math.sqrt((A / -l)) / math.sqrt(-V))
	tmp = 0
	if (V * l) <= -1e+274:
		tmp = t_0
	elif (V * l) <= -1e-317:
		tmp = c0_m * (math.sqrt(-A) / math.sqrt((l * -V)))
	elif (V * l) <= 0.0:
		tmp = t_0
	elif (V * l) <= 2e+304:
		tmp = c0_m * (math.sqrt(A) * (1.0 / math.sqrt((V * l))))
	else:
		tmp = math.sqrt(((c0_m * (A / l)) / (V / c0_m)))
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(c0_m * Float64(sqrt(Float64(A / Float64(-l))) / sqrt(Float64(-V))))
	tmp = 0.0
	if (Float64(V * l) <= -1e+274)
		tmp = t_0;
	elseif (Float64(V * l) <= -1e-317)
		tmp = Float64(c0_m * Float64(sqrt(Float64(-A)) / sqrt(Float64(l * Float64(-V)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	elseif (Float64(V * l) <= 2e+304)
		tmp = Float64(c0_m * Float64(sqrt(A) * Float64(1.0 / sqrt(Float64(V * l)))));
	else
		tmp = sqrt(Float64(Float64(c0_m * Float64(A / l)) / Float64(V / c0_m)));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = c0_m * (sqrt((A / -l)) / sqrt(-V));
	tmp = 0.0;
	if ((V * l) <= -1e+274)
		tmp = t_0;
	elseif ((V * l) <= -1e-317)
		tmp = c0_m * (sqrt(-A) / sqrt((l * -V)));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	elseif ((V * l) <= 2e+304)
		tmp = c0_m * (sqrt(A) * (1.0 / sqrt((V * l))));
	else
		tmp = sqrt(((c0_m * (A / l)) / (V / c0_m)));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(c0$95$m * N[(N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], -1e+274], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -1e-317], N[(c0$95$m * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], 2e+304], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(c0$95$m * N[(A / l), $MachinePrecision]), $MachinePrecision] / N[(V / c0$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -9.99999999999999921e273 or -1.00000023e-317 < (*.f64 V l) < 0.0

   1. Initial program 39.6%

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

      1. associate-/r* 54.2%

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

      2. div-inv 54.2%

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

   4. Applied egg-rr 54.2%

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

      1. associate-*l/ 54.2%

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

      2. div-inv 54.3%

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

      3. frac-2neg 54.3%

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

      4. sqrt-div 48.7%

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

      5. distribute-neg-frac 48.7%

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

   6. Applied egg-rr 48.7%

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

   if -9.99999999999999921e273 < (*.f64 V l) < -1.00000023e-317

   1. Initial program 89.4%

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

      1. frac-2neg 89.4%

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

      2. sqrt-div 98.9%

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

      3. *-commutative 98.9%

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

      4. distribute-rgt-neg-in 98.9%

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

   4. Applied egg-rr 98.9%

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

   if 0.0 < (*.f64 V l) < 1.9999999999999999e304

   1. Initial program 82.0%

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

      1. sqrt-div 99.1%

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

      2. div-inv 99.1%

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

   4. Applied egg-rr 99.1%

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

   if 1.9999999999999999e304 < (*.f64 V l)

   1. Initial program 23.3%

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

      1. add-sqr-sqrt 23.3%

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

      2. sqrt-unprod 23.3%

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

      3. *-commutative 23.3%

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

      4. *-commutative 23.3%

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

      5. swap-sqr 22.3%

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

      6. add-sqr-sqrt 22.3%

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

      7. pow2 22.3%

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

   4. Applied egg-rr 22.3%

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

      1. associate-*l/ 22.5%

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

      2. *-commutative 22.5%

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

      3. times-frac 22.9%

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

   6. Simplified 22.9%

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

      1. unpow2 22.9%

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

      2. *-un-lft-identity 22.9%

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

      3. times-frac 32.6%

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

   8. Applied egg-rr 32.6%

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

      1. /-rgt-identity 32.6%

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

      2. associate-*r* 37.4%

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

      3. clear-num 37.3%

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

      4. un-div-inv 37.4%

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

   10. Applied egg-rr 37.4%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+274}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-317}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+304}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \frac{1}{\sqrt{V \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{c0 \cdot \frac{A}{\ell}}{\frac{V}{c0}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-317}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-310} \lor \neg \left(V \cdot \ell \leq 2 \cdot 10^{+304}\right):\\ \;\;\;\;\sqrt{\frac{c0\_m \cdot \frac{A}{\ell}}{\frac{V}{c0\_m}}}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \left(\sqrt{A} \cdot \frac{1}{\sqrt{V \cdot \ell}}\right)\\ \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (*
  c0_s
  (if (<= (* V l) (- INFINITY))
    (* c0_m (/ (sqrt (/ A V)) (sqrt l)))
    (if (<= (* V l) -1e-317)
      (* c0_m (/ (sqrt (- A)) (sqrt (* l (- V)))))
      (if (or (<= (* V l) 5e-310) (not (<= (* V l) 2e+304)))
        (sqrt (/ (* c0_m (/ A l)) (/ V c0_m)))
        (* c0_m (* (sqrt A) (/ 1.0 (sqrt (* V l))))))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0_m * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= -1e-317) {
		tmp = c0_m * (sqrt(-A) / sqrt((l * -V)));
	} else if (((V * l) <= 5e-310) || !((V * l) <= 2e+304)) {
		tmp = sqrt(((c0_m * (A / l)) / (V / c0_m)));
	} else {
		tmp = c0_m * (sqrt(A) * (1.0 / sqrt((V * l))));
	}
	return c0_s * tmp;
}

