Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.4% → 89.8%

Time: 12.2s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.4% 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: 89.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+247}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (/ (sqrt (/ A V)) (sqrt l)))))
   (if (<= (* V l) -1e+247)
     t_0
     (if (<= (* V l) -1e-304)
       (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
       (if (<= (* V l) 0.0) t_0 (* c0 (/ (sqrt A) (sqrt (* V l)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (sqrt((A / V)) / sqrt(l));
	double tmp;
	if ((V * l) <= -1e+247) {
		tmp = t_0;
	} else if ((V * l) <= -1e-304) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this 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) :: t_0
    real(8) :: tmp
    t_0 = c0 * (sqrt((a / v)) / sqrt(l))
    if ((v * l) <= (-1d+247)) then
        tmp = t_0
    else if ((v * l) <= (-1d-304)) then
        tmp = c0 * (sqrt(-a) / sqrt((v * -l)))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	double tmp;
	if ((V * l) <= -1e+247) {
		tmp = t_0;
	} else if ((V * l) <= -1e-304) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	tmp = 0
	if (V * l) <= -1e+247:
		tmp = t_0
	elif (V * l) <= -1e-304:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = t_0
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)))
	tmp = 0.0
	if (Float64(V * l) <= -1e+247)
		tmp = t_0;
	elseif (Float64(V * l) <= -1e-304)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * (sqrt((A / V)) / sqrt(l));
	tmp = 0.0;
	if ((V * l) <= -1e+247)
		tmp = t_0;
	elseif ((V * l) <= -1e-304)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -1e+247], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -1e-304], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -9.99999999999999952e246 or -9.99999999999999971e-305 < (*.f64 V l) < 0.0

   1. Initial program 38.8%

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

      1. associate-/r* 65.1%

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

      2. sqrt-div 45.7%

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

      3. associate-*r/ 45.7%

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

   4. Applied egg-rr 45.7%

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

      1. *-commutative 45.7%

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

      2. associate-/l* 45.6%

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

      3. associate-/r/ 45.7%

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

   6. Simplified 45.7%

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

   if -9.99999999999999952e246 < (*.f64 V l) < -9.99999999999999971e-305

   1. Initial program 86.5%

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

      1. frac-2neg 86.5%

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

      2. sqrt-div 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. distribute-rgt-neg-out 99.4%

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

      2. *-commutative 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

   6. Simplified 99.4%

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

   if 0.0 < (*.f64 V l)

   1. Initial program 79.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. clear-num 74.8%

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

      3. sqrt-div 75.2%

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

      4. metadata-eval 75.2%

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

      5. div-inv 74.5%

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

      6. clear-num 74.6%

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

   4. Applied egg-rr 74.6%

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

   5. Taylor expanded in l around 0 79.5%

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

      1. frac-2neg 79.5%

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

      2. *-commutative 79.5%

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

      3. distribute-rgt-neg-out 79.5%

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

      4. sqrt-div 0.0%

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

      5. frac-2neg 0.0%

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

      6. clear-num 0.0%

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

      7. div-inv 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. sqr-neg 0.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 29.7%

