Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.7% → 93.1%

Time: 17.4s

Alternatives: 16

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: 93.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0_m = \left|c0\right| \\ c0_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\frac{c0\_m}{\frac{\sqrt{\ell}}{\frac{\sqrt{-A}}{\sqrt{-V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\sqrt{\frac{c0\_m}{\ell} \cdot \frac{\frac{c0\_m}{V}}{\frac{1}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+307}:\\ \;\;\;\;c0\_m \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \end{array} \]

FPCore

c0_m = (fabs.f64 c0)
c0_s = (copysign.f64 1 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-318)
    (/ c0_m (/ (sqrt l) (/ (sqrt (- A)) (sqrt (- V)))))
    (if (<= (* V l) 0.0)
      (sqrt (* (/ c0_m l) (/ (/ c0_m V) (/ 1.0 A))))
      (if (<= (* V l) 5e+307)
        (* c0_m (* (sqrt A) (pow (* V l) -0.5)))
        (/ c0_m (sqrt (/ l (/ A V)))))))))

C

c0_m = fabs(c0);
c0_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e-318) {
		tmp = c0_m / (sqrt(l) / (sqrt(-A) / sqrt(-V)));
	} else if ((V * l) <= 0.0) {
		tmp = sqrt(((c0_m / l) * ((c0_m / V) / (1.0 / A))));
	} else if ((V * l) <= 5e+307) {
		tmp = c0_m * (sqrt(A) * pow((V * l), -0.5));
	} else {
		tmp = c0_m / sqrt((l / (A / V)));
	}
	return c0_s * tmp;
}

Fortran

c0_m = abs(c0)
c0_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-1d-318)) then
        tmp = c0_m / (sqrt(l) / (sqrt(-a) / sqrt(-v)))
    else if ((v * l) <= 0.0d0) then
        tmp = sqrt(((c0_m / l) * ((c0_m / v) / (1.0d0 / a))))
    else if ((v * l) <= 5d+307) then
        tmp = c0_m * (sqrt(a) * ((v * l) ** (-0.5d0)))
    else
        tmp = c0_m / sqrt((l / (a / v)))
    end if
    code = c0_s * tmp
end function

Java

c0_m = Math.abs(c0);
c0_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e-318) {
		tmp = c0_m / (Math.sqrt(l) / (Math.sqrt(-A) / Math.sqrt(-V)));
	} else if ((V * l) <= 0.0) {
		tmp = Math.sqrt(((c0_m / l) * ((c0_m / V) / (1.0 / A))));
	} else if ((V * l) <= 5e+307) {
		tmp = c0_m * (Math.sqrt(A) * Math.pow((V * l), -0.5));
	} else {
		tmp = c0_m / Math.sqrt((l / (A / V)));
	}
	return c0_s * tmp;
}

Python

c0_m = math.fabs(c0)
c0_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	tmp = 0
	if (V * l) <= -1e-318:
		tmp = c0_m / (math.sqrt(l) / (math.sqrt(-A) / math.sqrt(-V)))
	elif (V * l) <= 0.0:
		tmp = math.sqrt(((c0_m / l) * ((c0_m / V) / (1.0 / A))))
	elif (V * l) <= 5e+307:
		tmp = c0_m * (math.sqrt(A) * math.pow((V * l), -0.5))
	else:
		tmp = c0_m / math.sqrt((l / (A / V)))
	return c0_s * tmp

Julia

c0_m = abs(c0)
c0_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e-318)
		tmp = Float64(c0_m / Float64(sqrt(l) / Float64(sqrt(Float64(-A)) / sqrt(Float64(-V)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = sqrt(Float64(Float64(c0_m / l) * Float64(Float64(c0_m / V) / Float64(1.0 / A))));
	elseif (Float64(V * l) <= 5e+307)
		tmp = Float64(c0_m * Float64(sqrt(A) * (Float64(V * l) ^ -0.5)));
	else
		tmp = Float64(c0_m / sqrt(Float64(l / Float64(A / V))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0_m = abs(c0);
c0_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e-318)
		tmp = c0_m / (sqrt(l) / (sqrt(-A) / sqrt(-V)));
	elseif ((V * l) <= 0.0)
		tmp = sqrt(((c0_m / l) * ((c0_m / V) / (1.0 / A))));
	elseif ((V * l) <= 5e+307)
		tmp = c0_m * (sqrt(A) * ((V * l) ^ -0.5));
	else
		tmp = c0_m / sqrt((l / (A / V)));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0_m = N[Abs[c0], $MachinePrecision]
c0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], -1e-318], N[(c0$95$m / N[(N[Sqrt[l], $MachinePrecision] / N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Sqrt[N[(N[(c0$95$m / l), $MachinePrecision] * N[(N[(c0$95$m / V), $MachinePrecision] / N[(1.0 / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+307], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] * N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0$95$m / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -9.9999875e-319

   1. Initial program 79.2%

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

      1. associate-/r* 75.1%

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

      2. sqrt-div 43.5%

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

      3. associate-*r/ 43.5%

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

   4. Applied egg-rr 43.5%

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

      1. associate-/l* 43.5%

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

   6. Simplified 43.5%

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

      1. frac-2neg 43.5%

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

      2. sqrt-div 50.6%

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

   8. Applied egg-rr 50.6%

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

   if -9.9999875e-319 < (*.f64 V l) < 0.0

   1. Initial program 50.8%

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

      1. associate-/r* 68.1%

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

      2. clear-num 68.1%

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

      3. sqrt-div 71.1%

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

      4. metadata-eval 71.1%

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

      5. div-inv 71.1%

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

      6. clear-num 71.1%

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

   4. Applied egg-rr 71.1%

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

      1. *-commutative 71.1%

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

      2. associate-*r/ 50.8%

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

      3. *-commutative 50.8%

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

      4. sqrt-div 23.9%

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

      5. clear-num 23.9%

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

      6. sqrt-div 50.8%

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

      7. add-sqr-sqrt 24.1%

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

      8. pow1/2 24.1%

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

      9. pow1/2 24.2%

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

      10. pow-prod-down 24.2%

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

   6. Applied egg-rr 25.9%

      \[\leadsto \color{blue}{{\left(\frac{{c0}^{2}}{V \cdot \frac{\ell}{A}}\right)}^{0.5}} \]
   7. Step-by-step derivation

      1. unpow1/2 25.9%

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

      2. associate-/r* 27.9%

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

   8. Simplified 27.9%

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

      1. div-inv 27.8%

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

      2. unpow2 27.8%

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

      3. associate-*l* 31.4%

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

   10. Applied egg-rr 31.4%

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

       1. div-inv 31.4%

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

       2. times-frac 35.0%

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

       3. un-div-inv 35.0%

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

   12. Applied egg-rr 35.0%

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

   if 0.0 < (*.f64 V l) < 5e307

   1. Initial program 82.2%

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

      1. associate-/r* 77.5%

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

      2. clear-num 77.3%

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

      3. sqrt-div 77.8%

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

      4. metadata-eval 77.8%

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

      5. div-inv 77.8%

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

      6. clear-num 77.8%

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

   4. Applied egg-rr 77.8%

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

      1. associate-*r/ 77.8%

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

      2. sqrt-prod 39.3%

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

      3. times-frac 37.4%

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

      4. metadata-eval 37.4%

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

      5. sqrt-div 37.4%

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

      6. clear-num 37.4%

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

      7. associate-/r/ 39.3%

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

      8. expm1-log1p-u 25.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)\right)} \]

      9. expm1-udef 13.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)} - 1} \]

      10. sqrt-undiv 33.4%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\frac{\ell}{\frac{A}{V}}}}}\right)} - 1 \]

      11. associate-/l* 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{\ell \cdot V}{A}}}}\right)} - 1 \]

      12. *-commutative 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell}}{A}}}\right)} - 1 \]

      13. *-un-lft-identity 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}}}\right)} - 1 \]

      14. times-frac 34.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{V}{1} \cdot \frac{\ell}{A}}}}\right)} - 1 \]

      15. /-rgt-identity 34.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{V} \cdot \frac{\ell}{A}}}\right)} - 1 \]

   6. Applied egg-rr 34.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 58.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)\right)} \]

      2. expm1-log1p 75.3%

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

   8. Simplified 75.3%

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

      1. clear-num 74.3%

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

      2. un-div-inv 75.9%

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

   10. Applied egg-rr 75.9%

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

       1. associate-/l* 82.6%

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

       2. sqrt-div 98.9%

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

       3. associate-/l* 97.1%

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

       4. *-commutative 97.1%

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

       5. div-inv 97.1%

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

       6. *-commutative 97.1%

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

       7. associate-*l* 98.9%

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

       8. pow1/2 98.9%

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

       9. pow-flip 99.0%

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

       10. metadata-eval 99.0%

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

   12. Applied egg-rr 99.0%

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

   if 5e307 < (*.f64 V l)

   1. Initial program 31.9%

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

      1. associate-/r* 79.1%

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

      2. clear-num 79.3%

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

      3. sqrt-div 79.3%

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

      4. metadata-eval 79.3%

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

      5. div-inv 79.1%

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

      6. clear-num 79.1%

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

   4. Applied egg-rr 79.1%

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

      1. associate-*r/ 79.2%

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

      2. sqrt-prod 37.2%

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

      3. times-frac 37.1%

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

      4. metadata-eval 37.1%

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

      5. sqrt-div 37.2%

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

      6. clear-num 37.2%

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

      7. associate-/r/ 37.1%

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

      8. expm1-log1p-u 28.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)\right)} \]

      9. expm1-udef 13.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)} - 1} \]

      10. sqrt-undiv 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\frac{\ell}{\frac{A}{V}}}}}\right)} - 1 \]

      11. associate-/l* 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{\ell \cdot V}{A}}}}\right)} - 1 \]

      12. *-commutative 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell}}{A}}}\right)} - 1 \]

      13. *-un-lft-identity 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}}}\right)} - 1 \]

      14. times-frac 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{V}{1} \cdot \frac{\ell}{A}}}}\right)} - 1 \]

      15. /-rgt-identity 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{V} \cdot \frac{\ell}{A}}}\right)} - 1 \]

   6. Applied egg-rr 35.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 66.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)\right)} \]

      2. expm1-log1p 79.2%

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

   8. Simplified 79.2%

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

      1. associate-*r/ 31.9%

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

      2. *-commutative 31.9%

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

      3. associate-/l* 79.3%

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

   10. Applied egg-rr 79.3%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\frac{\sqrt{-A}}{\sqrt{-V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\sqrt{\frac{c0}{\ell} \cdot \frac{\frac{c0}{V}}{\frac{1}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+307}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0_m = \left|c0\right| \\ c0_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 4 \cdot 10^{+290}\right):\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

c0_m = (fabs.f64 c0)
c0_s = (copysign.f64 1 c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (* c0_m (sqrt (/ A (* V l))))))
   (*
    c0_s
    (if (or (<= t_0 0.0) (not (<= t_0 4e+290)))
      (* c0_m (sqrt (/ (/ A V) l)))
      t_0))))

C

c0_m = fabs(c0);
c0_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 4e+290)) {
		tmp = c0_m * sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return c0_s * tmp;
}

Fortran

c0_m = abs(c0)
c0_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0_m * sqrt((a / (v * l)))
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 4d+290))) then
        tmp = c0_m * sqrt(((a / v) / l))
    else
        tmp = t_0
    end if
    code = c0_s * tmp
end function

Java

c0_m = Math.abs(c0);
c0_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * Math.sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 4e+290)) {
		tmp = c0_m * Math.sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return c0_s * tmp;
}

Python

c0_m = math.fabs(c0)
c0_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = c0_m * math.sqrt((A / (V * l)))
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 4e+290):
		tmp = c0_m * math.sqrt(((A / V) / l))
	else:
		tmp = t_0
	return c0_s * tmp