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0_m * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= -1e-317) {
		tmp = c0_m * (Math.sqrt(-A) / Math.sqrt((l * -V)));
	} else if (((V * l) <= 5e-310) || !((V * l) <= 2e+304)) {
		tmp = Math.sqrt(((c0_m * (A / l)) / (V / c0_m)));
	} else {
		tmp = c0_m * (Math.sqrt(A) * (1.0 / Math.sqrt((V * l))));
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0_m * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= -1e-317:
		tmp = c0_m * (math.sqrt(-A) / math.sqrt((l * -V)))
	elif ((V * l) <= 5e-310) or not ((V * l) <= 2e+304):
		tmp = math.sqrt(((c0_m * (A / l)) / (V / c0_m)))
	else:
		tmp = c0_m * (math.sqrt(A) * (1.0 / math.sqrt((V * l))))
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0_m * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= -1e-317)
		tmp = Float64(c0_m * Float64(sqrt(Float64(-A)) / sqrt(Float64(l * Float64(-V)))));
	elseif ((Float64(V * l) <= 5e-310) || !(Float64(V * l) <= 2e+304))
		tmp = sqrt(Float64(Float64(c0_m * Float64(A / l)) / Float64(V / c0_m)));
	else
		tmp = Float64(c0_m * Float64(sqrt(A) * Float64(1.0 / sqrt(Float64(V * l)))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0_m * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= -1e-317)
		tmp = c0_m * (sqrt(-A) / sqrt((l * -V)));
	elseif (((V * l) <= 5e-310) || ~(((V * l) <= 2e+304)))
		tmp = sqrt(((c0_m * (A / l)) / (V / c0_m)));
	else
		tmp = c0_m * (sqrt(A) * (1.0 / sqrt((V * l))));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0$95$m * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-317], N[(c0$95$m * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 5e-310], N[Not[LessEqual[N[(V * l), $MachinePrecision], 2e+304]], $MachinePrecision]], N[Sqrt[N[(N[(c0$95$m * N[(A / l), $MachinePrecision]), $MachinePrecision] / N[(V / c0$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 36.9%

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

      1. associate-/r* 47.6%

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

      2. sqrt-div 40.1%

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

      3. associate-*r/ 39.9%

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

   4. Applied egg-rr 39.9%

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

      1. associate-/l* 40.1%

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

   6. Simplified 40.1%

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

   if -inf.0 < (*.f64 V l) < -1.00000023e-317

   1. Initial program 87.7%

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

      1. frac-2neg 87.7%

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

      2. sqrt-div 98.9%

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

      3. *-commutative 98.9%

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

      4. distribute-rgt-neg-in 98.9%

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

   4. Applied egg-rr 98.9%

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

   if -1.00000023e-317 < (*.f64 V l) < 4.999999999999985e-310 or 1.9999999999999999e304 < (*.f64 V l)

   1. Initial program 34.7%

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

      1. add-sqr-sqrt 22.5%

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

      2. sqrt-unprod 22.6%

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

      3. *-commutative 22.6%

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

      4. *-commutative 22.6%

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

      5. swap-sqr 21.9%

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

      6. add-sqr-sqrt 21.9%

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

      7. pow2 21.9%

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

   4. Applied egg-rr 21.9%

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

      1. associate-*l/ 21.9%

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

      2. *-commutative 21.9%

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

      3. times-frac 28.4%

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

   6. Simplified 28.4%

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

      1. unpow2 28.4%

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

      2. *-un-lft-identity 28.4%

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

      3. times-frac 37.5%

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

   8. Applied egg-rr 37.5%

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

      1. /-rgt-identity 37.5%

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

      2. associate-*r* 42.9%

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

      3. clear-num 42.8%

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

      4. un-div-inv 44.5%

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

   10. Applied egg-rr 44.5%

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

   if 4.999999999999985e-310 < (*.f64 V l) < 1.9999999999999999e304

   1. Initial program 82.6%

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

      1. sqrt-div 99.4%

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

      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%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-317}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-310} \lor \neg \left(V \cdot \ell \leq 2 \cdot 10^{+304}\right):\\ \;\;\;\;\sqrt{\frac{c0 \cdot \frac{A}{\ell}}{\frac{V}{c0}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \frac{1}{\sqrt{V \cdot \ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{c0\_m \cdot \frac{A}{\ell}}{\frac{V}{c0\_m}}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+274}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-298}:\\ \;\;\;\;c0\_m \cdot \sqrt{A \cdot \frac{1}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-310} \lor \neg \left(V \cdot \ell \leq 2 \cdot 10^{+304}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \end{array} \]

