Henrywood and Agarwal, Equation (3)

Average Accuracy: 63.2% → 82.3%

Time: 17.2s

Precision: binary64

Cost: 14348

\[ \begin{array}{c}[V, l] = \mathsf{sort}([V, l])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+264}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-280}:\\ \;\;\;\;c0 \cdot \left({\left(V \cdot \left(-\ell\right)\right)}^{-0.5} \cdot \sqrt{-A}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \left({\left(\frac{-1}{V}\right)}^{0.5} \cdot {\left(\frac{-\ell}{A}\right)}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

FPCore

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

↓

(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -4e+264)
   (* c0 (/ (sqrt (/ (- A) l)) (sqrt (- V))))
   (if (<= (* V l) -1e-280)
     (* c0 (* (pow (* V (- l)) -0.5) (sqrt (- A))))
     (if (<= (* V l) 0.0)
       (* c0 (* (pow (/ -1.0 V) 0.5) (pow (/ (- l) A) -0.5)))
       (if (<= (* V l) 4e+306)
         (/ c0 (/ (sqrt (* V l)) (sqrt A)))
         (* c0 (sqrt (/ (/ A l) V))))))))

C

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

↓

double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -4e+264) {
		tmp = c0 * (sqrt((-A / l)) / sqrt(-V));
	} else if ((V * l) <= -1e-280) {
		tmp = c0 * (pow((V * -l), -0.5) * sqrt(-A));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (pow((-1.0 / V), 0.5) * pow((-l / A), -0.5));
	} else if ((V * l) <= 4e+306) {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	return tmp;
}

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

↓

real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-4d+264)) then
        tmp = c0 * (sqrt((-a / l)) / sqrt(-v))
    else if ((v * l) <= (-1d-280)) then
        tmp = c0 * (((v * -l) ** (-0.5d0)) * sqrt(-a))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 * ((((-1.0d0) / v) ** 0.5d0) * ((-l / a) ** (-0.5d0)))
    else if ((v * l) <= 4d+306) then
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    else
        tmp = c0 * sqrt(((a / l) / v))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -4e+264) {
		tmp = c0 * (Math.sqrt((-A / l)) / Math.sqrt(-V));
	} else if ((V * l) <= -1e-280) {
		tmp = c0 * (Math.pow((V * -l), -0.5) * Math.sqrt(-A));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (Math.pow((-1.0 / V), 0.5) * Math.pow((-l / A), -0.5));
	} else if ((V * l) <= 4e+306) {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	} else {
		tmp = c0 * Math.sqrt(((A / l) / V));
	}
	return tmp;
}

Python

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

↓

def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -4e+264:
		tmp = c0 * (math.sqrt((-A / l)) / math.sqrt(-V))
	elif (V * l) <= -1e-280:
		tmp = c0 * (math.pow((V * -l), -0.5) * math.sqrt(-A))
	elif (V * l) <= 0.0:
		tmp = c0 * (math.pow((-1.0 / V), 0.5) * math.pow((-l / A), -0.5))
	elif (V * l) <= 4e+306:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	else:
		tmp = c0 * math.sqrt(((A / l) / V))
	return tmp

Julia

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

↓

function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -4e+264)
		tmp = Float64(c0 * Float64(sqrt(Float64(Float64(-A) / l)) / sqrt(Float64(-V))));
	elseif (Float64(V * l) <= -1e-280)
		tmp = Float64(c0 * Float64((Float64(V * Float64(-l)) ^ -0.5) * sqrt(Float64(-A))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * Float64((Float64(-1.0 / V) ^ 0.5) * (Float64(Float64(-l) / A) ^ -0.5)));
	elseif (Float64(V * l) <= 4e+306)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / V)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -4e+264)
		tmp = c0 * (sqrt((-A / l)) / sqrt(-V));
	elseif ((V * l) <= -1e-280)
		tmp = c0 * (((V * -l) ^ -0.5) * sqrt(-A));
	elseif ((V * l) <= 0.0)
		tmp = c0 * (((-1.0 / V) ^ 0.5) * ((-l / A) ^ -0.5));
	elseif ((V * l) <= 4e+306)
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	else
		tmp = c0 * sqrt(((A / l) / V));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -4e+264], N[(c0 * N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-280], N[(c0 * N[(N[Power[N[(V * (-l)), $MachinePrecision], -0.5], $MachinePrecision] * N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[(N[Power[N[(-1.0 / V), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[((-l) / A), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e+306], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -4.00000000000000018e264