Julia

c0_m = abs(c0)
c0_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(c0_m * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 4e+290))
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = t_0;
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0_m = abs(c0);
c0_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = c0_m * sqrt((A / (V * l)));
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 4e+290)))
		tmp = c0_m * sqrt(((A / V) / l));
	else
		tmp = t_0;
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0_m = N[Abs[c0], $MachinePrecision]
c0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 4e+290]], $MachinePrecision]], N[(c0$95$m * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 0.0 or 4.00000000000000025e290 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 65.2%

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

      1. *-commutative 65.2%

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

      2. associate-/l/ 71.4%

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

   3. Simplified 71.4%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.00000000000000025e290

   1. Initial program 98.2%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0 \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 4 \cdot 10^{+290}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0_m = \left|c0\right| \\ c0_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+227}:\\ \;\;\;\;\frac{c0\_m \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-52}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+307}:\\ \;\;\;\;c0\_m \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \end{array} \]

FPCore

c0_m = (fabs.f64 c0)
c0_s = (copysign.f64 1 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+227)
    (/ (* c0_m (sqrt (/ A V))) (sqrt l))
    (if (<= (* V l) -1e-52)
      (* c0_m (sqrt (/ A (* V l))))
      (if (<= (* V l) 0.0)
        (/ c0_m (* (sqrt l) (sqrt (/ V A))))
        (if (<= (* V l) 5e+307)
          (* c0_m (* (sqrt A) (pow (* V l) -0.5)))
          (/ c0_m (sqrt (/ l (/ A V))))))))))

C

c0_m = fabs(c0);
c0_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+227) {
		tmp = (c0_m * sqrt((A / V))) / sqrt(l);
	} else if ((V * l) <= -1e-52) {
		tmp = c0_m * sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0_m / (sqrt(l) * sqrt((V / A)));
	} else if ((V * l) <= 5e+307) {
		tmp = c0_m * (sqrt(A) * pow((V * l), -0.5));
	} else {
		tmp = c0_m / sqrt((l / (A / V)));
	}
	return c0_s * tmp;
}

Fortran

c0_m = abs(c0)
c0_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-1d+227)) then
        tmp = (c0_m * sqrt((a / v))) / sqrt(l)
    else if ((v * l) <= (-1d-52)) then
        tmp = c0_m * sqrt((a / (v * l)))
    else if ((v * l) <= 0.0d0) then
        tmp = c0_m / (sqrt(l) * sqrt((v / a)))
    else if ((v * l) <= 5d+307) then
        tmp = c0_m * (sqrt(a) * ((v * l) ** (-0.5d0)))
    else
        tmp = c0_m / sqrt((l / (a / v)))
    end if
    code = c0_s * tmp
end function

Java

c0_m = Math.abs(c0);
c0_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+227) {
		tmp = (c0_m * Math.sqrt((A / V))) / Math.sqrt(l);
	} else if ((V * l) <= -1e-52) {
		tmp = c0_m * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0_m / (Math.sqrt(l) * Math.sqrt((V / A)));
	} else if ((V * l) <= 5e+307) {
		tmp = c0_m * (Math.sqrt(A) * Math.pow((V * l), -0.5));
	} else {
		tmp = c0_m / Math.sqrt((l / (A / V)));
	}
	return c0_s * tmp;
}

Python

c0_m = math.fabs(c0)
c0_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	tmp = 0
	if (V * l) <= -1e+227:
		tmp = (c0_m * math.sqrt((A / V))) / math.sqrt(l)
	elif (V * l) <= -1e-52:
		tmp = c0_m * math.sqrt((A / (V * l)))
	elif (V * l) <= 0.0:
		tmp = c0_m / (math.sqrt(l) * math.sqrt((V / A)))
	elif (V * l) <= 5e+307:
		tmp = c0_m * (math.sqrt(A) * math.pow((V * l), -0.5))
	else:
		tmp = c0_m / math.sqrt((l / (A / V)))
	return c0_s * tmp

Julia

c0_m = abs(c0)
c0_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e+227)
		tmp = Float64(Float64(c0_m * sqrt(Float64(A / V))) / sqrt(l));
	elseif (Float64(V * l) <= -1e-52)
		tmp = Float64(c0_m * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0_m / Float64(sqrt(l) * sqrt(Float64(V / A))));
	elseif (Float64(V * l) <= 5e+307)
		tmp = Float64(c0_m * Float64(sqrt(A) * (Float64(V * l) ^ -0.5)));
	else
		tmp = Float64(c0_m / sqrt(Float64(l / Float64(A / V))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0_m = abs(c0);
c0_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e+227)
		tmp = (c0_m * sqrt((A / V))) / sqrt(l);
	elseif ((V * l) <= -1e-52)
		tmp = c0_m * sqrt((A / (V * l)));
	elseif ((V * l) <= 0.0)
		tmp = c0_m / (sqrt(l) * sqrt((V / A)));
	elseif ((V * l) <= 5e+307)
		tmp = c0_m * (sqrt(A) * ((V * l) ^ -0.5));
	else
		tmp = c0_m / sqrt((l / (A / V)));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0_m = N[Abs[c0], $MachinePrecision]
c0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], -1e+227], N[(N[(c0$95$m * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-52], N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0$95$m / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+307], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] * N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0$95$m / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -1.0000000000000001e227

   1. Initial program 57.7%

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

      1. *-commutative 57.7%

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

      2. associate-/r* 57.7%

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

      3. sqrt-div 44.4%

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

      4. associate-*l/ 44.7%

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

   4. Applied egg-rr 44.7%

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

   if -1.0000000000000001e227 < (*.f64 V l) < -1e-52

   1. Initial program 85.3%

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

   if -1e-52 < (*.f64 V l) < 0.0

   1. Initial program 69.6%

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

      1. associate-/r* 76.3%

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

      2. clear-num 76.3%

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

      3. sqrt-div 77.5%

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

      4. metadata-eval 77.5%

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

      5. div-inv 77.5%

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

      6. clear-num 77.5%

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

   4. Applied egg-rr 77.5%

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

      1. un-div-inv 77.5%

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

      2. sqrt-prod 44.8%

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

      3. associate-/r* 44.8%

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

   6. Applied egg-rr 44.8%

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

      1. associate-/r* 44.8%

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

   8. Simplified 44.8%

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

   if 0.0 < (*.f64 V l) < 5e307

   1. Initial program 82.2%

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

      1. associate-/r* 77.5%

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

      2. clear-num 77.3%

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

      3. sqrt-div 77.8%

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

      4. metadata-eval 77.8%

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

      5. div-inv 77.8%

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

      6. clear-num 77.8%

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

   4. Applied egg-rr 77.8%

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

      1. associate-*r/ 77.8%

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

      2. sqrt-prod 39.3%

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

      3. times-frac 37.4%

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

      4. metadata-eval 37.4%

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

      5. sqrt-div 37.4%

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

      6. clear-num 37.4%

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

      7. associate-/r/ 39.3%

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

      8. expm1-log1p-u 25.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)\right)} \]

      9. expm1-udef 13.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)} - 1} \]

      10. sqrt-undiv 33.4%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\frac{\ell}{\frac{A}{V}}}}}\right)} - 1 \]

      11. associate-/l* 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{\ell \cdot V}{A}}}}\right)} - 1 \]

      12. *-commutative 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell}}{A}}}\right)} - 1 \]

      13. *-un-lft-identity 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}}}\right)} - 1 \]

      14. times-frac 34.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{V}{1} \cdot \frac{\ell}{A}}}}\right)} - 1 \]

      15. /-rgt-identity 34.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{V} \cdot \frac{\ell}{A}}}\right)} - 1 \]

   6. Applied egg-rr 34.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 58.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)\right)} \]

      2. expm1-log1p 75.3%

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

   8. Simplified 75.3%

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

      1. clear-num 74.3%

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

      2. un-div-inv 75.9%

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

   10. Applied egg-rr 75.9%

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

       1. associate-/l* 82.6%

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

       2. sqrt-div 98.9%

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

       3. associate-/l* 97.1%

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

       4. *-commutative 97.1%

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

       5. div-inv 97.1%

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

       6. *-commutative 97.1%

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

       7. associate-*l* 98.9%

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

       8. pow1/2 98.9%

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

       9. pow-flip 99.0%

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

       10. metadata-eval 99.0%

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

   12. Applied egg-rr 99.0%

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

   if 5e307 < (*.f64 V l)

   1. Initial program 31.9%

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

      1. associate-/r* 79.1%

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

      2. clear-num 79.3%

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

      3. sqrt-div 79.3%

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

      4. metadata-eval 79.3%

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

      5. div-inv 79.1%

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

      6. clear-num 79.1%

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

   4. Applied egg-rr 79.1%

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

      1. associate-*r/ 79.2%

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

      2. sqrt-prod 37.2%

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

      3. times-frac 37.1%

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

      4. metadata-eval 37.1%

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

      5. sqrt-div 37.2%

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

      6. clear-num 37.2%

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

      7. associate-/r/ 37.1%

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

      8. expm1-log1p-u 28.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)\right)} \]

      9. expm1-udef 13.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)} - 1} \]

      10. sqrt-undiv 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\frac{\ell}{\frac{A}{V}}}}}\right)} - 1 \]

      11. associate-/l* 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{\ell \cdot V}{A}}}}\right)} - 1 \]

      12. *-commutative 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell}}{A}}}\right)} - 1 \]

      13. *-un-lft-identity 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}}}\right)} - 1 \]

      14. times-frac 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{V}{1} \cdot \frac{\ell}{A}}}}\right)} - 1 \]

      15. /-rgt-identity 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{V} \cdot \frac{\ell}{A}}}\right)} - 1 \]

   6. Applied egg-rr 35.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 66.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)\right)} \]

      2. expm1-log1p 79.2%

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

   8. Simplified 79.2%

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

      1. associate-*r/ 31.9%

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

      2. *-commutative 31.9%

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

      3. associate-/l* 79.3%

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

   10. Applied egg-rr 79.3%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+227}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-52}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+307}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0_m = \left|c0\right| \\ c0_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+241}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-37}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+307}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \end{array} \end{array} \]

FPCore

c0_m = (fabs.f64 c0)
c0_s = (copysign.f64 1 c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (* c0_m (/ (sqrt (/ A V)) (sqrt l)))))
   (*
    c0_s
    (if (<= (* V l) -4e+241)
      t_0
      (if (<= (* V l) -1e-37)
        (* c0_m (sqrt (/ A (* V l))))
        (if (<= (* V l) 0.0)
          t_0
          (if (<= (* V l) 5e+307)
            (* c0_m (/ (sqrt A) (sqrt (* V l))))
            (/ c0_m (sqrt (/ l (/ A V)))))))))))

C

c0_m = fabs(c0);
c0_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * (sqrt((A / V)) / sqrt(l));
	double tmp;
	if ((V * l) <= -4e+241) {
		tmp = t_0;
	} else if ((V * l) <= -1e-37) {
		tmp = c0_m * sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 5e+307) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0_m / sqrt((l / (A / V)));
	}
	return c0_s * tmp;
}

Fortran

c0_m = abs(c0)
c0_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0_m * (sqrt((a / v)) / sqrt(l))
    if ((v * l) <= (-4d+241)) then
        tmp = t_0
    else if ((v * l) <= (-1d-37)) then
        tmp = c0_m * sqrt((a / (v * l)))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else if ((v * l) <= 5d+307) then
        tmp = c0_m * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0_m / sqrt((l / (a / v)))
    end if
    code = c0_s * tmp
end function

Java

c0_m = Math.abs(c0);
c0_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * (Math.sqrt((A / V)) / Math.sqrt(l));
	double tmp;
	if ((V * l) <= -4e+241) {
		tmp = t_0;
	} else if ((V * l) <= -1e-37) {
		tmp = c0_m * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 5e+307) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0_m / Math.sqrt((l / (A / V)));
	}
	return c0_s * tmp;
}