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

      15. sqr-neg 29.7%

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

      16. sqrt-unprod 45.8%

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

      17. add-sqr-sqrt 91.5%

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

   7. Applied egg-rr 91.5%

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

      1. *-commutative 91.5%

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

      2. neg-mul-1 91.5%

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

      3. associate-/r* 91.5%

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

      4. metadata-eval 91.5%

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

      5. /-rgt-identity 91.5%

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

      6. distribute-neg-frac 91.5%

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

      7. times-frac 91.6%

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

      8. neg-mul-1 91.6%

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

      9. remove-double-neg 91.6%

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

      10. *-rgt-identity 91.6%

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

   9. Simplified 91.6%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+247}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot \frac{V}{A}}\\ t_1 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{-300}:\\ \;\;\;\;\frac{c0}{t_0}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+189}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (* l (/ V A)))) (t_1 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_1 2e-300)
     (/ c0 t_0)
     (if (<= t_1 2e+189) t_1 (* c0 (/ 1.0 t_0))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((l * (V / A)));
	double t_1 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_1 <= 2e-300) {
		tmp = c0 / t_0;
	} else if (t_1 <= 2e+189) {
		tmp = t_1;
	} else {
		tmp = c0 * (1.0 / t_0);
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((l * (v / a)))
    t_1 = c0 * sqrt((a / (v * l)))
    if (t_1 <= 2d-300) then
        tmp = c0 / t_0
    else if (t_1 <= 2d+189) then
        tmp = t_1
    else
        tmp = c0 * (1.0d0 / t_0)
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((l * (V / A)));
	double t_1 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_1 <= 2e-300) {
		tmp = c0 / t_0;
	} else if (t_1 <= 2e+189) {
		tmp = t_1;
	} else {
		tmp = c0 * (1.0 / t_0);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((l * (V / A)))
	t_1 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_1 <= 2e-300:
		tmp = c0 / t_0
	elif t_1 <= 2e+189:
		tmp = t_1
	else:
		tmp = c0 * (1.0 / t_0)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(l * Float64(V / A)))
	t_1 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_1 <= 2e-300)
		tmp = Float64(c0 / t_0);
	elseif (t_1 <= 2e+189)
		tmp = t_1;
	else
		tmp = Float64(c0 * Float64(1.0 / t_0));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((l * (V / A)));
	t_1 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_1 <= 2e-300)
		tmp = c0 / t_0;
	elseif (t_1 <= 2e+189)
		tmp = t_1;
	else
		tmp = c0 * (1.0 / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-300], N[(c0 / t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 2e+189], t$95$1, N[(c0 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.00000000000000005e-300

   1. Initial program 69.5%

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

      1. associate-/r* 65.5%

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

      2. clear-num 65.2%

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

      3. sqrt-div 65.2%

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

      4. metadata-eval 65.2%

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

      5. div-inv 64.0%

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

      6. clear-num 64.1%

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

   4. Applied egg-rr 64.1%

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

   5. Taylor expanded in l around 0 69.2%

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

      1. un-div-inv 69.2%

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

      2. associate-*r/ 67.6%

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

   7. Applied egg-rr 67.6%

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

      1. *-commutative 67.6%

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

      2. associate-/r/ 65.2%

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

      3. associate-/l* 69.2%

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

      4. associate-*r/ 64.1%

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

   9. Simplified 64.1%

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

   if 2.00000000000000005e-300 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2e189

   1. Initial program 98.7%

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

   if 2e189 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 58.0%

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

      1. associate-/r* 70.9%

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

      2. clear-num 70.8%

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

      3. sqrt-div 73.5%

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

      4. metadata-eval 73.5%

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

      5. div-inv 73.6%

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

      6. clear-num 73.5%

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

   4. Applied egg-rr 73.5%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{-300}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{+189}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \ell \cdot \frac{V}{A}\\ t_1 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{-300}:\\ \;\;\;\;\frac{c0}{\sqrt{t_0}}\\ \mathbf{elif}\;t_1 \leq 10^{+165}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {t_0}^{-0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* l (/ V A))) (t_1 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_1 2e-300)
     (/ c0 (sqrt t_0))
     (if (<= t_1 1e+165) t_1 (* c0 (pow t_0 -0.5))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = l * (V / A);
	double t_1 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_1 <= 2e-300) {
		tmp = c0 / sqrt(t_0);
	} else if (t_1 <= 1e+165) {
		tmp = t_1;
	} else {
		tmp = c0 * pow(t_0, -0.5);
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = l * (v / a)
    t_1 = c0 * sqrt((a / (v * l)))
    if (t_1 <= 2d-300) then
        tmp = c0 / sqrt(t_0)
    else if (t_1 <= 1d+165) then
        tmp = t_1
    else
        tmp = c0 * (t_0 ** (-0.5d0))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = l * (V / A);
	double t_1 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_1 <= 2e-300) {
		tmp = c0 / Math.sqrt(t_0);
	} else if (t_1 <= 1e+165) {
		tmp = t_1;
	} else {
		tmp = c0 * Math.pow(t_0, -0.5);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = l * (V / A)
	t_1 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_1 <= 2e-300:
		tmp = c0 / math.sqrt(t_0)
	elif t_1 <= 1e+165:
		tmp = t_1
	else:
		tmp = c0 * math.pow(t_0, -0.5)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(l * Float64(V / A))
	t_1 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_1 <= 2e-300)
		tmp = Float64(c0 / sqrt(t_0));
	elseif (t_1 <= 1e+165)
		tmp = t_1;
	else
		tmp = Float64(c0 * (t_0 ^ -0.5));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = l * (V / A);
	t_1 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_1 <= 2e-300)
		tmp = c0 / sqrt(t_0);
	elseif (t_1 <= 1e+165)
		tmp = t_1;
	else
		tmp = c0 * (t_0 ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-300], N[(c0 / N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+165], t$95$1, N[(c0 * N[Power[t$95$0, -0.5], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.00000000000000005e-300