   1. Initial program 56.7%

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

   2. Applied egg-rr 70.4%

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

      [Start] 56.7	\[ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
      *-un-lft-identity [=>] 56.7	\[ c0 \cdot \sqrt{\frac{\color{blue}{1 \cdot A}}{V \cdot \ell}} \]
      times-frac [=>] 70.4	\[ c0 \cdot \sqrt{\color{blue}{\frac{1}{V} \cdot \frac{A}{\ell}}} \]

   3. Applied egg-rr 70.3%

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

      [Start] 70.4	\[ c0 \cdot \sqrt{\frac{1}{V} \cdot \frac{A}{\ell}} \]
      associate-*l/ [=>] 70.3	\[ c0 \cdot \sqrt{\color{blue}{\frac{1 \cdot \frac{A}{\ell}}{V}}} \]
      *-un-lft-identity [<=] 70.3	\[ c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{\ell}}}{V}} \]

   4. Applied egg-rr 37.0%

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

      [Start] 70.3	\[ c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}} \]
      frac-2neg [=>] 70.3	\[ c0 \cdot \sqrt{\color{blue}{\frac{-\frac{A}{\ell}}{-V}}} \]
      sqrt-div [=>] 37.0	\[ c0 \cdot \color{blue}{\frac{\sqrt{-\frac{A}{\ell}}}{\sqrt{-V}}} \]

   5. Simplified 37.0%

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

      [Start] 37.0	\[ c0 \cdot \frac{\sqrt{-\frac{A}{\ell}}}{\sqrt{-V}} \]
      distribute-neg-frac [=>] 37.0	\[ c0 \cdot \frac{\sqrt{\color{blue}{\frac{-A}{\ell}}}}{\sqrt{-V}} \]

   if -4.00000000000000018e264 < (*.f64 V l) < -9.9999999999999996e-281

   1. Initial program 76.6%

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

   2. Applied egg-rr 62.8%

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

      [Start] 76.6	\[ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
      pow1/2 [=>] 76.6	\[ c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}} \]
      clear-num [=>] 76.0	\[ c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5} \]
      inv-pow [=>] 76.0	\[ c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5} \]
      pow-pow [=>] 76.1	\[ c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}} \]
      associate-/l* [=>] 62.8	\[ c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)} \]
      metadata-eval [=>] 62.8	\[ c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{\color{blue}{-0.5}} \]

   3. Simplified 61.7%

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

      [Start] 62.8	\[ c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5} \]
      associate-/l* [<=] 76.1	\[ c0 \cdot {\color{blue}{\left(\frac{V \cdot \ell}{A}\right)}}^{-0.5} \]
      *-lft-identity [<=] 76.1	\[ c0 \cdot {\left(\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}\right)}^{-0.5} \]
      times-frac [=>] 61.7	\[ c0 \cdot {\color{blue}{\left(\frac{V}{1} \cdot \frac{\ell}{A}\right)}}^{-0.5} \]
      /-rgt-identity [=>] 61.7	\[ c0 \cdot {\left(\color{blue}{V} \cdot \frac{\ell}{A}\right)}^{-0.5} \]