Python

c0_m = math.fabs(c0)
c0_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = c0_m * (math.sqrt((A / V)) / math.sqrt(l))
	tmp = 0
	if (V * l) <= -4e+241:
		tmp = t_0
	elif (V * l) <= -1e-37:
		tmp = c0_m * math.sqrt((A / (V * l)))
	elif (V * l) <= 0.0:
		tmp = t_0
	elif (V * l) <= 5e+307:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0_m / math.sqrt((l / (A / V)))
	return c0_s * tmp

Julia

c0_m = abs(c0)
c0_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(c0_m * Float64(sqrt(Float64(A / V)) / sqrt(l)))
	tmp = 0.0
	if (Float64(V * l) <= -4e+241)
		tmp = t_0;
	elseif (Float64(V * l) <= -1e-37)
		tmp = Float64(c0_m * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	elseif (Float64(V * l) <= 5e+307)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0_m / sqrt(Float64(l / Float64(A / V))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0_m = abs(c0);
c0_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = c0_m * (sqrt((A / V)) / sqrt(l));
	tmp = 0.0;
	if ((V * l) <= -4e+241)
		tmp = t_0;
	elseif ((V * l) <= -1e-37)
		tmp = c0_m * sqrt((A / (V * l)));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	elseif ((V * l) <= 5e+307)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0_m / sqrt((l / (A / V)));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0_m = N[Abs[c0], $MachinePrecision]
c0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(c0$95$m * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], -4e+241], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -1e-37], N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], 5e+307], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0$95$m / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -4.0000000000000002e241 or -1.00000000000000007e-37 < (*.f64 V l) < 0.0

   1. Initial program 66.7%

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

      1. associate-/r* 72.1%

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

      2. sqrt-div 43.1%

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

      3. associate-*r/ 43.2%

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

   4. Applied egg-rr 43.2%

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

      1. *-commutative 43.2%

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

      2. associate-/l* 43.1%

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

      3. associate-/r/ 43.1%

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

   6. Simplified 43.1%

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

   if -4.0000000000000002e241 < (*.f64 V l) < -1.00000000000000007e-37

   1. Initial program 85.6%

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

   if 0.0 < (*.f64 V l) < 5e307

   1. Initial program 82.2%

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

      1. sqrt-div 98.9%

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

      2. associate-*r/ 97.1%

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

   4. Applied egg-rr 97.1%

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

      1. *-commutative 97.1%

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

      2. associate-/l* 94.3%

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

      3. associate-/r/ 98.9%

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

   6. Simplified 98.9%

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

   if 5e307 < (*.f64 V l)

   1. Initial program 31.9%

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

      1. associate-/r* 79.1%

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

      2. clear-num 79.3%

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

      3. sqrt-div 79.3%

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

      4. metadata-eval 79.3%

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

      5. div-inv 79.1%

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

      6. clear-num 79.1%

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

   4. Applied egg-rr 79.1%

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

      1. associate-*r/ 79.2%

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

      2. sqrt-prod 37.2%

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

      3. times-frac 37.1%

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

      4. metadata-eval 37.1%

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

      5. sqrt-div 37.2%

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

      6. clear-num 37.2%

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

      7. associate-/r/ 37.1%

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

      8. expm1-log1p-u 28.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)\right)} \]

      9. expm1-udef 13.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)} - 1} \]

      10. sqrt-undiv 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\frac{\ell}{\frac{A}{V}}}}}\right)} - 1 \]

      11. associate-/l* 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{\ell \cdot V}{A}}}}\right)} - 1 \]

      12. *-commutative 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell}}{A}}}\right)} - 1 \]

      13. *-un-lft-identity 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}}}\right)} - 1 \]

      14. times-frac 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{V}{1} \cdot \frac{\ell}{A}}}}\right)} - 1 \]

      15. /-rgt-identity 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{V} \cdot \frac{\ell}{A}}}\right)} - 1 \]

   6. Applied egg-rr 35.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 66.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)\right)} \]

      2. expm1-log1p 79.2%

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

   8. Simplified 79.2%

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

      1. associate-*r/ 31.9%

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

      2. *-commutative 31.9%

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

      3. associate-/l* 79.3%

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

   10. Applied egg-rr 79.3%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+241}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-37}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+307}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0_m = \left|c0\right| \\ c0_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+241}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-52}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+307}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \end{array} \]

FPCore

c0_m = (fabs.f64 c0)
c0_s = (copysign.f64 1 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) -4e+241)
    (* c0_m (/ (sqrt (/ A V)) (sqrt l)))
    (if (<= (* V l) -1e-52)
      (* c0_m (sqrt (/ A (* V l))))
      (if (<= (* V l) 0.0)
        (/ c0_m (* (sqrt l) (sqrt (/ V A))))
        (if (<= (* V l) 5e+307)
          (* c0_m (/ (sqrt A) (sqrt (* V l))))
          (/ c0_m (sqrt (/ l (/ A V))))))))))

C

c0_m = fabs(c0);
c0_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -4e+241) {
		tmp = c0_m * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= -1e-52) {
		tmp = c0_m * sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0_m / (sqrt(l) * sqrt((V / A)));
	} else if ((V * l) <= 5e+307) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0_m / sqrt((l / (A / V)));
	}
	return c0_s * tmp;
}

Fortran

c0_m = abs(c0)
c0_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-4d+241)) then
        tmp = c0_m * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= (-1d-52)) then
        tmp = c0_m * sqrt((a / (v * l)))
    else if ((v * l) <= 0.0d0) then
        tmp = c0_m / (sqrt(l) * sqrt((v / a)))
    else if ((v * l) <= 5d+307) then
        tmp = c0_m * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0_m / sqrt((l / (a / v)))
    end if
    code = c0_s * tmp
end function

Java

c0_m = Math.abs(c0);
c0_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -4e+241) {
		tmp = c0_m * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= -1e-52) {
		tmp = c0_m * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0_m / (Math.sqrt(l) * Math.sqrt((V / A)));
	} else if ((V * l) <= 5e+307) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0_m / Math.sqrt((l / (A / V)));
	}
	return c0_s * tmp;
}

Python

c0_m = math.fabs(c0)
c0_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	tmp = 0
	if (V * l) <= -4e+241:
		tmp = c0_m * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= -1e-52:
		tmp = c0_m * math.sqrt((A / (V * l)))
	elif (V * l) <= 0.0:
		tmp = c0_m / (math.sqrt(l) * math.sqrt((V / A)))
	elif (V * l) <= 5e+307:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0_m / math.sqrt((l / (A / V)))
	return c0_s * tmp

Julia

c0_m = abs(c0)
c0_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -4e+241)
		tmp = Float64(c0_m * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= -1e-52)
		tmp = Float64(c0_m * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0_m / Float64(sqrt(l) * sqrt(Float64(V / A))));
	elseif (Float64(V * l) <= 5e+307)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0_m / sqrt(Float64(l / Float64(A / V))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0_m = abs(c0);
c0_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -4e+241)
		tmp = c0_m * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= -1e-52)
		tmp = c0_m * sqrt((A / (V * l)));
	elseif ((V * l) <= 0.0)
		tmp = c0_m / (sqrt(l) * sqrt((V / A)));
	elseif ((V * l) <= 5e+307)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0_m / sqrt((l / (A / V)));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0_m = N[Abs[c0], $MachinePrecision]
c0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], -4e+241], 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-52], N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0$95$m / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+307], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0$95$m / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -4.0000000000000002e241

   1. Initial program 49.4%

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

      1. associate-/r* 49.4%

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

      2. sqrt-div 40.1%

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

      3. associate-*r/ 40.3%

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

   4. Applied egg-rr 40.3%

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

      1. *-commutative 40.3%

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

      2. associate-/l* 40.1%

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

      3. associate-/r/ 40.1%

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

   6. Simplified 40.1%

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

   if -4.0000000000000002e241 < (*.f64 V l) < -1e-52

   1. Initial program 86.2%

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

   if -1e-52 < (*.f64 V l) < 0.0

   1. Initial program 69.6%

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

      1. associate-/r* 76.3%

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

      2. clear-num 76.3%

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

      3. sqrt-div 77.5%

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

      4. metadata-eval 77.5%

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

      5. div-inv 77.5%

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

      6. clear-num 77.5%

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

   4. Applied egg-rr 77.5%

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

      1. un-div-inv 77.5%

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

      2. sqrt-prod 44.8%

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

      3. associate-/r* 44.8%

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

   6. Applied egg-rr 44.8%

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

      1. associate-/r* 44.8%

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

   8. Simplified 44.8%

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

   if 0.0 < (*.f64 V l) < 5e307

   1. Initial program 82.2%

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

      1. sqrt-div 98.9%

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

      2. associate-*r/ 97.1%

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

   4. Applied egg-rr 97.1%

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

      1. *-commutative 97.1%

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

      2. associate-/l* 94.3%

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

      3. associate-/r/ 98.9%

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

   6. Simplified 98.9%

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

   if 5e307 < (*.f64 V l)

   1. Initial program 31.9%

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

      1. associate-/r* 79.1%

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

      2. clear-num 79.3%

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

      3. sqrt-div 79.3%

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

      4. metadata-eval 79.3%

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

      5. div-inv 79.1%

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

      6. clear-num 79.1%

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

   4. Applied egg-rr 79.1%

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

      1. associate-*r/ 79.2%

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

      2. sqrt-prod 37.2%

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

      3. times-frac 37.1%

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

      4. metadata-eval 37.1%

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

      5. sqrt-div 37.2%

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

      6. clear-num 37.2%

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

      7. associate-/r/ 37.1%

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

      8. expm1-log1p-u 28.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)\right)} \]

      9. expm1-udef 13.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)} - 1} \]

      10. sqrt-undiv 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\frac{\ell}{\frac{A}{V}}}}}\right)} - 1 \]

      11. associate-/l* 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{\ell \cdot V}{A}}}}\right)} - 1 \]

      12. *-commutative 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell}}{A}}}\right)} - 1 \]

      13. *-un-lft-identity 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}}}\right)} - 1 \]

      14. times-frac 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{V}{1} \cdot \frac{\ell}{A}}}}\right)} - 1 \]

      15. /-rgt-identity 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{V} \cdot \frac{\ell}{A}}}\right)} - 1 \]

   6. Applied egg-rr 35.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 66.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)\right)} \]

      2. expm1-log1p 79.2%

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

   8. Simplified 79.2%

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

      1. associate-*r/ 31.9%

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

      2. *-commutative 31.9%

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

      3. associate-/l* 79.3%

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

   10. Applied egg-rr 79.3%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+241}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-52}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+307}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0_m = \left|c0\right| \\ c0_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+227}:\\ \;\;\;\;\frac{c0\_m}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-52}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+307}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \end{array} \]

FPCore

c0_m = (fabs.f64 c0)
c0_s = (copysign.f64 1 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+227)
    (/ c0_m (/ (sqrt l) (sqrt (/ A V))))
    (if (<= (* V l) -1e-52)
      (* c0_m (sqrt (/ A (* V l))))
      (if (<= (* V l) 0.0)
        (/ c0_m (* (sqrt l) (sqrt (/ V A))))
        (if (<= (* V l) 5e+307)
          (* c0_m (/ (sqrt A) (sqrt (* V l))))
          (/ c0_m (sqrt (/ l (/ A V))))))))))

C

c0_m = fabs(c0);
c0_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+227) {
		tmp = c0_m / (sqrt(l) / sqrt((A / V)));
	} else if ((V * l) <= -1e-52) {
		tmp = c0_m * sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0_m / (sqrt(l) * sqrt((V / A)));
	} else if ((V * l) <= 5e+307) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0_m / sqrt((l / (A / V)));
	}
	return c0_s * tmp;
}