   1. Initial program 69.5%

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

      1. associate-/r* 65.5%

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

      2. clear-num 65.2%

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

      3. sqrt-div 65.2%

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

      4. metadata-eval 65.2%

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

      5. div-inv 64.0%

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

      6. clear-num 64.1%

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

   4. Applied egg-rr 64.1%

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

   5. Taylor expanded in l around 0 69.2%

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

      1. un-div-inv 69.2%

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

      2. associate-*r/ 67.6%

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

   7. Applied egg-rr 67.6%

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

      1. *-commutative 67.6%

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

      2. associate-/r/ 65.2%

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

      3. associate-/l* 69.2%

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

      4. associate-*r/ 64.1%

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

   9. Simplified 64.1%

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

   if 2.00000000000000005e-300 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 9.99999999999999899e164

   1. Initial program 98.6%

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

   if 9.99999999999999899e164 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 59.9%

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

      1. associate-/r* 72.2%

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

      2. clear-num 72.1%

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

      3. sqrt-div 74.7%

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

      4. metadata-eval 74.7%

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

      5. div-inv 74.7%

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

      6. clear-num 74.6%

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

   4. Applied egg-rr 74.6%

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

   5. Taylor expanded in l around 0 62.3%

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

      1. pow1/2 62.3%

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

      2. associate-*r/ 73.6%

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

      3. pow-flip 73.6%

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

      4. metadata-eval 73.6%

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

   7. Applied egg-rr 73.6%

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

      1. *-commutative 73.6%

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

      2. associate-*l/ 62.4%

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

      3. associate-*r/ 74.7%

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

   9. Simplified 74.7%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{-300}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{+165}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-300} \lor \neg \left(t_0 \leq 10^{+165}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (or (<= t_0 2e-300) (not (<= t_0 1e+165)))
     (* c0 (sqrt (/ (/ A V) l)))
     t_0)))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 2e-300) || !(t_0 <= 1e+165)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this 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) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if ((t_0 <= 2d-300) .or. (.not. (t_0 <= 1d+165))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 2e-300) || !(t_0 <= 1e+165)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if (t_0 <= 2e-300) or not (t_0 <= 1e+165):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if ((t_0 <= 2e-300) || !(t_0 <= 1e+165))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if ((t_0 <= 2e-300) || ~((t_0 <= 1e+165)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 2e-300], N[Not[LessEqual[t$95$0, 1e+165]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.00000000000000005e-300 or 9.99999999999999899e164 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 67.3%

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

      1. associate-/r* 67.1%

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

   3. Simplified 67.1%

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

   if 2.00000000000000005e-300 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 9.99999999999999899e164

   1. Initial program 98.6%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{-300} \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{+165}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-300} \lor \neg \left(t_0 \leq 2 \cdot 10^{+189}\right):\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (or (<= t_0 2e-300) (not (<= t_0 2e+189)))
     (/ c0 (sqrt (* l (/ V A))))
     t_0)))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 2e-300) || !(t_0 <= 2e+189)) {
		tmp = c0 / sqrt((l * (V / A)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this 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) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if ((t_0 <= 2d-300) .or. (.not. (t_0 <= 2d+189))) then
        tmp = c0 / sqrt((l * (v / a)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 2e-300) || !(t_0 <= 2e+189)) {
		tmp = c0 / Math.sqrt((l * (V / A)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if (t_0 <= 2e-300) or not (t_0 <= 2e+189):
		tmp = c0 / math.sqrt((l * (V / A)))
	else:
		tmp = t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if ((t_0 <= 2e-300) || !(t_0 <= 2e+189))
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if ((t_0 <= 2e-300) || ~((t_0 <= 2e+189)))
		tmp = c0 / sqrt((l * (V / A)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 2e-300], N[Not[LessEqual[t$95$0, 2e+189]], $MachinePrecision]], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.00000000000000005e-300 or 2e189 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 66.9%

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

      1. associate-/r* 66.7%

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

      2. clear-num 66.5%

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

      3. sqrt-div 67.0%

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

      4. metadata-eval 67.0%

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

      5. div-inv 66.2%

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

      6. clear-num 66.2%

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

   4. Applied egg-rr 66.2%

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

   5. Taylor expanded in l around 0 67.3%

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

      1. un-div-inv 67.3%

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

      2. associate-*r/ 69.2%

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

   7. Applied egg-rr 69.2%

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

      1. *-commutative 69.2%

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

      2. associate-/r/ 67.1%

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

      3. associate-/l* 67.3%

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

      4. associate-*r/ 66.2%

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

   9. Simplified 66.2%

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

   if 2.00000000000000005e-300 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2e189