   4. Taylor expanded in A around -inf 84.4%

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

   5. Simplified 91.4%

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

      [Start] 84.4	\[ c0 \cdot e^{-0.5 \cdot \left(\log \left(-1 \cdot \left(V \cdot \ell\right)\right) + \log \left(\frac{-1}{A}\right)\right)} \]
      distribute-lft-in [=>] 84.4	\[ c0 \cdot e^{\color{blue}{-0.5 \cdot \log \left(-1 \cdot \left(V \cdot \ell\right)\right) + -0.5 \cdot \log \left(\frac{-1}{A}\right)}} \]
      exp-sum [=>] 85.0	\[ c0 \cdot \color{blue}{\left(e^{-0.5 \cdot \log \left(-1 \cdot \left(V \cdot \ell\right)\right)} \cdot e^{-0.5 \cdot \log \left(\frac{-1}{A}\right)}\right)} \]
      *-commutative [=>] 85.0	\[ c0 \cdot \left(e^{\color{blue}{\log \left(-1 \cdot \left(V \cdot \ell\right)\right) \cdot -0.5}} \cdot e^{-0.5 \cdot \log \left(\frac{-1}{A}\right)}\right) \]
      neg-mul-1 [<=] 85.0	\[ c0 \cdot \left(e^{\log \color{blue}{\left(-V \cdot \ell\right)} \cdot -0.5} \cdot e^{-0.5 \cdot \log \left(\frac{-1}{A}\right)}\right) \]
      exp-to-pow [=>] 86.0	\[ c0 \cdot \left(\color{blue}{{\left(-V \cdot \ell\right)}^{-0.5}} \cdot e^{-0.5 \cdot \log \left(\frac{-1}{A}\right)}\right) \]
      distribute-rgt-neg-in [=>] 86.0	\[ c0 \cdot \left({\color{blue}{\left(V \cdot \left(-\ell\right)\right)}}^{-0.5} \cdot e^{-0.5 \cdot \log \left(\frac{-1}{A}\right)}\right) \]
      *-commutative [=>] 86.0	\[ c0 \cdot \left({\left(V \cdot \left(-\ell\right)\right)}^{-0.5} \cdot e^{\color{blue}{\log \left(\frac{-1}{A}\right) \cdot -0.5}}\right) \]
      exp-to-pow [=>] 91.4	\[ c0 \cdot \left({\left(V \cdot \left(-\ell\right)\right)}^{-0.5} \cdot \color{blue}{{\left(\frac{-1}{A}\right)}^{-0.5}}\right) \]

   6. Applied egg-rr 91.5%

      \[\leadsto c0 \cdot \left({\left(V \cdot \left(-\ell\right)\right)}^{-0.5} \cdot \color{blue}{\left(0 + \sqrt{A \cdot -1}\right)}\right) \]
      Step-by-step derivation