Fortran

c0_m = abs(c0)
c0_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-1d+227)) then
        tmp = c0_m / (sqrt(l) / sqrt((a / v)))
    else if ((v * l) <= (-1d-52)) then
        tmp = c0_m * sqrt((a / (v * l)))
    else if ((v * l) <= 0.0d0) then
        tmp = c0_m / (sqrt(l) * sqrt((v / a)))
    else if ((v * l) <= 5d+307) then
        tmp = c0_m * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0_m / sqrt((l / (a / v)))
    end if
    code = c0_s * tmp
end function

Java

c0_m = Math.abs(c0);
c0_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+227) {
		tmp = c0_m / (Math.sqrt(l) / Math.sqrt((A / V)));
	} else if ((V * l) <= -1e-52) {
		tmp = c0_m * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0_m / (Math.sqrt(l) * Math.sqrt((V / A)));
	} else if ((V * l) <= 5e+307) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0_m / Math.sqrt((l / (A / V)));
	}
	return c0_s * tmp;
}

Python

c0_m = math.fabs(c0)
c0_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	tmp = 0
	if (V * l) <= -1e+227:
		tmp = c0_m / (math.sqrt(l) / math.sqrt((A / V)))
	elif (V * l) <= -1e-52:
		tmp = c0_m * math.sqrt((A / (V * l)))
	elif (V * l) <= 0.0:
		tmp = c0_m / (math.sqrt(l) * math.sqrt((V / A)))
	elif (V * l) <= 5e+307:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0_m / math.sqrt((l / (A / V)))
	return c0_s * tmp

Julia

c0_m = abs(c0)
c0_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e+227)
		tmp = Float64(c0_m / Float64(sqrt(l) / sqrt(Float64(A / V))));
	elseif (Float64(V * l) <= -1e-52)
		tmp = Float64(c0_m * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0_m / Float64(sqrt(l) * sqrt(Float64(V / A))));
	elseif (Float64(V * l) <= 5e+307)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0_m / sqrt(Float64(l / Float64(A / V))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0_m = abs(c0);
c0_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e+227)
		tmp = c0_m / (sqrt(l) / sqrt((A / V)));
	elseif ((V * l) <= -1e-52)
		tmp = c0_m * sqrt((A / (V * l)));
	elseif ((V * l) <= 0.0)
		tmp = c0_m / (sqrt(l) * sqrt((V / A)));
	elseif ((V * l) <= 5e+307)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0_m / sqrt((l / (A / V)));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0_m = N[Abs[c0], $MachinePrecision]
c0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], -1e+227], N[(c0$95$m / N[(N[Sqrt[l], $MachinePrecision] / N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-52], N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0$95$m / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+307], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0$95$m / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -1.0000000000000001e227

   1. Initial program 57.7%

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

      1. associate-/r* 57.7%

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

      2. sqrt-div 44.4%

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

      3. associate-*r/ 44.7%

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

   4. Applied egg-rr 44.7%

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

      1. associate-/l* 44.5%

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

   6. Simplified 44.5%

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

   if -1.0000000000000001e227 < (*.f64 V l) < -1e-52

   1. Initial program 85.3%

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

   if -1e-52 < (*.f64 V l) < 0.0

   1. Initial program 69.6%

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

      1. associate-/r* 76.3%

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

      2. clear-num 76.3%

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

      3. sqrt-div 77.5%

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

      4. metadata-eval 77.5%

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

      5. div-inv 77.5%

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

      6. clear-num 77.5%

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

   4. Applied egg-rr 77.5%

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

      1. un-div-inv 77.5%

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

      2. sqrt-prod 44.8%

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

      3. associate-/r* 44.8%

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

   6. Applied egg-rr 44.8%

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

      1. associate-/r* 44.8%

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

   8. Simplified 44.8%

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

   if 0.0 < (*.f64 V l) < 5e307

   1. Initial program 82.2%

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

      1. sqrt-div 98.9%

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

      2. associate-*r/ 97.1%

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

   4. Applied egg-rr 97.1%

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

      1. *-commutative 97.1%

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

      2. associate-/l* 94.3%

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

      3. associate-/r/ 98.9%

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

   6. Simplified 98.9%

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

   if 5e307 < (*.f64 V l)

   1. Initial program 31.9%

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

      1. associate-/r* 79.1%

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

      2. clear-num 79.3%

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

      3. sqrt-div 79.3%

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

      4. metadata-eval 79.3%

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

      5. div-inv 79.1%

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

      6. clear-num 79.1%

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

   4. Applied egg-rr 79.1%

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

      1. associate-*r/ 79.2%

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

      2. sqrt-prod 37.2%

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

      3. times-frac 37.1%

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

      4. metadata-eval 37.1%

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

      5. sqrt-div 37.2%

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

      6. clear-num 37.2%

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

      7. associate-/r/ 37.1%

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

      8. expm1-log1p-u 28.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)\right)} \]

      9. expm1-udef 13.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)} - 1} \]

      10. sqrt-undiv 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\frac{\ell}{\frac{A}{V}}}}}\right)} - 1 \]

      11. associate-/l* 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{\ell \cdot V}{A}}}}\right)} - 1 \]

      12. *-commutative 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell}}{A}}}\right)} - 1 \]

      13. *-un-lft-identity 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}}}\right)} - 1 \]

      14. times-frac 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{V}{1} \cdot \frac{\ell}{A}}}}\right)} - 1 \]

      15. /-rgt-identity 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{V} \cdot \frac{\ell}{A}}}\right)} - 1 \]

   6. Applied egg-rr 35.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 66.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)\right)} \]

      2. expm1-log1p 79.2%

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

   8. Simplified 79.2%

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

      1. associate-*r/ 31.9%

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

      2. *-commutative 31.9%

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

      3. associate-/l* 79.3%

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

   10. Applied egg-rr 79.3%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+227}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-52}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+307}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0_m = \left|c0\right| \\ c0_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+227}:\\ \;\;\;\;\frac{c0\_m}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-52}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{c0\_m}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \end{array} \]

FPCore

c0_m = (fabs.f64 c0)
c0_s = (copysign.f64 1 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+227)
    (/ c0_m (/ (sqrt l) (sqrt (/ A V))))
    (if (<= (* V l) -1e-52)
      (* c0_m (sqrt (/ A (* V l))))
      (if (<= (* V l) 0.0)
        (/ c0_m (* (sqrt l) (sqrt (/ V A))))
        (if (<= (* V l) 5e+307)
          (/ c0_m (/ (sqrt (* V l)) (sqrt A)))
          (/ c0_m (sqrt (/ l (/ A V))))))))))

C

c0_m = fabs(c0);
c0_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+227) {
		tmp = c0_m / (sqrt(l) / sqrt((A / V)));
	} else if ((V * l) <= -1e-52) {
		tmp = c0_m * sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0_m / (sqrt(l) * sqrt((V / A)));
	} else if ((V * l) <= 5e+307) {
		tmp = c0_m / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = c0_m / sqrt((l / (A / V)));
	}
	return c0_s * tmp;
}

Fortran

c0_m = abs(c0)
c0_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-1d+227)) then
        tmp = c0_m / (sqrt(l) / sqrt((a / v)))
    else if ((v * l) <= (-1d-52)) then
        tmp = c0_m * sqrt((a / (v * l)))
    else if ((v * l) <= 0.0d0) then
        tmp = c0_m / (sqrt(l) * sqrt((v / a)))
    else if ((v * l) <= 5d+307) then
        tmp = c0_m / (sqrt((v * l)) / sqrt(a))
    else
        tmp = c0_m / sqrt((l / (a / v)))
    end if
    code = c0_s * tmp
end function

Java

c0_m = Math.abs(c0);
c0_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+227) {
		tmp = c0_m / (Math.sqrt(l) / Math.sqrt((A / V)));
	} else if ((V * l) <= -1e-52) {
		tmp = c0_m * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0_m / (Math.sqrt(l) * Math.sqrt((V / A)));
	} else if ((V * l) <= 5e+307) {
		tmp = c0_m / (Math.sqrt((V * l)) / Math.sqrt(A));
	} else {
		tmp = c0_m / Math.sqrt((l / (A / V)));
	}
	return c0_s * tmp;
}

Python

c0_m = math.fabs(c0)
c0_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	tmp = 0
	if (V * l) <= -1e+227:
		tmp = c0_m / (math.sqrt(l) / math.sqrt((A / V)))
	elif (V * l) <= -1e-52:
		tmp = c0_m * math.sqrt((A / (V * l)))
	elif (V * l) <= 0.0:
		tmp = c0_m / (math.sqrt(l) * math.sqrt((V / A)))
	elif (V * l) <= 5e+307:
		tmp = c0_m / (math.sqrt((V * l)) / math.sqrt(A))
	else:
		tmp = c0_m / math.sqrt((l / (A / V)))
	return c0_s * tmp

Julia

c0_m = abs(c0)
c0_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e+227)
		tmp = Float64(c0_m / Float64(sqrt(l) / sqrt(Float64(A / V))));
	elseif (Float64(V * l) <= -1e-52)
		tmp = Float64(c0_m * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0_m / Float64(sqrt(l) * sqrt(Float64(V / A))));
	elseif (Float64(V * l) <= 5e+307)
		tmp = Float64(c0_m / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	else
		tmp = Float64(c0_m / sqrt(Float64(l / Float64(A / V))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0_m = abs(c0);
c0_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e+227)
		tmp = c0_m / (sqrt(l) / sqrt((A / V)));
	elseif ((V * l) <= -1e-52)
		tmp = c0_m * sqrt((A / (V * l)));
	elseif ((V * l) <= 0.0)
		tmp = c0_m / (sqrt(l) * sqrt((V / A)));
	elseif ((V * l) <= 5e+307)
		tmp = c0_m / (sqrt((V * l)) / sqrt(A));
	else
		tmp = c0_m / sqrt((l / (A / V)));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0_m = N[Abs[c0], $MachinePrecision]
c0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], -1e+227], N[(c0$95$m / N[(N[Sqrt[l], $MachinePrecision] / N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-52], N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0$95$m / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+307], N[(c0$95$m / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0$95$m / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -1.0000000000000001e227

   1. Initial program 57.7%

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

      1. associate-/r* 57.7%

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

      2. sqrt-div 44.4%

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

      3. associate-*r/ 44.7%

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

   4. Applied egg-rr 44.7%

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

      1. associate-/l* 44.5%

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

   6. Simplified 44.5%

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

   if -1.0000000000000001e227 < (*.f64 V l) < -1e-52

   1. Initial program 85.3%

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

   if -1e-52 < (*.f64 V l) < 0.0

   1. Initial program 69.6%

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

      1. associate-/r* 76.3%

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

      2. clear-num 76.3%

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

      3. sqrt-div 77.5%

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

      4. metadata-eval 77.5%

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

      5. div-inv 77.5%

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

      6. clear-num 77.5%

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

   4. Applied egg-rr 77.5%

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

      1. un-div-inv 77.5%

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

      2. sqrt-prod 44.8%

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

      3. associate-/r* 44.8%

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

   6. Applied egg-rr 44.8%

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

      1. associate-/r* 44.8%

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

   8. Simplified 44.8%

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

   if 0.0 < (*.f64 V l) < 5e307

   1. Initial program 82.2%

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

      1. associate-/r* 77.5%

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

      2. clear-num 77.3%

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

      3. sqrt-div 77.8%

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

      4. metadata-eval 77.8%

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

      5. div-inv 77.8%

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

      6. clear-num 77.8%

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

   4. Applied egg-rr 77.8%

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

      1. associate-*r/ 77.8%

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

      2. sqrt-prod 39.3%

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

      3. times-frac 37.4%

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

      4. metadata-eval 37.4%

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

      5. sqrt-div 37.4%

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

      6. clear-num 37.4%

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

      7. sqrt-div 44.3%

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

      8. frac-times 45.1%

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

      9. sqrt-prod 97.1%

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

   6. Applied egg-rr 97.1%

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

      1. associate-/l* 98.9%

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

      2. *-commutative 98.9%

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

   8. Simplified 98.9%

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

   if 5e307 < (*.f64 V l)