   1. Initial program 98.7%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{-300} \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{+189}\right):\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-300}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 10^{+297}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 2e-300)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 1e+297) t_0 (/ c0 (sqrt (* V (/ l A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 2e-300) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 1e+297) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this 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) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 2d-300) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 1d+297) then
        tmp = t_0
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 2e-300) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 1e+297) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 2e-300:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 1e+297:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 2e-300)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 1e+297)
		tmp = t_0;
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 2e-300)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 1e+297)
		tmp = t_0;
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-300], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+297], t$95$0, N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.00000000000000005e-300

   1. Initial program 69.5%

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

      1. associate-/r* 65.5%

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

   3. Simplified 65.5%

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

   if 2.00000000000000005e-300 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1e297

   1. Initial program 98.1%

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

   if 1e297 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 46.8%

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

      1. associate-/r* 64.7%

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

      2. clear-num 64.6%

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

      3. sqrt-div 68.2%

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

      4. metadata-eval 68.2%

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

      5. div-inv 68.2%

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

      6. clear-num 68.2%

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

   4. Applied egg-rr 68.2%

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

   5. Taylor expanded in l around 0 50.2%

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

      1. un-div-inv 50.2%

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

      2. associate-*r/ 66.9%

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

   7. Applied egg-rr 66.9%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{-300}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{+297}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{\sqrt{-A}}{\sqrt{-V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -4e-310)
   (* (/ (/ (sqrt (- A)) (sqrt (- V))) (sqrt l)) c0)
   (* c0 (/ (sqrt A) (sqrt (* V l))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -4e-310) {
		tmp = ((sqrt(-A) / sqrt(-V)) / sqrt(l)) * c0;
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this 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 (a <= (-4d-310)) then
        tmp = ((sqrt(-a) / sqrt(-v)) / sqrt(l)) * c0
    else
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -4e-310) {
		tmp = ((Math.sqrt(-A) / Math.sqrt(-V)) / Math.sqrt(l)) * c0;
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -4e-310:
		tmp = ((math.sqrt(-A) / math.sqrt(-V)) / math.sqrt(l)) * c0
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -4e-310)
		tmp = Float64(Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(-V))) / sqrt(l)) * c0);
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (A <= -4e-310)
		tmp = ((sqrt(-A) / sqrt(-V)) / sqrt(l)) * c0;
	else
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -4e-310], N[(N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -3.999999999999988e-310

   1. Initial program 74.6%

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

      1. associate-/r* 72.7%

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

      2. sqrt-div 39.7%

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

      3. associate-*r/ 36.2%

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

   4. Applied egg-rr 36.2%

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

      1. *-commutative 36.2%

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

      2. associate-/l* 39.6%

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

      3. associate-/r/ 39.7%

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

   6. Simplified 39.7%

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

      1. frac-2neg 39.7%

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

      2. sqrt-div 45.2%

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

   8. Applied egg-rr 45.2%

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

   if -3.999999999999988e-310 < A

   1. Initial program 76.1%

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

      1. associate-/r* 73.7%

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

      2. clear-num 73.4%

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

      3. sqrt-div 73.8%

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

      4. metadata-eval 73.8%

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

      5. div-inv 73.1%

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

      6. clear-num 73.2%

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

   4. Applied egg-rr 73.2%

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

   5. Taylor expanded in l around 0 76.3%

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

      1. frac-2neg 76.3%

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

      2. *-commutative 76.3%

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

      3. distribute-rgt-neg-out 76.3%

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

      4. sqrt-div 0.0%

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

      5. frac-2neg 0.0%

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

      6. clear-num 0.0%

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

      7. div-inv 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. sqr-neg 0.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 29.6%