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 (sqrt (/ (* c0_m (/ A l)) (/ V c0_m)))))
   (*
    c0_s
    (if (<= (* V l) -1e+274)
      t_0
      (if (<= (* V l) -5e-298)
        (* c0_m (sqrt (* A (/ 1.0 (* V l)))))
        (if (or (<= (* V l) 5e-310) (not (<= (* V l) 2e+304)))
          t_0
          (* c0_m (/ (sqrt A) (sqrt (* 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 = sqrt(((c0_m * (A / l)) / (V / c0_m)));
	double tmp;
	if ((V * l) <= -1e+274) {
		tmp = t_0;
	} else if ((V * l) <= -5e-298) {
		tmp = c0_m * sqrt((A * (1.0 / (V * l))));
	} else if (((V * l) <= 5e-310) || !((V * l) <= 2e+304)) {
		tmp = t_0;
	} else {
		tmp = c0_m * (sqrt(A) / sqrt((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 = sqrt(((c0_m * (a / l)) / (v / c0_m)))
    if ((v * l) <= (-1d+274)) then
        tmp = t_0
    else if ((v * l) <= (-5d-298)) then
        tmp = c0_m * sqrt((a * (1.0d0 / (v * l))))
    else if (((v * l) <= 5d-310) .or. (.not. ((v * l) <= 2d+304))) then
        tmp = t_0
    else
        tmp = c0_m * (sqrt(a) / sqrt((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 = Math.sqrt(((c0_m * (A / l)) / (V / c0_m)));
	double tmp;
	if ((V * l) <= -1e+274) {
		tmp = t_0;
	} else if ((V * l) <= -5e-298) {
		tmp = c0_m * Math.sqrt((A * (1.0 / (V * l))));
	} else if (((V * l) <= 5e-310) || !((V * l) <= 2e+304)) {
		tmp = t_0;
	} else {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((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 = math.sqrt(((c0_m * (A / l)) / (V / c0_m)))
	tmp = 0
	if (V * l) <= -1e+274:
		tmp = t_0
	elif (V * l) <= -5e-298:
		tmp = c0_m * math.sqrt((A * (1.0 / (V * l))))
	elif ((V * l) <= 5e-310) or not ((V * l) <= 2e+304):
		tmp = t_0
	else:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((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 = sqrt(Float64(Float64(c0_m * Float64(A / l)) / Float64(V / c0_m)))
	tmp = 0.0
	if (Float64(V * l) <= -1e+274)
		tmp = t_0;
	elseif (Float64(V * l) <= -5e-298)
		tmp = Float64(c0_m * sqrt(Float64(A * Float64(1.0 / Float64(V * l)))));
	elseif ((Float64(V * l) <= 5e-310) || !(Float64(V * l) <= 2e+304))
		tmp = t_0;
	else
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(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 = sqrt(((c0_m * (A / l)) / (V / c0_m)));
	tmp = 0.0;
	if ((V * l) <= -1e+274)
		tmp = t_0;
	elseif ((V * l) <= -5e-298)
		tmp = c0_m * sqrt((A * (1.0 / (V * l))));
	elseif (((V * l) <= 5e-310) || ~(((V * l) <= 2e+304)))
		tmp = t_0;
	else
		tmp = c0_m * (sqrt(A) / sqrt((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[Sqrt[N[(N[(c0$95$m * N[(A / l), $MachinePrecision]), $MachinePrecision] / N[(V / c0$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], -1e+274], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -5e-298], N[(c0$95$m * N[Sqrt[N[(A * N[(1.0 / N[(V * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 5e-310], N[Not[LessEqual[N[(V * l), $MachinePrecision], 2e+304]], $MachinePrecision]], t$95$0, N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -9.99999999999999921e273 or -5.0000000000000002e-298 < (*.f64 V l) < 4.999999999999985e-310 or 1.9999999999999999e304 < (*.f64 V l)