   1. Initial program 31.9%

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

      1. associate-/r* 79.1%

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

      2. clear-num 79.3%

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

      3. sqrt-div 79.3%

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

      4. metadata-eval 79.3%

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

      5. div-inv 79.1%

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

      6. clear-num 79.1%

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

   4. Applied egg-rr 79.1%

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

      1. associate-*r/ 79.2%

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

      2. sqrt-prod 37.2%

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

      3. times-frac 37.1%

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

      4. metadata-eval 37.1%

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

      5. sqrt-div 37.2%

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

      6. clear-num 37.2%

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

      7. associate-/r/ 37.1%

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

      8. expm1-log1p-u 28.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)\right)} \]

      9. expm1-udef 13.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)} - 1} \]

      10. sqrt-undiv 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\frac{\ell}{\frac{A}{V}}}}}\right)} - 1 \]

      11. associate-/l* 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{\ell \cdot V}{A}}}}\right)} - 1 \]

      12. *-commutative 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell}}{A}}}\right)} - 1 \]

      13. *-un-lft-identity 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}}}\right)} - 1 \]

      14. times-frac 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{V}{1} \cdot \frac{\ell}{A}}}}\right)} - 1 \]

      15. /-rgt-identity 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{V} \cdot \frac{\ell}{A}}}\right)} - 1 \]

   6. Applied egg-rr 35.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 66.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)\right)} \]

      2. expm1-log1p 79.2%

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

   8. Simplified 79.2%

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

      1. associate-*r/ 31.9%

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

      2. *-commutative 31.9%

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

      3. associate-/l* 79.3%

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

   10. Applied egg-rr 79.3%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+227}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-52}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0_m = \left|c0\right| \\ c0_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+227}:\\ \;\;\;\;\frac{c0\_m \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-52}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{c0\_m}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \end{array} \]

FPCore

c0_m = (fabs.f64 c0)
c0_s = (copysign.f64 1 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+227)
    (/ (* c0_m (sqrt (/ A V))) (sqrt l))
    (if (<= (* V l) -1e-52)
      (* c0_m (sqrt (/ A (* V l))))
      (if (<= (* V l) 0.0)
        (/ c0_m (* (sqrt l) (sqrt (/ V A))))
        (if (<= (* V l) 5e+307)
          (/ c0_m (/ (sqrt (* V l)) (sqrt A)))
          (/ c0_m (sqrt (/ l (/ A V))))))))))

C

c0_m = fabs(c0);
c0_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+227) {
		tmp = (c0_m * sqrt((A / V))) / sqrt(l);
	} else if ((V * l) <= -1e-52) {
		tmp = c0_m * sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0_m / (sqrt(l) * sqrt((V / A)));
	} else if ((V * l) <= 5e+307) {
		tmp = c0_m / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = c0_m / sqrt((l / (A / V)));
	}
	return c0_s * tmp;
}

Fortran

c0_m = abs(c0)
c0_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-1d+227)) then
        tmp = (c0_m * sqrt((a / v))) / sqrt(l)
    else if ((v * l) <= (-1d-52)) then
        tmp = c0_m * sqrt((a / (v * l)))
    else if ((v * l) <= 0.0d0) then
        tmp = c0_m / (sqrt(l) * sqrt((v / a)))
    else if ((v * l) <= 5d+307) then
        tmp = c0_m / (sqrt((v * l)) / sqrt(a))
    else
        tmp = c0_m / sqrt((l / (a / v)))
    end if
    code = c0_s * tmp
end function

Java

c0_m = Math.abs(c0);
c0_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+227) {
		tmp = (c0_m * Math.sqrt((A / V))) / Math.sqrt(l);
	} else if ((V * l) <= -1e-52) {
		tmp = c0_m * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0_m / (Math.sqrt(l) * Math.sqrt((V / A)));
	} else if ((V * l) <= 5e+307) {
		tmp = c0_m / (Math.sqrt((V * l)) / Math.sqrt(A));
	} else {
		tmp = c0_m / Math.sqrt((l / (A / V)));
	}
	return c0_s * tmp;
}

Python

c0_m = math.fabs(c0)
c0_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	tmp = 0
	if (V * l) <= -1e+227:
		tmp = (c0_m * math.sqrt((A / V))) / math.sqrt(l)
	elif (V * l) <= -1e-52:
		tmp = c0_m * math.sqrt((A / (V * l)))
	elif (V * l) <= 0.0:
		tmp = c0_m / (math.sqrt(l) * math.sqrt((V / A)))
	elif (V * l) <= 5e+307:
		tmp = c0_m / (math.sqrt((V * l)) / math.sqrt(A))
	else:
		tmp = c0_m / math.sqrt((l / (A / V)))
	return c0_s * tmp

Julia

c0_m = abs(c0)
c0_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e+227)
		tmp = Float64(Float64(c0_m * sqrt(Float64(A / V))) / sqrt(l));
	elseif (Float64(V * l) <= -1e-52)
		tmp = Float64(c0_m * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0_m / Float64(sqrt(l) * sqrt(Float64(V / A))));
	elseif (Float64(V * l) <= 5e+307)
		tmp = Float64(c0_m / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	else
		tmp = Float64(c0_m / sqrt(Float64(l / Float64(A / V))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0_m = abs(c0);
c0_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e+227)
		tmp = (c0_m * sqrt((A / V))) / sqrt(l);
	elseif ((V * l) <= -1e-52)
		tmp = c0_m * sqrt((A / (V * l)));
	elseif ((V * l) <= 0.0)
		tmp = c0_m / (sqrt(l) * sqrt((V / A)));
	elseif ((V * l) <= 5e+307)
		tmp = c0_m / (sqrt((V * l)) / sqrt(A));
	else
		tmp = c0_m / sqrt((l / (A / V)));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0_m = N[Abs[c0], $MachinePrecision]
c0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], -1e+227], N[(N[(c0$95$m * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-52], N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0$95$m / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+307], N[(c0$95$m / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0$95$m / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -1.0000000000000001e227

   1. Initial program 57.7%

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

      1. *-commutative 57.7%

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

      2. associate-/r* 57.7%

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

      3. sqrt-div 44.4%

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

      4. associate-*l/ 44.7%

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

   4. Applied egg-rr 44.7%

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

   if -1.0000000000000001e227 < (*.f64 V l) < -1e-52

   1. Initial program 85.3%

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

   if -1e-52 < (*.f64 V l) < 0.0

   1. Initial program 69.6%

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

      1. associate-/r* 76.3%

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

      2. clear-num 76.3%

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

      3. sqrt-div 77.5%

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

      4. metadata-eval 77.5%

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

      5. div-inv 77.5%

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

      6. clear-num 77.5%

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

   4. Applied egg-rr 77.5%

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

      1. un-div-inv 77.5%

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

      2. sqrt-prod 44.8%

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

      3. associate-/r* 44.8%

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

   6. Applied egg-rr 44.8%

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

      1. associate-/r* 44.8%

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

   8. Simplified 44.8%

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

   if 0.0 < (*.f64 V l) < 5e307

   1. Initial program 82.2%

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

      1. associate-/r* 77.5%

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

      2. clear-num 77.3%

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

      3. sqrt-div 77.8%

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

      4. metadata-eval 77.8%

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

      5. div-inv 77.8%

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

      6. clear-num 77.8%

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

   4. Applied egg-rr 77.8%

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

      1. associate-*r/ 77.8%

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

      2. sqrt-prod 39.3%

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

      3. times-frac 37.4%

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

      4. metadata-eval 37.4%

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

      5. sqrt-div 37.4%

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

      6. clear-num 37.4%

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

      7. sqrt-div 44.3%

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

      8. frac-times 45.1%

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

      9. sqrt-prod 97.1%

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

   6. Applied egg-rr 97.1%

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

      1. associate-/l* 98.9%

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

      2. *-commutative 98.9%

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

   8. Simplified 98.9%

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

   if 5e307 < (*.f64 V l)

   1. Initial program 31.9%

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

      1. associate-/r* 79.1%

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

      2. clear-num 79.3%

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

      3. sqrt-div 79.3%

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

      4. metadata-eval 79.3%

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

      5. div-inv 79.1%

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

      6. clear-num 79.1%

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

   4. Applied egg-rr 79.1%

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

      1. associate-*r/ 79.2%

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

      2. sqrt-prod 37.2%

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

      3. times-frac 37.1%

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

      4. metadata-eval 37.1%

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

      5. sqrt-div 37.2%

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

      6. clear-num 37.2%

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

      7. associate-/r/ 37.1%

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

      8. expm1-log1p-u 28.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)\right)} \]

      9. expm1-udef 13.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)} - 1} \]

      10. sqrt-undiv 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\frac{\ell}{\frac{A}{V}}}}}\right)} - 1 \]

      11. associate-/l* 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{\ell \cdot V}{A}}}}\right)} - 1 \]

      12. *-commutative 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell}}{A}}}\right)} - 1 \]

      13. *-un-lft-identity 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}}}\right)} - 1 \]

      14. times-frac 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{V}{1} \cdot \frac{\ell}{A}}}}\right)} - 1 \]

      15. /-rgt-identity 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{V} \cdot \frac{\ell}{A}}}\right)} - 1 \]

   6. Applied egg-rr 35.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 66.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)\right)} \]

      2. expm1-log1p 79.2%

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

   8. Simplified 79.2%

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

      1. associate-*r/ 31.9%

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

      2. *-commutative 31.9%

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

      3. associate-/l* 79.3%

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

   10. Applied egg-rr 79.3%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+227}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-52}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0_m = \left|c0\right| \\ c0_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{-311}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\sqrt{\frac{c0\_m}{\ell} \cdot \frac{\frac{c0\_m}{V}}{\frac{1}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+307}:\\ \;\;\;\;c0\_m \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \end{array} \]

FPCore

c0_m = (fabs.f64 c0)
c0_s = (copysign.f64 1 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) -5e-311)
    (* c0_m (/ (sqrt (- A)) (sqrt (* V (- l)))))
    (if (<= (* V l) 0.0)
      (sqrt (* (/ c0_m l) (/ (/ c0_m V) (/ 1.0 A))))
      (if (<= (* V l) 5e+307)
        (* c0_m (* (sqrt A) (pow (* V l) -0.5)))
        (/ c0_m (sqrt (/ l (/ A V)))))))))

C

c0_m = fabs(c0);
c0_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e-311) {
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = sqrt(((c0_m / l) * ((c0_m / V) / (1.0 / A))));
	} else if ((V * l) <= 5e+307) {
		tmp = c0_m * (sqrt(A) * pow((V * l), -0.5));
	} else {
		tmp = c0_m / sqrt((l / (A / V)));
	}
	return c0_s * tmp;
}

Fortran

c0_m = abs(c0)
c0_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-5d-311)) then
        tmp = c0_m * (sqrt(-a) / sqrt((v * -l)))
    else if ((v * l) <= 0.0d0) then
        tmp = sqrt(((c0_m / l) * ((c0_m / v) / (1.0d0 / a))))
    else if ((v * l) <= 5d+307) then
        tmp = c0_m * (sqrt(a) * ((v * l) ** (-0.5d0)))
    else
        tmp = c0_m / sqrt((l / (a / v)))
    end if
    code = c0_s * tmp
end function

Java

c0_m = Math.abs(c0);
c0_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e-311) {
		tmp = c0_m * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = Math.sqrt(((c0_m / l) * ((c0_m / V) / (1.0 / A))));
	} else if ((V * l) <= 5e+307) {
		tmp = c0_m * (Math.sqrt(A) * Math.pow((V * l), -0.5));
	} else {
		tmp = c0_m / Math.sqrt((l / (A / V)));
	}
	return c0_s * tmp;
}