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

      15. sqr-neg 29.6%

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

      16. sqrt-unprod 44.7%

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

      17. add-sqr-sqrt 87.6%

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

   7. Applied egg-rr 87.6%

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

      1. *-commutative 87.6%

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

      2. neg-mul-1 87.6%

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

      3. associate-/r* 87.6%

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

      4. metadata-eval 87.6%

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

      5. /-rgt-identity 87.6%

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

      6. distribute-neg-frac 87.6%

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

      7. times-frac 87.6%

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

      8. neg-mul-1 87.6%

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

      9. remove-double-neg 87.6%

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

      10. *-rgt-identity 87.6%

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

   9. Simplified 87.6%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{\sqrt{-A}}{\sqrt{-V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+97}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-81}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (/ (sqrt (/ A V)) (sqrt l)))))
   (if (<= (* V l) -5e+97)
     t_0
     (if (<= (* V l) -2e-81)
       (* c0 (/ 1.0 (sqrt (/ (* V l) A))))
       (if (<= (* V l) 0.0) t_0 (* c0 (/ (sqrt A) (sqrt (* V l)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (sqrt((A / V)) / sqrt(l));
	double tmp;
	if ((V * l) <= -5e+97) {
		tmp = t_0;
	} else if ((V * l) <= -2e-81) {
		tmp = c0 * (1.0 / sqrt(((V * l) / A)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this 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) :: t_0
    real(8) :: tmp
    t_0 = c0 * (sqrt((a / v)) / sqrt(l))
    if ((v * l) <= (-5d+97)) then
        tmp = t_0
    else if ((v * l) <= (-2d-81)) then
        tmp = c0 * (1.0d0 / sqrt(((v * l) / a)))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	double tmp;
	if ((V * l) <= -5e+97) {
		tmp = t_0;
	} else if ((V * l) <= -2e-81) {
		tmp = c0 * (1.0 / Math.sqrt(((V * l) / A)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	tmp = 0
	if (V * l) <= -5e+97:
		tmp = t_0
	elif (V * l) <= -2e-81:
		tmp = c0 * (1.0 / math.sqrt(((V * l) / A)))
	elif (V * l) <= 0.0:
		tmp = t_0
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)))
	tmp = 0.0
	if (Float64(V * l) <= -5e+97)
		tmp = t_0;
	elseif (Float64(V * l) <= -2e-81)
		tmp = Float64(c0 * Float64(1.0 / sqrt(Float64(Float64(V * l) / A))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * (sqrt((A / V)) / sqrt(l));
	tmp = 0.0;
	if ((V * l) <= -5e+97)
		tmp = t_0;
	elseif ((V * l) <= -2e-81)
		tmp = c0 * (1.0 / sqrt(((V * l) / A)));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	else
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e+97], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -2e-81], N[(c0 * N[(1.0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -4.99999999999999999e97 or -1.9999999999999999e-81 < (*.f64 V l) < 0.0

   1. Initial program 62.9%

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

      1. associate-/r* 70.0%

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

      2. sqrt-div 43.0%

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

      3. associate-*r/ 42.1%

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

   4. Applied egg-rr 42.1%

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

      1. *-commutative 42.1%

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

      2. associate-/l* 42.9%

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

      3. associate-/r/ 43.0%

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

   6. Simplified 43.0%

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

   if -4.99999999999999999e97 < (*.f64 V l) < -1.9999999999999999e-81

   1. Initial program 94.6%

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

      1. associate-/r* 75.2%

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

      2. clear-num 75.2%

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

      3. sqrt-div 76.3%

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

      4. metadata-eval 76.3%

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

      5. div-inv 76.3%

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

      6. clear-num 76.7%

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

   4. Applied egg-rr 76.7%

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

   5. Taylor expanded in l around 0 95.6%

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

   if 0.0 < (*.f64 V l)

   1. Initial program 79.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. clear-num 74.8%

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

      3. sqrt-div 75.2%

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

      4. metadata-eval 75.2%

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

      5. div-inv 74.5%

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

      6. clear-num 74.6%

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

   4. Applied egg-rr 74.6%

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

   5. Taylor expanded in l around 0 79.5%

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

      1. frac-2neg 79.5%

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

      2. *-commutative 79.5%

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

      3. distribute-rgt-neg-out 79.5%

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

      4. sqrt-div 0.0%

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

      5. frac-2neg 0.0%

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

      6. clear-num 0.0%

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

      7. div-inv 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. sqr-neg 0.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 29.7%