   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 25.8%

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

      2. sqrt-unprod 25.9%

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

      3. *-commutative 25.9%

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

      4. *-commutative 25.9%

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

      5. swap-sqr 25.2%

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

      6. add-sqr-sqrt 25.2%

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

      7. pow2 25.2%

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

   4. Applied egg-rr 25.2%

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

      1. associate-*l/ 23.9%

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

      2. *-commutative 23.9%

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

      3. times-frac 31.1%

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

   6. Simplified 31.1%

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

      1. unpow2 31.1%

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

      2. *-un-lft-identity 31.1%

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

      3. times-frac 40.1%

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

   8. Applied egg-rr 40.1%

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

      1. /-rgt-identity 40.1%

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

      2. associate-*r* 45.1%

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

      3. clear-num 45.0%

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

      4. un-div-inv 46.2%

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

   10. Applied egg-rr 46.2%

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

   if -9.99999999999999921e273 < (*.f64 V l) < -5.0000000000000002e-298

   1. Initial program 90.4%

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

      1. clear-num 89.1%

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

      2. associate-/r/ 90.4%

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

   4. Applied egg-rr 90.4%

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

   if 4.999999999999985e-310 < (*.f64 V l) < 1.9999999999999999e304

   1. Initial program 82.6%

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

      1. sqrt-div 99.4%

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

      2. associate-*r/ 96.4%

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

   4. Applied egg-rr 96.4%

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

      1. associate-/l* 99.4%

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

   6. Simplified 99.4%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+274}:\\ \;\;\;\;\sqrt{\frac{c0 \cdot \frac{A}{\ell}}{\frac{V}{c0}}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-298}:\\ \;\;\;\;c0 \cdot \sqrt{A \cdot \frac{1}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-310} \lor \neg \left(V \cdot \ell \leq 2 \cdot 10^{+304}\right):\\ \;\;\;\;\sqrt{\frac{c0 \cdot \frac{A}{\ell}}{\frac{V}{c0}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 10^{-312}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+304}:\\ \;\;\;\;c0\_m \cdot \left(\sqrt{A} \cdot \frac{1}{\sqrt{V \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{c0\_m \cdot \frac{A}{\ell}}{\frac{V}{c0\_m}}}\\ \end{array} \end{array} \]

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) 1e-312)
    (/ c0_m (* (sqrt l) (sqrt (/ V A))))
    (if (<= (* V l) 2e+304)
      (* c0_m (* (sqrt A) (/ 1.0 (sqrt (* V l)))))
      (sqrt (/ (* c0_m (/ A l)) (/ V c0_m)))))))

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) <= 1e-312) {
		tmp = c0_m / (sqrt(l) * sqrt((V / A)));
	} else if ((V * l) <= 2e+304) {
		tmp = c0_m * (sqrt(A) * (1.0 / sqrt((V * l))));
	} else {
		tmp = sqrt(((c0_m * (A / l)) / (V / c0_m)));
	}
	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) :: tmp
    if ((v * l) <= 1d-312) then
        tmp = c0_m / (sqrt(l) * sqrt((v / a)))
    else if ((v * l) <= 2d+304) then
        tmp = c0_m * (sqrt(a) * (1.0d0 / sqrt((v * l))))
    else
        tmp = sqrt(((c0_m * (a / l)) / (v / c0_m)))
    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 tmp;
	if ((V * l) <= 1e-312) {
		tmp = c0_m / (Math.sqrt(l) * Math.sqrt((V / A)));
	} else if ((V * l) <= 2e+304) {
		tmp = c0_m * (Math.sqrt(A) * (1.0 / Math.sqrt((V * l))));
	} else {
		tmp = Math.sqrt(((c0_m * (A / l)) / (V / c0_m)));
	}
	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) <= 1e-312:
		tmp = c0_m / (math.sqrt(l) * math.sqrt((V / A)))
	elif (V * l) <= 2e+304:
		tmp = c0_m * (math.sqrt(A) * (1.0 / math.sqrt((V * l))))
	else:
		tmp = math.sqrt(((c0_m * (A / l)) / (V / c0_m)))
	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) <= 1e-312)
		tmp = Float64(c0_m / Float64(sqrt(l) * sqrt(Float64(V / A))));
	elseif (Float64(V * l) <= 2e+304)
		tmp = Float64(c0_m * Float64(sqrt(A) * Float64(1.0 / sqrt(Float64(V * l)))));
	else
		tmp = sqrt(Float64(Float64(c0_m * Float64(A / l)) / Float64(V / c0_m)));
	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) <= 1e-312)
		tmp = c0_m / (sqrt(l) * sqrt((V / A)));
	elseif ((V * l) <= 2e+304)
		tmp = c0_m * (sqrt(A) * (1.0 / sqrt((V * l))));
	else
		tmp = sqrt(((c0_m * (A / l)) / (V / c0_m)));
	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], 1e-312], N[(c0$95$m / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e+304], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(c0$95$m * N[(A / l), $MachinePrecision]), $MachinePrecision] / N[(V / c0$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < 9.9999999999847e-313

   1. Initial program 70.9%

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

      1. associate-/r* 73.1%

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

      2. div-inv 73.0%

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

   4. Applied egg-rr 73.0%

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

      1. un-div-inv 73.1%

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

      2. associate-/r* 70.9%

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

      3. sqrt-undiv 5.1%

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

      4. clear-num 5.1%

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

      5. un-div-inv 5.1%

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

      6. sqrt-undiv 70.0%

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

      7. associate-/l* 70.4%

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

   6. Applied egg-rr 70.4%

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

      1. *-commutative 70.4%

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

      2. associate-*l/ 70.0%

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

      3. associate-*r/ 72.6%

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

   8. Simplified 72.6%

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

      1. *-commutative 72.6%

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

      2. sqrt-prod 42.7%

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

   10. Applied egg-rr 42.7%

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

   if 9.9999999999847e-313 < (*.f64 V l) < 1.9999999999999999e304

   1. Initial program 81.8%

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

      1. sqrt-div 99.3%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

   if 1.9999999999999999e304 < (*.f64 V l)