Python

c0_m = math.fabs(c0)
c0_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	tmp = 0
	if (V * l) <= -5e-311:
		tmp = c0_m * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = math.sqrt(((c0_m / l) * ((c0_m / V) / (1.0 / A))))
	elif (V * l) <= 5e+307:
		tmp = c0_m * (math.sqrt(A) * math.pow((V * l), -0.5))
	else:
		tmp = c0_m / math.sqrt((l / (A / V)))
	return c0_s * tmp

Julia

c0_m = abs(c0)
c0_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -5e-311)
		tmp = Float64(c0_m * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = sqrt(Float64(Float64(c0_m / l) * Float64(Float64(c0_m / V) / Float64(1.0 / A))));
	elseif (Float64(V * l) <= 5e+307)
		tmp = Float64(c0_m * Float64(sqrt(A) * (Float64(V * l) ^ -0.5)));
	else
		tmp = Float64(c0_m / sqrt(Float64(l / Float64(A / V))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0_m = abs(c0);
c0_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -5e-311)
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = sqrt(((c0_m / l) * ((c0_m / V) / (1.0 / A))));
	elseif ((V * l) <= 5e+307)
		tmp = c0_m * (sqrt(A) * ((V * l) ^ -0.5));
	else
		tmp = c0_m / sqrt((l / (A / V)));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0_m = N[Abs[c0], $MachinePrecision]
c0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], -5e-311], N[(c0$95$m * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Sqrt[N[(N[(c0$95$m / l), $MachinePrecision] * N[(N[(c0$95$m / V), $MachinePrecision] / N[(1.0 / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+307], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] * N[Power[N[(V * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0$95$m / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -5.00000000000023e-311

   1. Initial program 79.9%

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

      1. frac-2neg 79.9%

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

      2. sqrt-div 93.1%

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

      3. distribute-rgt-neg-in 93.1%

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

   4. Applied egg-rr 93.1%

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

      1. distribute-rgt-neg-out 93.1%

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

      2. *-commutative 93.1%

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

      3. distribute-rgt-neg-in 93.1%

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

   6. Simplified 93.1%

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

   if -5.00000000000023e-311 < (*.f64 V l) < 0.0

   1. Initial program 49.2%

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

      1. associate-/r* 65.8%

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

      2. clear-num 65.8%

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

      3. sqrt-div 68.7%

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

      4. metadata-eval 68.7%

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

      5. div-inv 68.7%

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

      6. clear-num 68.7%

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

   4. Applied egg-rr 68.7%

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

      1. *-commutative 68.7%

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

      2. associate-*r/ 49.2%

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

      3. *-commutative 49.2%

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

      4. sqrt-div 23.1%

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

      5. clear-num 23.1%

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

      6. sqrt-div 49.2%

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

      7. add-sqr-sqrt 23.3%

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

      8. pow1/2 23.3%

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

      9. pow1/2 23.4%

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

      10. pow-prod-down 23.4%

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

   6. Applied egg-rr 25.0%

      \[\leadsto \color{blue}{{\left(\frac{{c0}^{2}}{V \cdot \frac{\ell}{A}}\right)}^{0.5}} \]
   7. Step-by-step derivation

      1. unpow1/2 25.0%

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

      2. associate-/r* 26.9%

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

   8. Simplified 26.9%

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

      1. div-inv 26.9%

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

      2. unpow2 26.9%

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

      3. associate-*l* 30.3%

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

   10. Applied egg-rr 30.3%

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

       1. div-inv 30.3%

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

       2. times-frac 33.8%

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

       3. un-div-inv 33.8%

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

   12. Applied egg-rr 33.8%

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

   if 0.0 < (*.f64 V l) < 5e307

   1. Initial program 82.2%

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

      1. associate-/r* 77.5%

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

      2. clear-num 77.3%

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

      3. sqrt-div 77.8%

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

      4. metadata-eval 77.8%

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

      5. div-inv 77.8%

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

      6. clear-num 77.8%

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

   4. Applied egg-rr 77.8%

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

      1. associate-*r/ 77.8%

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

      2. sqrt-prod 39.3%

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

      3. times-frac 37.4%

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

      4. metadata-eval 37.4%

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

      5. sqrt-div 37.4%

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

      6. clear-num 37.4%

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

      7. associate-/r/ 39.3%

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

      8. expm1-log1p-u 25.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)\right)} \]

      9. expm1-udef 13.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)} - 1} \]

      10. sqrt-undiv 33.4%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\frac{\ell}{\frac{A}{V}}}}}\right)} - 1 \]

      11. associate-/l* 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{\ell \cdot V}{A}}}}\right)} - 1 \]

      12. *-commutative 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell}}{A}}}\right)} - 1 \]

      13. *-un-lft-identity 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}}}\right)} - 1 \]

      14. times-frac 34.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{V}{1} \cdot \frac{\ell}{A}}}}\right)} - 1 \]

      15. /-rgt-identity 34.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{V} \cdot \frac{\ell}{A}}}\right)} - 1 \]

   6. Applied egg-rr 34.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 58.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)\right)} \]

      2. expm1-log1p 75.3%

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

   8. Simplified 75.3%

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

      1. clear-num 74.3%

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

      2. un-div-inv 75.9%

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

   10. Applied egg-rr 75.9%

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

       1. associate-/l* 82.6%

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

       2. sqrt-div 98.9%

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

       3. associate-/l* 97.1%

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

       4. *-commutative 97.1%

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

       5. div-inv 97.1%

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

       6. *-commutative 97.1%

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

       7. associate-*l* 98.9%

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

       8. pow1/2 98.9%

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

       9. pow-flip 99.0%

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

       10. metadata-eval 99.0%

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

   12. Applied egg-rr 99.0%

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

   if 5e307 < (*.f64 V l)

   1. Initial program 31.9%

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

      1. associate-/r* 79.1%

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

      2. clear-num 79.3%

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

      3. sqrt-div 79.3%

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

      4. metadata-eval 79.3%

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

      5. div-inv 79.1%

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

      6. clear-num 79.1%

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

   4. Applied egg-rr 79.1%

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

      1. associate-*r/ 79.2%

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

      2. sqrt-prod 37.2%

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

      3. times-frac 37.1%

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

      4. metadata-eval 37.1%

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

      5. sqrt-div 37.2%

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

      6. clear-num 37.2%

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

      7. associate-/r/ 37.1%

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

      8. expm1-log1p-u 28.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)\right)} \]

      9. expm1-udef 13.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)} - 1} \]

      10. sqrt-undiv 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\frac{\ell}{\frac{A}{V}}}}}\right)} - 1 \]

      11. associate-/l* 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{\ell \cdot V}{A}}}}\right)} - 1 \]

      12. *-commutative 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell}}{A}}}\right)} - 1 \]

      13. *-un-lft-identity 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}}}\right)} - 1 \]

      14. times-frac 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{V}{1} \cdot \frac{\ell}{A}}}}\right)} - 1 \]

      15. /-rgt-identity 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{V} \cdot \frac{\ell}{A}}}\right)} - 1 \]

   6. Applied egg-rr 35.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 66.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)\right)} \]

      2. expm1-log1p 79.2%

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

   8. Simplified 79.2%

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

      1. associate-*r/ 31.9%

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

      2. *-commutative 31.9%

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

      3. associate-/l* 79.3%

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

   10. Applied egg-rr 79.3%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{-311}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\sqrt{\frac{c0}{\ell} \cdot \frac{\frac{c0}{V}}{\frac{1}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+307}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot {\left(V \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0_m = \left|c0\right| \\ c0_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{-260}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\sqrt{\frac{c0\_m}{\ell} \cdot \frac{\frac{c0\_m}{V}}{\frac{1}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+307}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \end{array} \]

FPCore

c0_m = (fabs.f64 c0)
c0_s = (copysign.f64 1 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-260)
    (* c0_m (sqrt (/ A (* V l))))
    (if (<= (* V l) 0.0)
      (sqrt (* (/ c0_m l) (/ (/ c0_m V) (/ 1.0 A))))
      (if (<= (* V l) 5e+307)
        (* c0_m (/ (sqrt A) (sqrt (* V l))))
        (/ c0_m (sqrt (/ l (/ A V)))))))))

C

c0_m = fabs(c0);
c0_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e-260) {
		tmp = c0_m * sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = sqrt(((c0_m / l) * ((c0_m / V) / (1.0 / A))));
	} else if ((V * l) <= 5e+307) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0_m / sqrt((l / (A / V)));
	}
	return c0_s * tmp;
}

Fortran

c0_m = abs(c0)
c0_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-1d-260)) then
        tmp = c0_m * sqrt((a / (v * l)))
    else if ((v * l) <= 0.0d0) then
        tmp = sqrt(((c0_m / l) * ((c0_m / v) / (1.0d0 / a))))
    else if ((v * l) <= 5d+307) then
        tmp = c0_m * (sqrt(a) / sqrt((v * l)))
    else
        tmp = c0_m / sqrt((l / (a / v)))
    end if
    code = c0_s * tmp
end function

Java

c0_m = Math.abs(c0);
c0_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e-260) {
		tmp = c0_m * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 0.0) {
		tmp = Math.sqrt(((c0_m / l) * ((c0_m / V) / (1.0 / A))));
	} else if ((V * l) <= 5e+307) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0_m / Math.sqrt((l / (A / V)));
	}
	return c0_s * tmp;
}

Python

c0_m = math.fabs(c0)
c0_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	tmp = 0
	if (V * l) <= -1e-260:
		tmp = c0_m * math.sqrt((A / (V * l)))
	elif (V * l) <= 0.0:
		tmp = math.sqrt(((c0_m / l) * ((c0_m / V) / (1.0 / A))))
	elif (V * l) <= 5e+307:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = c0_m / math.sqrt((l / (A / V)))
	return c0_s * tmp

Julia

c0_m = abs(c0)
c0_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e-260)
		tmp = Float64(c0_m * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 0.0)
		tmp = sqrt(Float64(Float64(c0_m / l) * Float64(Float64(c0_m / V) / Float64(1.0 / A))));
	elseif (Float64(V * l) <= 5e+307)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0_m / sqrt(Float64(l / Float64(A / V))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0_m = abs(c0);
c0_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e-260)
		tmp = c0_m * sqrt((A / (V * l)));
	elseif ((V * l) <= 0.0)
		tmp = sqrt(((c0_m / l) * ((c0_m / V) / (1.0 / A))));
	elseif ((V * l) <= 5e+307)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	else
		tmp = c0_m / sqrt((l / (A / V)));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0_m = N[Abs[c0], $MachinePrecision]
c0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], -1e-260], N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Sqrt[N[(N[(c0$95$m / l), $MachinePrecision] * N[(N[(c0$95$m / V), $MachinePrecision] / N[(1.0 / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 5e+307], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0$95$m / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -9.99999999999999961e-261

   1. Initial program 81.5%

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

   if -9.99999999999999961e-261 < (*.f64 V l) < 0.0

   1. Initial program 49.8%

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

      1. associate-/r* 63.5%

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

      2. clear-num 63.4%

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

      3. sqrt-div 65.9%

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

      4. metadata-eval 65.9%

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

      5. div-inv 65.9%

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

      6. clear-num 65.9%

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

   4. Applied egg-rr 65.9%

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

      1. *-commutative 65.9%

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

      2. associate-*r/ 49.8%

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

      3. *-commutative 49.8%

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

      4. sqrt-div 19.0%

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

      5. clear-num 19.0%

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

      6. sqrt-div 49.8%

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

      7. add-sqr-sqrt 19.2%

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

      8. pow1/2 19.2%

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

      9. pow1/2 19.3%

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

      10. pow-prod-down 19.4%

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

   6. Applied egg-rr 20.7%

      \[\leadsto \color{blue}{{\left(\frac{{c0}^{2}}{V \cdot \frac{\ell}{A}}\right)}^{0.5}} \]
   7. Step-by-step derivation