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

      15. sqr-neg 29.7%

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

      16. sqrt-unprod 45.8%

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

      17. add-sqr-sqrt 91.5%

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

   7. Applied egg-rr 91.5%

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

      1. *-commutative 91.5%

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

      2. neg-mul-1 91.5%

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

      3. associate-/r* 91.5%

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

      4. metadata-eval 91.5%

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

      5. /-rgt-identity 91.5%

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

      6. distribute-neg-frac 91.5%

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

      7. times-frac 91.6%

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

      8. neg-mul-1 91.6%

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

      9. remove-double-neg 91.6%

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

      10. *-rgt-identity 91.6%

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

   9. Simplified 91.6%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+97}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-81}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+97}:\\ \;\;\;\;c0 \cdot \frac{t_0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-83}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (/ A V))))
   (if (<= (* V l) -5e+97)
     (* c0 (/ t_0 (sqrt l)))
     (if (<= (* V l) -5e-83)
       (* c0 (/ 1.0 (sqrt (/ (* V l) A))))
       (if (<= (* V l) 0.0)
         (/ c0 (/ (sqrt l) t_0))
         (* c0 (/ (sqrt A) (sqrt (* V l)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / V));
	double tmp;
	if ((V * l) <= -5e+97) {
		tmp = c0 * (t_0 / sqrt(l));
	} else if ((V * l) <= -5e-83) {
		tmp = c0 * (1.0 / sqrt(((V * l) / A)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (sqrt(l) / t_0);
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a / v))
    if ((v * l) <= (-5d+97)) then
        tmp = c0 * (t_0 / sqrt(l))
    else if ((v * l) <= (-5d-83)) then
        tmp = c0 * (1.0d0 / sqrt(((v * l) / a)))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 / (sqrt(l) / t_0)
    else
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / V));
	double tmp;
	if ((V * l) <= -5e+97) {
		tmp = c0 * (t_0 / Math.sqrt(l));
	} else if ((V * l) <= -5e-83) {
		tmp = c0 * (1.0 / Math.sqrt(((V * l) / A)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (Math.sqrt(l) / t_0);
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / V))
	tmp = 0
	if (V * l) <= -5e+97:
		tmp = c0 * (t_0 / math.sqrt(l))
	elif (V * l) <= -5e-83:
		tmp = c0 * (1.0 / math.sqrt(((V * l) / A)))
	elif (V * l) <= 0.0:
		tmp = c0 / (math.sqrt(l) / t_0)
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(A / V))
	tmp = 0.0
	if (Float64(V * l) <= -5e+97)
		tmp = Float64(c0 * Float64(t_0 / sqrt(l)));
	elseif (Float64(V * l) <= -5e-83)
		tmp = Float64(c0 * Float64(1.0 / sqrt(Float64(Float64(V * l) / A))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 / Float64(sqrt(l) / t_0));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = sqrt((A / V));
	tmp = 0.0;
	if ((V * l) <= -5e+97)
		tmp = c0 * (t_0 / sqrt(l));
	elseif ((V * l) <= -5e-83)
		tmp = c0 * (1.0 / sqrt(((V * l) / A)));
	elseif ((V * l) <= 0.0)
		tmp = c0 / (sqrt(l) / t_0);
	else
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e+97], N[(c0 * N[(t$95$0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-83], N[(c0 * N[(1.0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -4.99999999999999999e97

   1. Initial program 58.5%

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

      1. associate-/r* 61.0%

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

      2. sqrt-div 42.9%

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

      3. associate-*r/ 40.6%

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

   4. Applied egg-rr 40.6%

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

      1. *-commutative 40.6%

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

      2. associate-/l* 42.6%

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

      3. associate-/r/ 42.9%

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

   6. Simplified 42.9%

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

   if -4.99999999999999999e97 < (*.f64 V l) < -5e-83

   1. Initial program 94.9%

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

      1. associate-/r* 76.4%

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

      2. clear-num 76.4%

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

      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 77.4%

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

      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. Taylor expanded in l around 0 95.8%

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

   if -5e-83 < (*.f64 V l) < 0.0

   1. Initial program 64.7%

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

      1. associate-/r* 75.4%

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

      2. sqrt-div 44.7%

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

      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* 44.7%

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

   6. Simplified 44.7%

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

   if 0.0 < (*.f64 V l)

   1. Initial program 79.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. clear-num 74.8%

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

      3. sqrt-div 75.2%

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

      4. metadata-eval 75.2%

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

      5. div-inv 74.5%

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

      6. clear-num 74.6%

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

   4. Applied egg-rr 74.6%

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

   5. Taylor expanded in l around 0 79.5%

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

      1. frac-2neg 79.5%

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

      2. *-commutative 79.5%

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

      3. distribute-rgt-neg-out 79.5%

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

      4. sqrt-div 0.0%

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

      5. frac-2neg 0.0%

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

      6. clear-num 0.0%

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

      7. div-inv 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. sqr-neg 0.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 29.7%