   1. Initial program 23.3%

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

      1. add-sqr-sqrt 23.3%

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

      2. sqrt-unprod 23.3%

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

      3. *-commutative 23.3%

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

      4. *-commutative 23.3%

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

      5. swap-sqr 22.3%

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

      6. add-sqr-sqrt 22.3%

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

      7. pow2 22.3%

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

   4. Applied egg-rr 22.3%

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

      1. associate-*l/ 22.5%

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

      2. *-commutative 22.5%

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

      3. times-frac 22.9%

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

   6. Simplified 22.9%

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

      1. unpow2 22.9%

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

      2. *-un-lft-identity 22.9%

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

      3. times-frac 32.6%

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

   8. Applied egg-rr 32.6%

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

      1. /-rgt-identity 32.6%

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

      2. associate-*r* 37.4%

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

      3. clear-num 37.3%

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

      4. un-div-inv 37.4%

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

   10. Applied egg-rr 37.4%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 10^{-312}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+304}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \frac{1}{\sqrt{V \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{c0 \cdot \frac{A}{\ell}}{\frac{V}{c0}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 10^{-312}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+304}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{c0\_m \cdot \frac{A}{\ell}}{\frac{V}{c0\_m}}}\\ \end{array} \end{array} \]

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) 1e-312)
    (/ c0_m (* (sqrt l) (sqrt (/ V A))))
    (if (<= (* V l) 2e+304)
      (* c0_m (/ (sqrt A) (sqrt (* V l))))
      (sqrt (/ (* c0_m (/ A l)) (/ V c0_m)))))))

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) <= 1e-312) {
		tmp = c0_m / (sqrt(l) * sqrt((V / A)));
	} else if ((V * l) <= 2e+304) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = sqrt(((c0_m * (A / l)) / (V / c0_m)));
	}
	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) :: tmp
    if ((v * l) <= 1d-312) then
        tmp = c0_m / (sqrt(l) * sqrt((v / a)))
    else if ((v * l) <= 2d+304) then
        tmp = c0_m * (sqrt(a) / sqrt((v * l)))
    else
        tmp = sqrt(((c0_m * (a / l)) / (v / c0_m)))
    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 tmp;
	if ((V * l) <= 1e-312) {
		tmp = c0_m / (Math.sqrt(l) * Math.sqrt((V / A)));
	} else if ((V * l) <= 2e+304) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = Math.sqrt(((c0_m * (A / l)) / (V / c0_m)));
	}
	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) <= 1e-312:
		tmp = c0_m / (math.sqrt(l) * math.sqrt((V / A)))
	elif (V * l) <= 2e+304:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = math.sqrt(((c0_m * (A / l)) / (V / c0_m)))
	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) <= 1e-312)
		tmp = Float64(c0_m / Float64(sqrt(l) * sqrt(Float64(V / A))));
	elseif (Float64(V * l) <= 2e+304)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = sqrt(Float64(Float64(c0_m * Float64(A / l)) / Float64(V / c0_m)));
	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) <= 1e-312)
		tmp = c0_m / (sqrt(l) * sqrt((V / A)));
	elseif ((V * l) <= 2e+304)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	else
		tmp = sqrt(((c0_m * (A / l)) / (V / c0_m)));
	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], 1e-312], N[(c0$95$m / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e+304], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(c0$95$m * N[(A / l), $MachinePrecision]), $MachinePrecision] / N[(V / c0$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < 9.9999999999847e-313

   1. Initial program 70.9%

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

      1. associate-/r* 73.1%

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

      2. div-inv 73.0%

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

   4. Applied egg-rr 73.0%

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

      1. un-div-inv 73.1%

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

      2. associate-/r* 70.9%

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

      3. sqrt-undiv 5.1%

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

      4. clear-num 5.1%

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

      5. un-div-inv 5.1%

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

      6. sqrt-undiv 70.0%

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

      7. associate-/l* 70.4%

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

   6. Applied egg-rr 70.4%

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

      1. *-commutative 70.4%

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

      2. associate-*l/ 70.0%

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

      3. associate-*r/ 72.6%

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

   8. Simplified 72.6%

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

      1. *-commutative 72.6%

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

      2. sqrt-prod 42.7%

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

   10. Applied egg-rr 42.7%

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

   if 9.9999999999847e-313 < (*.f64 V l) < 1.9999999999999999e304

   1. Initial program 81.8%

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

      1. sqrt-div 99.3%

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

      2. associate-*r/ 96.4%

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

   4. Applied egg-rr 96.4%

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

      1. associate-/l* 99.3%

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

   6. Simplified 99.3%

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

   if 1.9999999999999999e304 < (*.f64 V l)