      1. unpow1/2 20.7%

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

      2. associate-/r* 22.3%

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

   8. Simplified 22.3%

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

      1. div-inv 22.2%

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

      2. unpow2 22.2%

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

      3. associate-*l* 25.0%

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

   10. Applied egg-rr 25.0%

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

       1. div-inv 25.0%

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

       2. times-frac 27.9%

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

       3. un-div-inv 27.9%

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

   12. Applied egg-rr 27.9%

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

   if 0.0 < (*.f64 V l) < 5e307

   1. Initial program 82.2%

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

      1. sqrt-div 98.9%

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

      2. associate-*r/ 97.1%

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

   4. Applied egg-rr 97.1%

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

      1. *-commutative 97.1%

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

      2. associate-/l* 94.3%

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

      3. associate-/r/ 98.9%

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

   6. Simplified 98.9%

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

   if 5e307 < (*.f64 V l)

   1. Initial program 31.9%

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

      1. associate-/r* 79.1%

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

      2. clear-num 79.3%

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

      3. sqrt-div 79.3%

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

      4. metadata-eval 79.3%

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

      5. div-inv 79.1%

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

      6. clear-num 79.1%

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

   4. Applied egg-rr 79.1%

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

      1. associate-*r/ 79.2%

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

      2. sqrt-prod 37.2%

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

      3. times-frac 37.1%

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

      4. metadata-eval 37.1%

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

      5. sqrt-div 37.2%

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

      6. clear-num 37.2%

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

      7. associate-/r/ 37.1%

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

      8. expm1-log1p-u 28.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)\right)} \]

      9. expm1-udef 13.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)} - 1} \]

      10. sqrt-undiv 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\frac{\ell}{\frac{A}{V}}}}}\right)} - 1 \]

      11. associate-/l* 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{\ell \cdot V}{A}}}}\right)} - 1 \]

      12. *-commutative 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell}}{A}}}\right)} - 1 \]

      13. *-un-lft-identity 31.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}}}\right)} - 1 \]

      14. times-frac 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{V}{1} \cdot \frac{\ell}{A}}}}\right)} - 1 \]

      15. /-rgt-identity 35.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{V} \cdot \frac{\ell}{A}}}\right)} - 1 \]

   6. Applied egg-rr 35.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 66.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)\right)} \]

      2. expm1-log1p 79.2%

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

   8. Simplified 79.2%

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

      1. associate-*r/ 31.9%

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

      2. *-commutative 31.9%

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

      3. associate-/l* 79.3%

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

   10. Applied egg-rr 79.3%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{-260}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\sqrt{\frac{c0}{\ell} \cdot \frac{\frac{c0}{V}}{\frac{1}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+307}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0_m = \left|c0\right| \\ c0_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{+54}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

c0_m = (fabs.f64 c0)
c0_s = (copysign.f64 1 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 (<= (* c0_m (sqrt (/ A (* V l)))) 1e+54)
    (* c0_m (sqrt (/ (/ A V) l)))
    (/ 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 tmp;
	if ((c0_m * sqrt((A / (V * l)))) <= 1e+54) {
		tmp = c0_m * sqrt(((A / V) / l));
	} 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) :: tmp
    if ((c0_m * sqrt((a / (v * l)))) <= 1d+54) then
        tmp = c0_m * sqrt(((a / v) / l))
    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 tmp;
	if ((c0_m * Math.sqrt((A / (V * l)))) <= 1e+54) {
		tmp = c0_m * Math.sqrt(((A / V) / l));
	} 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):
	tmp = 0
	if (c0_m * math.sqrt((A / (V * l)))) <= 1e+54:
		tmp = c0_m * math.sqrt(((A / V) / l))
	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)
	tmp = 0.0
	if (Float64(c0_m * sqrt(Float64(A / Float64(V * l)))) <= 1e+54)
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / V) / l)));
	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)
	tmp = 0.0;
	if ((c0_m * sqrt((A / (V * l)))) <= 1e+54)
		tmp = c0_m * sqrt(((A / V) / l));
	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_] := N[(c0$95$s * If[LessEqual[N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+54], N[(c0$95$m * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0$95$m / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.0000000000000001e54

   1. Initial program 72.7%

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

      1. *-commutative 72.7%

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

      2. associate-/l/ 76.5%

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

   3. Simplified 76.5%

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

   if 1.0000000000000001e54 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 73.4%

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

      1. associate-/r* 73.4%

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

      2. clear-num 73.4%

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

      3. sqrt-div 76.6%

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

      4. metadata-eval 76.6%

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

      5. div-inv 76.6%

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

      6. clear-num 76.6%

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

   4. Applied egg-rr 76.6%

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

      1. associate-*r/ 76.6%

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

      2. sqrt-prod 37.1%

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

      3. times-frac 37.1%

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

      4. metadata-eval 37.1%

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

      5. sqrt-div 37.1%

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

      6. clear-num 37.1%

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

      7. associate-/r/ 37.0%

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

      8. expm1-log1p-u 35.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)\right)} \]

      9. expm1-udef 35.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)} - 1} \]

      10. sqrt-undiv 72.4%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\frac{\ell}{\frac{A}{V}}}}}\right)} - 1 \]

      11. associate-/l* 71.4%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{\ell \cdot V}{A}}}}\right)} - 1 \]

      12. *-commutative 71.4%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell}}{A}}}\right)} - 1 \]

      13. *-un-lft-identity 71.4%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}}}\right)} - 1 \]

      14. times-frac 73.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{V}{1} \cdot \frac{\ell}{A}}}}\right)} - 1 \]

      15. /-rgt-identity 73.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{V} \cdot \frac{\ell}{A}}}\right)} - 1 \]

   6. Applied egg-rr 73.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 73.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)\right)} \]

      2. expm1-log1p 77.6%

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

   8. Simplified 77.6%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{+54}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 86.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} c0_m = \left|c0\right| \\ c0_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 4 \cdot 10^{+298}\right):\\ \;\;\;\;\sqrt{\frac{c0\_m}{\ell} \cdot \frac{\frac{c0\_m}{V}}{\frac{1}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \end{array} \end{array} \end{array} \]

FPCore

c0_m = (fabs.f64 c0)
c0_s = (copysign.f64 1 c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (*
    c0_s
    (if (or (<= t_0 0.0) (not (<= t_0 4e+298)))
      (sqrt (* (/ c0_m l) (/ (/ c0_m V) (/ 1.0 A))))
      (* c0_m (sqrt t_0))))))

C

c0_m = fabs(c0);
c0_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 4e+298)) {
		tmp = sqrt(((c0_m / l) * ((c0_m / V) / (1.0 / A))));
	} else {
		tmp = c0_m * sqrt(t_0);
	}
	return c0_s * tmp;
}

Fortran

c0_m = abs(c0)
c0_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 4d+298))) then
        tmp = sqrt(((c0_m / l) * ((c0_m / v) / (1.0d0 / a))))
    else
        tmp = c0_m * sqrt(t_0)
    end if
    code = c0_s * tmp
end function

Java

c0_m = Math.abs(c0);
c0_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 4e+298)) {
		tmp = Math.sqrt(((c0_m / l) * ((c0_m / V) / (1.0 / A))));
	} else {
		tmp = c0_m * Math.sqrt(t_0);
	}
	return c0_s * tmp;
}

Python

c0_m = math.fabs(c0)
c0_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 4e+298):
		tmp = math.sqrt(((c0_m / l) * ((c0_m / V) / (1.0 / A))))
	else:
		tmp = c0_m * math.sqrt(t_0)
	return c0_s * tmp

Julia

c0_m = abs(c0)
c0_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 4e+298))
		tmp = sqrt(Float64(Float64(c0_m / l) * Float64(Float64(c0_m / V) / Float64(1.0 / A))));
	else
		tmp = Float64(c0_m * sqrt(t_0));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0_m = abs(c0);
c0_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 4e+298)))
		tmp = sqrt(((c0_m / l) * ((c0_m / V) / (1.0 / A))));
	else
		tmp = c0_m * sqrt(t_0);
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0_m = N[Abs[c0], $MachinePrecision]
c0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 4e+298]], $MachinePrecision]], N[Sqrt[N[(N[(c0$95$m / l), $MachinePrecision] * N[(N[(c0$95$m / V), $MachinePrecision] / N[(1.0 / A), $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 3.9999999999999998e298 < (/.f64 A (*.f64 V l))

   1. Initial program 41.4%

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

      1. associate-/r* 55.2%

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

      2. clear-num 55.2%

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

      3. sqrt-div 57.1%

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

      4. metadata-eval 57.1%

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

      5. div-inv 57.0%

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

      6. clear-num 57.0%

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

   4. Applied egg-rr 57.0%

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

      1. *-commutative 57.0%

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

      2. associate-*r/ 42.5%

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

      3. *-commutative 42.5%

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

      4. sqrt-div 35.8%

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

      5. clear-num 35.8%

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

      6. sqrt-div 41.4%

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

      7. add-sqr-sqrt 32.0%

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

      8. pow1/2 32.0%

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

      9. pow1/2 32.1%

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

      10. pow-prod-down 32.1%

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

   6. Applied egg-rr 33.1%

      \[\leadsto \color{blue}{{\left(\frac{{c0}^{2}}{V \cdot \frac{\ell}{A}}\right)}^{0.5}} \]
   7. Step-by-step derivation

      1. unpow1/2 33.1%

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

      2. associate-/r* 35.4%

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

   8. Simplified 35.4%

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

      1. div-inv 35.4%

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

      2. unpow2 35.4%

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

      3. associate-*l* 39.6%

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

   10. Applied egg-rr 39.6%

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

       1. div-inv 39.5%

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

       2. times-frac 44.6%

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

       3. un-div-inv 44.6%

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

   12. Applied egg-rr 44.6%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 3.9999999999999998e298

   1. Initial program 99.0%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 4 \cdot 10^{+298}\right):\\ \;\;\;\;\sqrt{\frac{c0}{\ell} \cdot \frac{\frac{c0}{V}}{\frac{1}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 83.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} c0_m = \left|c0\right| \\ c0_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 4 \cdot 10^{+298}\right):\\ \;\;\;\;\sqrt{\frac{c0\_m}{V} \cdot \left(c0\_m \cdot \frac{A}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \end{array} \end{array} \end{array} \]

FPCore

c0_m = (fabs.f64 c0)
c0_s = (copysign.f64 1 c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (*
    c0_s
    (if (or (<= t_0 0.0) (not (<= t_0 4e+298)))
      (sqrt (* (/ c0_m V) (* c0_m (/ A l))))
      (* c0_m (sqrt t_0))))))

C

c0_m = fabs(c0);
c0_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 4e+298)) {
		tmp = sqrt(((c0_m / V) * (c0_m * (A / l))));
	} else {
		tmp = c0_m * sqrt(t_0);
	}
	return c0_s * tmp;
}

Fortran

c0_m = abs(c0)
c0_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 4d+298))) then
        tmp = sqrt(((c0_m / v) * (c0_m * (a / l))))
    else
        tmp = c0_m * sqrt(t_0)
    end if
    code = c0_s * tmp
end function

Java

c0_m = Math.abs(c0);
c0_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 4e+298)) {
		tmp = Math.sqrt(((c0_m / V) * (c0_m * (A / l))));
	} else {
		tmp = c0_m * Math.sqrt(t_0);
	}
	return c0_s * tmp;
}

Python

c0_m = math.fabs(c0)
c0_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 4e+298):
		tmp = math.sqrt(((c0_m / V) * (c0_m * (A / l))))
	else:
		tmp = c0_m * math.sqrt(t_0)
	return c0_s * tmp

Julia

c0_m = abs(c0)
c0_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 4e+298))
		tmp = sqrt(Float64(Float64(c0_m / V) * Float64(c0_m * Float64(A / l))));
	else
		tmp = Float64(c0_m * sqrt(t_0));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0_m = abs(c0);
c0_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 4e+298)))
		tmp = sqrt(((c0_m / V) * (c0_m * (A / l))));
	else
		tmp = c0_m * sqrt(t_0);
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0_m = N[Abs[c0], $MachinePrecision]
c0_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 4e+298]], $MachinePrecision]], N[Sqrt[N[(N[(c0$95$m / V), $MachinePrecision] * N[(c0$95$m * N[(A / l), $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 3.9999999999999998e298 < (/.f64 A (*.f64 V l))