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

      15. sqr-neg 29.7%

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

      16. sqrt-unprod 45.8%

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

      17. add-sqr-sqrt 91.5%

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

   7. Applied egg-rr 91.5%

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

      1. *-commutative 91.5%

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

      2. neg-mul-1 91.5%

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

      3. associate-/r* 91.5%

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

      4. metadata-eval 91.5%

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

      5. /-rgt-identity 91.5%

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

      6. distribute-neg-frac 91.5%

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

      7. times-frac 91.6%

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

      8. neg-mul-1 91.6%

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

      9. remove-double-neg 91.6%

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

      10. *-rgt-identity 91.6%

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

   9. Simplified 91.6%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+97}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-83}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+97}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-81}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}}}{\sqrt{\frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -5e+97)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* V l) -1e-81)
     (* c0 (/ 1.0 (sqrt (/ (* V l) A))))
     (if (<= (* V l) 0.0)
       (/ (/ c0 (sqrt l)) (sqrt (/ V A)))
       (* c0 (/ (sqrt A) (sqrt (* V l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+97) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= -1e-81) {
		tmp = c0 * (1.0 / sqrt(((V * l) / A)));
	} else if ((V * l) <= 0.0) {
		tmp = (c0 / sqrt(l)) / sqrt((V / A));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this 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) <= (-5d+97)) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= (-1d-81)) then
        tmp = c0 * (1.0d0 / sqrt(((v * l) / a)))
    else if ((v * l) <= 0.0d0) then
        tmp = (c0 / sqrt(l)) / sqrt((v / a))
    else
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+97) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= -1e-81) {
		tmp = c0 * (1.0 / Math.sqrt(((V * l) / A)));
	} else if ((V * l) <= 0.0) {
		tmp = (c0 / Math.sqrt(l)) / Math.sqrt((V / A));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -5e+97:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= -1e-81:
		tmp = c0 * (1.0 / math.sqrt(((V * l) / A)))
	elif (V * l) <= 0.0:
		tmp = (c0 / math.sqrt(l)) / math.sqrt((V / A))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -5e+97)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= -1e-81)
		tmp = Float64(c0 * Float64(1.0 / sqrt(Float64(Float64(V * l) / A))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(Float64(c0 / sqrt(l)) / sqrt(Float64(V / A)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -5e+97)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= -1e-81)
		tmp = c0 * (1.0 / sqrt(((V * l) / A)));
	elseif ((V * l) <= 0.0)
		tmp = (c0 / sqrt(l)) / sqrt((V / A));
	else
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -5e+97], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-81], N[(c0 * N[(1.0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -4.99999999999999999e97

   1. Initial program 58.5%

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

      1. associate-/r* 61.0%

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

      2. sqrt-div 42.9%

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

      3. associate-*r/ 40.6%

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

   4. Applied egg-rr 40.6%

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

      1. *-commutative 40.6%

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

      2. associate-/l* 42.6%

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

      3. associate-/r/ 42.9%

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

   6. Simplified 42.9%

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

   if -4.99999999999999999e97 < (*.f64 V l) < -9.9999999999999996e-82

   1. Initial program 94.8%

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

      1. associate-/r* 75.8%

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

      2. clear-num 75.8%

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

      3. sqrt-div 76.9%

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

      4. metadata-eval 76.9%

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

      5. div-inv 76.9%

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

      6. clear-num 77.3%

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

   4. Applied egg-rr 77.3%

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

   5. Taylor expanded in l around 0 95.7%

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

   if -9.9999999999999996e-82 < (*.f64 V l) < 0.0

   1. Initial program 65.3%

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

      1. associate-/r* 75.9%

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

      2. clear-num 75.8%

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

      3. sqrt-div 75.8%

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

      4. metadata-eval 75.8%

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

      5. div-inv 74.5%

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

      6. clear-num 74.5%

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

   4. Applied egg-rr 74.5%

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

   5. Taylor expanded in l around 0 65.3%

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

      1. associate-/l* 72.9%

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

      2. *-un-lft-identity 72.9%

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

      3. div-inv 72.8%

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

      4. times-frac 65.3%

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

   7. Applied egg-rr 65.3%

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

      1. add-sqr-sqrt 40.4%

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

      2. sqrt-unprod 32.4%

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

      3. un-div-inv 32.4%

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

      4. un-div-inv 32.4%

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

      5. frac-times 27.5%

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

      6. add-sqr-sqrt 27.5%

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

      7. frac-times 33.4%

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

      8. *-un-lft-identity 33.4%

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

      9. div-inv 33.4%

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

      10. div-inv 33.3%

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

      11. clear-num 33.4%

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

      12. add-sqr-sqrt 33.3%

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

      13. frac-times 35.1%

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

   9. Applied egg-rr 43.9%

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

   if 0.0 < (*.f64 V l)