   1. Initial program 23.3%

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

      1. add-sqr-sqrt 23.3%

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

      2. sqrt-unprod 23.3%

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

      3. *-commutative 23.3%

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

      4. *-commutative 23.3%

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

      5. swap-sqr 22.3%

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

      6. add-sqr-sqrt 22.3%

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

      7. pow2 22.3%

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

   4. Applied egg-rr 22.3%

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

      1. associate-*l/ 22.5%

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

      2. *-commutative 22.5%

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

      3. times-frac 22.9%

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

   6. Simplified 22.9%

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

      1. unpow2 22.9%

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

      2. *-un-lft-identity 22.9%

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

      3. times-frac 32.6%

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

   8. Applied egg-rr 32.6%

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

      1. /-rgt-identity 32.6%

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

      2. associate-*r* 37.4%

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

      3. clear-num 37.3%

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

      4. un-div-inv 37.4%

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

   10. Applied egg-rr 37.4%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 10^{-312}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{c0 \cdot \frac{A}{\ell}}{\frac{V}{c0}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 10^{-312}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+304}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{c0\_m \cdot \frac{A}{\ell}}{\frac{V}{c0\_m}}}\\ \end{array} \end{array} \]

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) 1e-312)
    (* c0_m (/ (sqrt (/ A V)) (sqrt l)))
    (if (<= (* V l) 2e+304)
      (* c0_m (/ (sqrt A) (sqrt (* V l))))
      (sqrt (/ (* c0_m (/ A l)) (/ V c0_m)))))))

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) <= 1e-312) {
		tmp = c0_m * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= 2e+304) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = sqrt(((c0_m * (A / l)) / (V / c0_m)));
	}
	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) :: tmp
    if ((v * l) <= 1d-312) then
        tmp = c0_m * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= 2d+304) then
        tmp = c0_m * (sqrt(a) / sqrt((v * l)))
    else
        tmp = sqrt(((c0_m * (a / l)) / (v / c0_m)))
    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 tmp;
	if ((V * l) <= 1e-312) {
		tmp = c0_m * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= 2e+304) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = Math.sqrt(((c0_m * (A / l)) / (V / c0_m)));
	}
	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) <= 1e-312:
		tmp = c0_m * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= 2e+304:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = math.sqrt(((c0_m * (A / l)) / (V / c0_m)))
	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) <= 1e-312)
		tmp = Float64(c0_m * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= 2e+304)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = sqrt(Float64(Float64(c0_m * Float64(A / l)) / Float64(V / c0_m)));
	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) <= 1e-312)
		tmp = c0_m * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= 2e+304)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	else
		tmp = sqrt(((c0_m * (A / l)) / (V / c0_m)));
	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], 1e-312], N[(c0$95$m * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e+304], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(c0$95$m * N[(A / l), $MachinePrecision]), $MachinePrecision] / N[(V / c0$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < 9.9999999999847e-313

   1. Initial program 70.9%

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

      1. associate-/r* 73.1%

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

      2. sqrt-div 42.3%

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

      3. associate-*r/ 39.8%

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

   4. Applied egg-rr 39.8%

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

      1. associate-/l* 42.3%

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

   6. Simplified 42.3%

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

   if 9.9999999999847e-313 < (*.f64 V l) < 1.9999999999999999e304

   1. Initial program 81.8%

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

      1. sqrt-div 99.3%

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

      2. associate-*r/ 96.4%

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

   4. Applied egg-rr 96.4%

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

      1. associate-/l* 99.3%

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

   6. Simplified 99.3%

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

   if 1.9999999999999999e304 < (*.f64 V l)

   1. Initial program 23.3%

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

      1. add-sqr-sqrt 23.3%

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

      2. sqrt-unprod 23.3%

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

      3. *-commutative 23.3%

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

      4. *-commutative 23.3%

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

      5. swap-sqr 22.3%

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

      6. add-sqr-sqrt 22.3%

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

      7. pow2 22.3%

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

   4. Applied egg-rr 22.3%

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

      1. associate-*l/ 22.5%

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

      2. *-commutative 22.5%

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

      3. times-frac 22.9%

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

   6. Simplified 22.9%

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

      1. unpow2 22.9%

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

      2. *-un-lft-identity 22.9%

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

      3. times-frac 32.6%

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

   8. Applied egg-rr 32.6%

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

      1. /-rgt-identity 32.6%

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

      2. associate-*r* 37.4%

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

      3. clear-num 37.3%

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

      4. un-div-inv 37.4%

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

   10. Applied egg-rr 37.4%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 10^{-312}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{c0 \cdot \frac{A}{\ell}}{\frac{V}{c0}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 2 \cdot 10^{+296}\right):\\ \;\;\;\;\sqrt{\frac{c0\_m \cdot \frac{A}{\ell}}{\frac{V}{c0\_m}}}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \end{array} \end{array} \end{array} \]