      [Start] 91.4	\[ c0 \cdot \left({\left(V \cdot \left(-\ell\right)\right)}^{-0.5} \cdot {\left(\frac{-1}{A}\right)}^{-0.5}\right) \]
      add-log-exp [=>] 15.2	\[ c0 \cdot \left({\left(V \cdot \left(-\ell\right)\right)}^{-0.5} \cdot \color{blue}{\log \left(e^{{\left(\frac{-1}{A}\right)}^{-0.5}}\right)}\right) \]
      *-un-lft-identity [=>] 15.2	\[ c0 \cdot \left({\left(V \cdot \left(-\ell\right)\right)}^{-0.5} \cdot \log \color{blue}{\left(1 \cdot e^{{\left(\frac{-1}{A}\right)}^{-0.5}}\right)}\right) \]
      log-prod [=>] 15.2	\[ c0 \cdot \left({\left(V \cdot \left(-\ell\right)\right)}^{-0.5} \cdot \color{blue}{\left(\log 1 + \log \left(e^{{\left(\frac{-1}{A}\right)}^{-0.5}}\right)\right)}\right) \]
      metadata-eval [=>] 15.2	\[ c0 \cdot \left({\left(V \cdot \left(-\ell\right)\right)}^{-0.5} \cdot \left(\color{blue}{0} + \log \left(e^{{\left(\frac{-1}{A}\right)}^{-0.5}}\right)\right)\right) \]
      add-log-exp [<=] 91.4	\[ c0 \cdot \left({\left(V \cdot \left(-\ell\right)\right)}^{-0.5} \cdot \left(0 + \color{blue}{{\left(\frac{-1}{A}\right)}^{-0.5}}\right)\right) \]
      add-sqr-sqrt [=>] 91.2	\[ c0 \cdot \left({\left(V \cdot \left(-\ell\right)\right)}^{-0.5} \cdot \left(0 + \color{blue}{\sqrt{{\left(\frac{-1}{A}\right)}^{-0.5}} \cdot \sqrt{{\left(\frac{-1}{A}\right)}^{-0.5}}}\right)\right) \]
      sqrt-unprod [=>] 91.4	\[ c0 \cdot \left({\left(V \cdot \left(-\ell\right)\right)}^{-0.5} \cdot \left(0 + \color{blue}{\sqrt{{\left(\frac{-1}{A}\right)}^{-0.5} \cdot {\left(\frac{-1}{A}\right)}^{-0.5}}}\right)\right) \]
      pow-prod-up [=>] 91.4	\[ c0 \cdot \left({\left(V \cdot \left(-\ell\right)\right)}^{-0.5} \cdot \left(0 + \sqrt{\color{blue}{{\left(\frac{-1}{A}\right)}^{\left(-0.5 + -0.5\right)}}}\right)\right) \]
      metadata-eval [=>] 91.4	\[ c0 \cdot \left({\left(V \cdot \left(-\ell\right)\right)}^{-0.5} \cdot \left(0 + \sqrt{{\left(\frac{-1}{A}\right)}^{\color{blue}{-1}}}\right)\right) \]
      inv-pow [<=] 91.4	\[ c0 \cdot \left({\left(V \cdot \left(-\ell\right)\right)}^{-0.5} \cdot \left(0 + \sqrt{\color{blue}{\frac{1}{\frac{-1}{A}}}}\right)\right) \]
      clear-num [<=] 91.5	\[ c0 \cdot \left({\left(V \cdot \left(-\ell\right)\right)}^{-0.5} \cdot \left(0 + \sqrt{\color{blue}{\frac{A}{-1}}}\right)\right) \]
      div-inv [=>] 91.5	\[ c0 \cdot \left({\left(V \cdot \left(-\ell\right)\right)}^{-0.5} \cdot \left(0 + \sqrt{\color{blue}{A \cdot \frac{1}{-1}}}\right)\right) \]
      metadata-eval [=>] 91.5	\[ c0 \cdot \left({\left(V \cdot \left(-\ell\right)\right)}^{-0.5} \cdot \left(0 + \sqrt{A \cdot \color{blue}{-1}}\right)\right) \]

   7. Simplified 91.5%

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

      [Start] 91.5	\[ c0 \cdot \left({\left(V \cdot \left(-\ell\right)\right)}^{-0.5} \cdot \left(0 + \sqrt{A \cdot -1}\right)\right) \]
      +-lft-identity [=>] 91.5	\[ c0 \cdot \left({\left(V \cdot \left(-\ell\right)\right)}^{-0.5} \cdot \color{blue}{\sqrt{A \cdot -1}}\right) \]
      *-commutative [=>] 91.5	\[ c0 \cdot \left({\left(V \cdot \left(-\ell\right)\right)}^{-0.5} \cdot \sqrt{\color{blue}{-1 \cdot A}}\right) \]
      neg-mul-1 [<=] 91.5	\[ c0 \cdot \left({\left(V \cdot \left(-\ell\right)\right)}^{-0.5} \cdot \sqrt{\color{blue}{-A}}\right) \]

   if -9.9999999999999996e-281 < (*.f64 V l) < 0.0

   1. Initial program 3.3%

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

   2. Applied egg-rr 23.2%

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

      [Start] 3.3	\[ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
      pow1/2 [=>] 3.3	\[ c0 \cdot \color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{0.5}} \]
      clear-num [=>] 3.3	\[ c0 \cdot {\color{blue}{\left(\frac{1}{\frac{V \cdot \ell}{A}}\right)}}^{0.5} \]
      inv-pow [=>] 3.3	\[ c0 \cdot {\color{blue}{\left({\left(\frac{V \cdot \ell}{A}\right)}^{-1}\right)}}^{0.5} \]
      pow-pow [=>] 3.3	\[ c0 \cdot \color{blue}{{\left(\frac{V \cdot \ell}{A}\right)}^{\left(-1 \cdot 0.5\right)}} \]
      associate-/l* [=>] 23.2	\[ c0 \cdot {\color{blue}{\left(\frac{V}{\frac{A}{\ell}}\right)}}^{\left(-1 \cdot 0.5\right)} \]
      metadata-eval [=>] 23.2	\[ c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{\color{blue}{-0.5}} \]