   1. Initial program 41.4%

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

      1. associate-/r* 55.2%

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

      2. clear-num 55.2%

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

      3. sqrt-div 57.1%

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

      4. metadata-eval 57.1%

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

      5. div-inv 57.0%

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

      6. clear-num 57.0%

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

   4. Applied egg-rr 57.0%

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

      1. *-commutative 57.0%

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

      2. associate-*r/ 42.5%

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

      3. *-commutative 42.5%

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

      4. sqrt-div 35.8%

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

      5. clear-num 35.8%

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

      6. sqrt-div 41.4%

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

      7. add-sqr-sqrt 32.0%

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

      8. pow1/2 32.0%

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

      9. pow1/2 32.1%

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

      10. pow-prod-down 32.1%

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

   6. Applied egg-rr 33.1%

      \[\leadsto \color{blue}{{\left(\frac{{c0}^{2}}{V \cdot \frac{\ell}{A}}\right)}^{0.5}} \]
   7. Step-by-step derivation

      1. unpow1/2 33.1%

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

      2. associate-/r* 35.4%

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

   8. Simplified 35.4%

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

      1. div-inv 35.4%

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

      2. unpow2 35.4%

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

      3. associate-*l* 39.6%

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

   10. Applied egg-rr 39.6%

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

       1. div-inv 39.6%

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

       2. *-commutative 39.6%

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

       3. clear-num 39.6%

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

       4. associate-*l* 40.3%

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

       5. un-div-inv 40.3%

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

   12. Applied egg-rr 40.3%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 3.9999999999999998e298

   1. Initial program 99.0%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 4 \cdot 10^{+298}\right):\\ \;\;\;\;\sqrt{\frac{c0}{V} \cdot \left(c0 \cdot \frac{A}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} c0_m = \left|c0\right| \\ c0_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.8 \cdot 10^{+125}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \end{array} \end{array} \]

FPCore

c0_m = (fabs.f64 c0)
c0_s = (copysign.f64 1 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 (<= l 1.8e+125)
    (/ c0_m (sqrt (* l (/ V A))))
    (/ c0_m (sqrt (/ V (/ A 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 (l <= 1.8e+125) {
		tmp = c0_m / sqrt((l * (V / A)));
	} else {
		tmp = c0_m / sqrt((V / (A / 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) :: tmp
    if (l <= 1.8d+125) then
        tmp = c0_m / sqrt((l * (v / a)))
    else
        tmp = c0_m / sqrt((v / (a / 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 tmp;
	if (l <= 1.8e+125) {
		tmp = c0_m / Math.sqrt((l * (V / A)));
	} else {
		tmp = c0_m / Math.sqrt((V / (A / 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 l <= 1.8e+125:
		tmp = c0_m / math.sqrt((l * (V / A)))
	else:
		tmp = c0_m / math.sqrt((V / (A / 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 (l <= 1.8e+125)
		tmp = Float64(c0_m / sqrt(Float64(l * Float64(V / A))));
	else
		tmp = Float64(c0_m / sqrt(Float64(V / Float64(A / 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 (l <= 1.8e+125)
		tmp = c0_m / sqrt((l * (V / A)));
	else
		tmp = c0_m / sqrt((V / (A / 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[l, 1.8e+125], N[(c0$95$m / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0$95$m / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.8000000000000002e125

   1. Initial program 75.1%

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

      1. associate-/r* 79.2%

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

      2. clear-num 79.1%

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

      3. sqrt-div 80.1%

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

      4. metadata-eval 80.1%

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

      5. div-inv 79.8%

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

      6. clear-num 79.8%

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

   4. Applied egg-rr 79.8%

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

      1. associate-*r/ 79.8%

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

      2. sqrt-prod 35.2%

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

      3. times-frac 34.9%

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

      4. metadata-eval 34.9%

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

      5. sqrt-div 34.5%

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

      6. clear-num 34.5%

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

      7. associate-/r/ 34.8%

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

      8. expm1-log1p-u 20.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)\right)} \]

      9. expm1-udef 12.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)} - 1} \]

      10. sqrt-undiv 35.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\frac{\ell}{\frac{A}{V}}}}}\right)} - 1 \]

      11. associate-/l* 34.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{\ell \cdot V}{A}}}}\right)} - 1 \]

      12. *-commutative 34.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell}}{A}}}\right)} - 1 \]

      13. *-un-lft-identity 34.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}}}\right)} - 1 \]

      14. times-frac 35.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{V}{1} \cdot \frac{\ell}{A}}}}\right)} - 1 \]

      15. /-rgt-identity 35.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{V} \cdot \frac{\ell}{A}}}\right)} - 1 \]

   6. Applied egg-rr 35.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 54.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)\right)} \]

      2. expm1-log1p 75.2%

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

   8. Simplified 75.2%

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

   9. Taylor expanded in V around 0 75.6%

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

       1. associate-*l/ 79.8%

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

   11. Simplified 79.8%

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

   if 1.8000000000000002e125 < l

   1. Initial program 58.3%

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

      1. associate-/r* 52.2%

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

      2. clear-num 51.6%

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

      3. sqrt-div 51.6%

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

      4. metadata-eval 51.6%

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

      5. div-inv 51.5%

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

      6. clear-num 51.5%

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

   4. Applied egg-rr 51.5%

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

      1. associate-*r/ 51.5%

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

      2. sqrt-prod 79.3%

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

      3. times-frac 76.2%

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

      4. metadata-eval 76.2%

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

      5. sqrt-div 76.5%

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

      6. clear-num 76.4%

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

      7. associate-/r/ 79.2%

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

      8. expm1-log1p-u 66.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)\right)} \]

      9. expm1-udef 28.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)} - 1} \]

      10. sqrt-undiv 16.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\frac{\ell}{\frac{A}{V}}}}}\right)} - 1 \]

      11. associate-/l* 25.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{\ell \cdot V}{A}}}}\right)} - 1 \]

      12. *-commutative 25.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell}}{A}}}\right)} - 1 \]

      13. *-un-lft-identity 25.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}}}\right)} - 1 \]

      14. times-frac 25.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{V}{1} \cdot \frac{\ell}{A}}}}\right)} - 1 \]

      15. /-rgt-identity 25.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{V} \cdot \frac{\ell}{A}}}\right)} - 1 \]

   6. Applied egg-rr 25.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 47.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)\right)} \]

      2. expm1-log1p 63.2%

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

   8. Simplified 63.2%

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

      1. clear-num 63.2%

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

      2. un-div-inv 63.1%

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

   10. Applied egg-rr 63.1%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.8 \cdot 10^{+125}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} c0_m = \left|c0\right| \\ c0_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 3.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \end{array} \end{array} \]

FPCore

c0_m = (fabs.f64 c0)
c0_s = (copysign.f64 1 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 (<= l 3.5e+130)
    (/ c0_m (sqrt (/ l (/ A V))))
    (/ c0_m (sqrt (/ V (/ A 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 (l <= 3.5e+130) {
		tmp = c0_m / sqrt((l / (A / V)));
	} else {
		tmp = c0_m / sqrt((V / (A / 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) :: tmp
    if (l <= 3.5d+130) then
        tmp = c0_m / sqrt((l / (a / v)))
    else
        tmp = c0_m / sqrt((v / (a / 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 tmp;
	if (l <= 3.5e+130) {
		tmp = c0_m / Math.sqrt((l / (A / V)));
	} else {
		tmp = c0_m / Math.sqrt((V / (A / 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 l <= 3.5e+130:
		tmp = c0_m / math.sqrt((l / (A / V)))
	else:
		tmp = c0_m / math.sqrt((V / (A / 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 (l <= 3.5e+130)
		tmp = Float64(c0_m / sqrt(Float64(l / Float64(A / V))));
	else
		tmp = Float64(c0_m / sqrt(Float64(V / Float64(A / 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 (l <= 3.5e+130)
		tmp = c0_m / sqrt((l / (A / V)));
	else
		tmp = c0_m / sqrt((V / (A / 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[l, 3.5e+130], N[(c0$95$m / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0$95$m / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3.5000000000000001e130

   1. Initial program 75.1%

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

      1. associate-/r* 79.2%

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

      2. clear-num 79.1%

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

      3. sqrt-div 80.1%

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

      4. metadata-eval 80.1%

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

      5. div-inv 79.8%

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

      6. clear-num 79.8%

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

   4. Applied egg-rr 79.8%

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

      1. associate-*r/ 79.8%

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

      2. sqrt-prod 35.2%

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

      3. times-frac 34.9%

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

      4. metadata-eval 34.9%

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

      5. sqrt-div 34.5%

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

      6. clear-num 34.5%

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

      7. associate-/r/ 34.8%

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

      8. expm1-log1p-u 20.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)\right)} \]

      9. expm1-udef 12.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)} - 1} \]

      10. sqrt-undiv 35.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\frac{\ell}{\frac{A}{V}}}}}\right)} - 1 \]

      11. associate-/l* 34.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{\ell \cdot V}{A}}}}\right)} - 1 \]

      12. *-commutative 34.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell}}{A}}}\right)} - 1 \]

      13. *-un-lft-identity 34.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}}}\right)} - 1 \]

      14. times-frac 35.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{V}{1} \cdot \frac{\ell}{A}}}}\right)} - 1 \]

      15. /-rgt-identity 35.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{V} \cdot \frac{\ell}{A}}}\right)} - 1 \]

   6. Applied egg-rr 35.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 54.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)\right)} \]

      2. expm1-log1p 75.2%

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

   8. Simplified 75.2%

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

      1. associate-*r/ 75.6%

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

      2. *-commutative 75.6%

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

      3. associate-/l* 80.1%

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

   10. Applied egg-rr 80.1%

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

   if 3.5000000000000001e130 < l

   1. Initial program 58.3%

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

      1. associate-/r* 52.2%

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

      2. clear-num 51.6%

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

      3. sqrt-div 51.6%

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

      4. metadata-eval 51.6%

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

      5. div-inv 51.5%

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

      6. clear-num 51.5%

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

   4. Applied egg-rr 51.5%

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

      1. associate-*r/ 51.5%

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

      2. sqrt-prod 79.3%

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

      3. times-frac 76.2%

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

      4. metadata-eval 76.2%

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

      5. sqrt-div 76.5%

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

      6. clear-num 76.4%

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

      7. associate-/r/ 79.2%

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

      8. expm1-log1p-u 66.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)\right)} \]

      9. expm1-udef 28.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\right)} - 1} \]

      10. sqrt-undiv 16.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\color{blue}{\sqrt{\frac{\ell}{\frac{A}{V}}}}}\right)} - 1 \]

      11. associate-/l* 25.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{\ell \cdot V}{A}}}}\right)} - 1 \]

      12. *-commutative 25.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{\color{blue}{V \cdot \ell}}{A}}}\right)} - 1 \]

      13. *-un-lft-identity 25.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}}}\right)} - 1 \]

      14. times-frac 25.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{\frac{V}{1} \cdot \frac{\ell}{A}}}}\right)} - 1 \]

      15. /-rgt-identity 25.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\color{blue}{V} \cdot \frac{\ell}{A}}}\right)} - 1 \]

   6. Applied egg-rr 25.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 47.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\right)\right)} \]

      2. expm1-log1p 63.2%

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

   8. Simplified 63.2%

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

      1. clear-num 63.2%

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

      2. un-div-inv 63.1%

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

   10. Applied egg-rr 63.1%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 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 1 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 72.9%

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

3. Final simplification 72.9%

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

Reproduce

herbie shell --seed 2024040 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))