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 2e+296)))
      (sqrt (/ (* c0_m (/ A l)) (/ V c0_m)))
      (* 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 <= 2e+296)) {
		tmp = sqrt(((c0_m * (A / l)) / (V / c0_m)));
	} 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 <= 2d+296))) then
        tmp = sqrt(((c0_m * (a / l)) / (v / c0_m)))
    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 <= 2e+296)) {
		tmp = Math.sqrt(((c0_m * (A / l)) / (V / c0_m)));
	} 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 <= 2e+296):
		tmp = math.sqrt(((c0_m * (A / l)) / (V / c0_m)))
	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 <= 2e+296))
		tmp = sqrt(Float64(Float64(c0_m * Float64(A / l)) / Float64(V / c0_m)));
	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 <= 2e+296)))
		tmp = sqrt(((c0_m * (A / l)) / (V / c0_m)));
	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, 2e+296]], $MachinePrecision]], N[Sqrt[N[(N[(c0$95$m * N[(A / l), $MachinePrecision]), $MachinePrecision] / N[(V / c0$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(c0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 1.99999999999999996e296 < (/.f64 A (*.f64 V l))

   1. Initial program 32.4%

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

      1. add-sqr-sqrt 22.9%

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

      2. sqrt-unprod 22.9%

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

      3. *-commutative 22.9%

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

      4. *-commutative 22.9%

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

      5. swap-sqr 22.3%

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

      6. add-sqr-sqrt 22.3%

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

      7. pow2 22.3%

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

   4. Applied egg-rr 22.3%

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

      1. associate-*l/ 22.5%

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

      2. *-commutative 22.5%

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

      3. times-frac 26.4%

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

   6. Simplified 26.4%

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

      1. unpow2 26.4%

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

      2. *-un-lft-identity 26.4%

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

      3. times-frac 32.8%

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

   8. Applied egg-rr 32.8%

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

      1. /-rgt-identity 32.8%

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

      2. associate-*r* 36.4%

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

      3. clear-num 36.4%

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

      4. un-div-inv 37.2%

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

   10. Applied egg-rr 37.2%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 1.99999999999999996e296

   1. Initial program 99.2%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+296}\right):\\ \;\;\;\;\sqrt{\frac{c0 \cdot \frac{A}{\ell}}{\frac{V}{c0}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 2 \cdot 10^{+296}\right):\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \frac{c0\_m}{\frac{V}{c0\_m}}}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \end{array} \end{array} \end{array} \]

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 2e+296)))
      (sqrt (* (/ A l) (/ c0_m (/ V c0_m))))
      (* 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 <= 2e+296)) {
		tmp = sqrt(((A / l) * (c0_m / (V / c0_m))));
	} 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 <= 2d+296))) then
        tmp = sqrt(((a / l) * (c0_m / (v / c0_m))))
    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 <= 2e+296)) {
		tmp = Math.sqrt(((A / l) * (c0_m / (V / c0_m))));
	} 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 <= 2e+296):
		tmp = math.sqrt(((A / l) * (c0_m / (V / c0_m))))
	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 <= 2e+296))
		tmp = sqrt(Float64(Float64(A / l) * Float64(c0_m / Float64(V / c0_m))));
	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 <= 2e+296)))
		tmp = sqrt(((A / l) * (c0_m / (V / c0_m))));
	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, 2e+296]], $MachinePrecision]], N[Sqrt[N[(N[(A / l), $MachinePrecision] * N[(c0$95$m / N[(V / c0$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(c0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 1.99999999999999996e296 < (/.f64 A (*.f64 V l))

   1. Initial program 32.4%

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

      1. add-sqr-sqrt 22.9%

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

      2. sqrt-unprod 22.9%

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

      3. *-commutative 22.9%

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

      4. *-commutative 22.9%

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

      5. swap-sqr 22.3%

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

      6. add-sqr-sqrt 22.3%

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

      7. pow2 22.3%

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

   4. Applied egg-rr 22.3%

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

      1. associate-*l/ 22.5%

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

      2. *-commutative 22.5%

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

      3. times-frac 26.4%

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

   6. Simplified 26.4%

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

      1. unpow2 26.4%

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

      2. *-un-lft-identity 26.4%

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

      3. times-frac 32.8%

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

   8. Applied egg-rr 32.8%

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

      1. /-rgt-identity 32.8%

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

      2. clear-num 32.8%

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

      3. un-div-inv 32.8%

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

   10. Applied egg-rr 32.8%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 1.99999999999999996e296

   1. Initial program 99.2%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+296}\right):\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \frac{c0}{\frac{V}{c0}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 2 \cdot 10^{+296}\right):\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \end{array} \end{array} \end{array} \]