   3. Simplified 23.3%

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

      [Start] 23.2	\[ c0 \cdot {\left(\frac{V}{\frac{A}{\ell}}\right)}^{-0.5} \]
      associate-/l* [<=] 3.3	\[ c0 \cdot {\color{blue}{\left(\frac{V \cdot \ell}{A}\right)}}^{-0.5} \]
      *-lft-identity [<=] 3.3	\[ c0 \cdot {\left(\frac{V \cdot \ell}{\color{blue}{1 \cdot A}}\right)}^{-0.5} \]
      times-frac [=>] 23.3	\[ c0 \cdot {\color{blue}{\left(\frac{V}{1} \cdot \frac{\ell}{A}\right)}}^{-0.5} \]
      /-rgt-identity [=>] 23.3	\[ c0 \cdot {\left(\color{blue}{V} \cdot \frac{\ell}{A}\right)}^{-0.5} \]

   4. Taylor expanded in V around -inf 29.8%

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

   5. Simplified 32.8%

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

      [Start] 29.8	\[ c0 \cdot e^{-0.5 \cdot \left(\log \left(-1 \cdot \frac{\ell}{A}\right) + -1 \cdot \log \left(\frac{-1}{V}\right)\right)} \]
      +-commutative [=>] 29.8	\[ c0 \cdot e^{-0.5 \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{-1}{V}\right) + \log \left(-1 \cdot \frac{\ell}{A}\right)\right)}} \]
      distribute-lft-in [=>] 29.8	\[ c0 \cdot e^{\color{blue}{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{V}\right)\right) + -0.5 \cdot \log \left(-1 \cdot \frac{\ell}{A}\right)}} \]
      *-commutative [<=] 29.8	\[ c0 \cdot e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{V}\right)\right) + \color{blue}{\log \left(-1 \cdot \frac{\ell}{A}\right) \cdot -0.5}} \]
      exp-sum [=>] 30.6	\[ c0 \cdot \color{blue}{\left(e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{V}\right)\right)} \cdot e^{\log \left(-1 \cdot \frac{\ell}{A}\right) \cdot -0.5}\right)} \]
      associate-*r* [=>] 30.6	\[ c0 \cdot \left(e^{\color{blue}{\left(-0.5 \cdot -1\right) \cdot \log \left(\frac{-1}{V}\right)}} \cdot e^{\log \left(-1 \cdot \frac{\ell}{A}\right) \cdot -0.5}\right) \]
      metadata-eval [=>] 30.6	\[ c0 \cdot \left(e^{\color{blue}{0.5} \cdot \log \left(\frac{-1}{V}\right)} \cdot e^{\log \left(-1 \cdot \frac{\ell}{A}\right) \cdot -0.5}\right) \]
      *-commutative [<=] 30.6	\[ c0 \cdot \left(e^{\color{blue}{\log \left(\frac{-1}{V}\right) \cdot 0.5}} \cdot e^{\log \left(-1 \cdot \frac{\ell}{A}\right) \cdot -0.5}\right) \]
      exp-to-pow [=>] 30.2	\[ c0 \cdot \left(\color{blue}{{\left(\frac{-1}{V}\right)}^{0.5}} \cdot e^{\log \left(-1 \cdot \frac{\ell}{A}\right) \cdot -0.5}\right) \]
      exp-to-pow [=>] 32.8	\[ c0 \cdot \left({\left(\frac{-1}{V}\right)}^{0.5} \cdot \color{blue}{{\left(-1 \cdot \frac{\ell}{A}\right)}^{-0.5}}\right) \]
      associate-*r/ [=>] 32.8	\[ c0 \cdot \left({\left(\frac{-1}{V}\right)}^{0.5} \cdot {\color{blue}{\left(\frac{-1 \cdot \ell}{A}\right)}}^{-0.5}\right) \]
      neg-mul-1 [<=] 32.8	\[ c0 \cdot \left({\left(\frac{-1}{V}\right)}^{0.5} \cdot {\left(\frac{\color{blue}{-\ell}}{A}\right)}^{-0.5}\right) \]

   if 0.0 < (*.f64 V l) < 4.00000000000000007e306

   1. Initial program 71.5%

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

   2. Applied egg-rr 85.3%

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

      [Start] 71.5	\[ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
      sqrt-div [=>] 90.0	\[ c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]
      associate-*r/ [=>] 85.3	\[ \color{blue}{\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}} \]

   3. Simplified 90.0%

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

      [Start] 85.3	\[ \frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}} \]
      associate-/l* [=>] 90.0	\[ \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]

   if 4.00000000000000007e306 < (*.f64 V l)

   1. Initial program 20.3%

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

   2. Applied egg-rr 52.5%

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

      [Start] 20.3	\[ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
      *-un-lft-identity [=>] 20.3	\[ c0 \cdot \sqrt{\frac{\color{blue}{1 \cdot A}}{V \cdot \ell}} \]
      times-frac [=>] 52.5	\[ c0 \cdot \sqrt{\color{blue}{\frac{1}{V} \cdot \frac{A}{\ell}}} \]

   3. Applied egg-rr 52.5%

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

      [Start] 52.5	\[ c0 \cdot \sqrt{\frac{1}{V} \cdot \frac{A}{\ell}} \]
      associate-*l/ [=>] 52.5	\[ c0 \cdot \sqrt{\color{blue}{\frac{1 \cdot \frac{A}{\ell}}{V}}} \]
      *-un-lft-identity [<=] 52.5	\[ c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{\ell}}}{V}} \]

3. Recombined 5 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+264}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-280}:\\ \;\;\;\;c0 \cdot \left({\left(V \cdot \left(-\ell\right)\right)}^{-0.5} \cdot \sqrt{-A}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \left({\left(\frac{-1}{V}\right)}^{0.5} \cdot {\left(\frac{-\ell}{A}\right)}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	72.4%
Cost	34640

\[\begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ t_1 := \frac{A}{V \cdot \ell}\\ t_2 := c0 \cdot \sqrt{t_1}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-248}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_2 \leq 10^{+305}:\\ \;\;\;\;c0 \cdot {t_1}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \frac{c0}{\frac{V}{\frac{c0}{\ell}}}}\\ \end{array} \]

Alternative 2
Accuracy	82.5%
Cost	20356

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \left(\left({\ell}^{-0.5} \cdot {\left(\frac{-1}{V}\right)}^{0.5}\right) \cdot {\left(\frac{-1}{A}\right)}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 3
Accuracy	77.0%
Cost	14288

\[\begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+195}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-74}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\frac{V}{\frac{1}{\ell}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{A} \cdot \frac{c0}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 4
Accuracy	78.3%
Cost	14288

\[\begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+195}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-74}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\frac{V}{\frac{1}{\ell}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 5
Accuracy	82.2%
Cost	14288

\[\begin{array}{l} t_0 := \frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+262}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-305}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 6
Accuracy	81.9%
Cost	14288

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-239}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 7
Accuracy	82.1%
Cost	14288

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+264}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-305}:\\ \;\;\;\;c0 \cdot \left({\left(V \cdot \left(-\ell\right)\right)}^{-0.5} \cdot \sqrt{-A}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 8
Accuracy	69.7%
Cost	7752

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{+301}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \end{array} \]

Alternative 9
Accuracy	69.6%
Cost	7752

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{+301}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \frac{c0}{\frac{V}{\frac{c0}{\ell}}}}\\ \end{array} \]

Alternative 10
Accuracy	69.5%
Cost	7688

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{+301}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \]

Alternative 11
Accuracy	69.1%
Cost	7625

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 4 \cdot 10^{+301}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \end{array} \]

Alternative 12
Accuracy	69.1%
Cost	7624

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{+301}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]

Alternative 13
Accuracy	69.5%
Cost	7624

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{+301}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]

Alternative 14
Accuracy	63.2%
Cost	6848

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

Reproduce

herbie shell --seed 2023157 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))