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 2e+296)))
      (* c0_m (sqrt (/ (/ A l) V)))
      (* c0_m (sqrt t_0))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 2e+296)) {
		tmp = c0_m * sqrt(((A / l) / V));
	} else {
		tmp = c0_m * sqrt(t_0);
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 2d+296))) then
        tmp = c0_m * sqrt(((a / l) / v))
    else
        tmp = c0_m * sqrt(t_0)
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 2e+296)) {
		tmp = c0_m * Math.sqrt(((A / l) / V));
	} else {
		tmp = c0_m * Math.sqrt(t_0);
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 2e+296):
		tmp = c0_m * math.sqrt(((A / l) / V))
	else:
		tmp = c0_m * math.sqrt(t_0)
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 2e+296))
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / l) / V)));
	else
		tmp = Float64(c0_m * sqrt(t_0));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 2e+296)))
		tmp = c0_m * sqrt(((A / l) / V));
	else
		tmp = c0_m * sqrt(t_0);
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 2e+296]], $MachinePrecision]], N[(c0$95$m * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 1.99999999999999996e296 < (/.f64 A (*.f64 V l))

   1. Initial program 32.4%

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

   3. Taylor expanded in c0 around 0 32.4%

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

      1. *-commutative 32.4%

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

      2. associate-/r* 44.9%

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

   5. Simplified 44.9%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 1.99999999999999996e296

   1. Initial program 99.2%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+296}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 10^{+293}\right):\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \end{array} \end{array} \end{array} \]

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 0.0) (not (<= t_0 1e+293)))
      (* 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 <= 0.0) || !(t_0 <= 1e+293)) {
		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 <= 0.0d0) .or. (.not. (t_0 <= 1d+293))) 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 <= 0.0) || !(t_0 <= 1e+293)) {
		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 <= 0.0) or not (t_0 <= 1e+293):
		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 <= 0.0) || !(t_0 <= 1e+293))
		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 <= 0.0) || ~((t_0 <= 1e+293)))
		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, 0.0], N[Not[LessEqual[t$95$0, 1e+293]], $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)) < 0.0 or 9.9999999999999992e292 < (/.f64 A (*.f64 V l))

   1. Initial program 33.6%

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

      1. associate-/r* 45.9%

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

   3. Simplified 45.9%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 9.9999999999999992e292

   1. Initial program 99.2%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 10^{+293}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+296}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \end{array} \]

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 0.0)
      (* c0_m (sqrt (/ (/ A l) V)))
      (if (<= t_0 2e+296)
        (* c0_m (sqrt t_0))
        (/ c0_m (sqrt (* V (/ l 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 <= 0.0) {
		tmp = c0_m * sqrt(((A / l) / V));
	} else if (t_0 <= 2e+296) {
		tmp = c0_m * sqrt(t_0);
	} else {
		tmp = c0_m / sqrt((V * (l / 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 <= 0.0d0) then
        tmp = c0_m * sqrt(((a / l) / v))
    else if (t_0 <= 2d+296) then
        tmp = c0_m * sqrt(t_0)
    else
        tmp = c0_m / sqrt((v * (l / 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 <= 0.0) {
		tmp = c0_m * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 2e+296) {
		tmp = c0_m * Math.sqrt(t_0);
	} else {
		tmp = c0_m / Math.sqrt((V * (l / 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 <= 0.0:
		tmp = c0_m * math.sqrt(((A / l) / V))
	elif t_0 <= 2e+296:
		tmp = c0_m * math.sqrt(t_0)
	else:
		tmp = c0_m / math.sqrt((V * (l / 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 <= 0.0)
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 2e+296)
		tmp = Float64(c0_m * sqrt(t_0));
	else
		tmp = Float64(c0_m / sqrt(Float64(V * Float64(l / 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 <= 0.0)
		tmp = c0_m * sqrt(((A / l) / V));
	elseif (t_0 <= 2e+296)
		tmp = c0_m * sqrt(t_0);
	else
		tmp = c0_m / sqrt((V * (l / 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, 0.0], N[(c0$95$m * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+296], N[(c0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0$95$m / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 29.8%

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

   3. Taylor expanded in c0 around 0 29.8%

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

      1. *-commutative 29.8%

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

      2. associate-/r* 44.3%

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

   5. Simplified 44.3%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 1.99999999999999996e296

   1. Initial program 99.2%

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

   if 1.99999999999999996e296 < (/.f64 A (*.f64 V l))

   1. Initial program 34.9%

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

      1. associate-/r* 45.4%

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

      2. div-inv 45.4%

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

   4. Applied egg-rr 45.4%

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

      1. un-div-inv 45.4%

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

      2. associate-/r* 34.9%

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

      3. sqrt-undiv 33.7%

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

      4. clear-num 33.6%

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

      5. un-div-inv 33.6%

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

      6. sqrt-undiv 35.5%

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

      7. associate-/l* 45.5%

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

   6. Applied egg-rr 45.5%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+296}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.7% 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 71.3%

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

Reproduce

herbie shell --seed 2024125 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))