   1. Initial program 79.3%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   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. clear-num 74.8%

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

      3. sqrt-div 75.2%

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

      4. metadata-eval 75.2%

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

      5. div-inv 74.5%

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

      6. clear-num 74.6%

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

   4. Applied egg-rr 74.6%

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

   5. Taylor expanded in l around 0 79.5%

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

      1. frac-2neg 79.5%

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

      2. *-commutative 79.5%

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

      3. distribute-rgt-neg-out 79.5%

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

      4. sqrt-div 0.0%

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

      5. frac-2neg 0.0%

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

      6. clear-num 0.0%

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

      7. div-inv 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. sqr-neg 0.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 29.7%

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

      15. sqr-neg 29.7%

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

      16. sqrt-unprod 45.8%

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

      17. add-sqr-sqrt 91.5%

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

   7. Applied egg-rr 91.5%

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

      1. *-commutative 91.5%

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

      2. neg-mul-1 91.5%

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

      3. associate-/r* 91.5%

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

      4. metadata-eval 91.5%

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

      5. /-rgt-identity 91.5%

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

      6. distribute-neg-frac 91.5%

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

      7. times-frac 91.6%

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

      8. neg-mul-1 91.6%

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

      9. remove-double-neg 91.6%

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

      10. *-rgt-identity 91.6%

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

   9. Simplified 91.6%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+97}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-81}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}}}{\sqrt{\frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{-222}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-320}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -4e-222)
   (* c0 (/ 1.0 (sqrt (/ (* V l) A))))
   (if (<= (* V l) 2e-320)
     (/ c0 (sqrt (* l (/ V A))))
     (* c0 (/ (sqrt A) (sqrt (* V l)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -4e-222) {
		tmp = c0 * (1.0 / sqrt(((V * l) / A)));
	} else if ((V * l) <= 2e-320) {
		tmp = c0 / sqrt((l * (V / A)));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this 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-222)) then
        tmp = c0 * (1.0d0 / sqrt(((v * l) / a)))
    else if ((v * l) <= 2d-320) then
        tmp = c0 / sqrt((l * (v / a)))
    else
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -4e-222) {
		tmp = c0 * (1.0 / Math.sqrt(((V * l) / A)));
	} else if ((V * l) <= 2e-320) {
		tmp = c0 / Math.sqrt((l * (V / A)));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -4e-222:
		tmp = c0 * (1.0 / math.sqrt(((V * l) / A)))
	elif (V * l) <= 2e-320:
		tmp = c0 / math.sqrt((l * (V / A)))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -4e-222)
		tmp = Float64(c0 * Float64(1.0 / sqrt(Float64(Float64(V * l) / A))));
	elseif (Float64(V * l) <= 2e-320)
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -4e-222)
		tmp = c0 * (1.0 / sqrt(((V * l) / A)));
	elseif ((V * l) <= 2e-320)
		tmp = c0 / sqrt((l * (V / A)));
	else
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -4e-222], N[(c0 * N[(1.0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e-320], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -4.00000000000000019e-222

   1. Initial program 78.5%

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

      1. associate-/r* 70.2%

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

      2. clear-num 70.2%

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

      3. sqrt-div 70.6%

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

      4. metadata-eval 70.6%

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

      5. div-inv 69.8%

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

      6. clear-num 69.9%

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

   4. Applied egg-rr 69.9%

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

   5. Taylor expanded in l around 0 78.8%

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

   if -4.00000000000000019e-222 < (*.f64 V l) < 1.99998e-320

   1. Initial program 50.1%

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

      1. associate-/r* 76.6%

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

      2. clear-num 76.5%

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

      3. sqrt-div 76.5%

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

      4. metadata-eval 76.5%

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

      5. div-inv 76.5%

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

      6. clear-num 76.5%

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

   4. Applied egg-rr 76.5%

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

   5. Taylor expanded in l around 0 50.1%

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

      1. un-div-inv 50.1%

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

      2. associate-*r/ 76.6%

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

   7. Applied egg-rr 76.6%

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

      1. *-commutative 76.6%

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

      2. associate-/r/ 76.6%

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

      3. associate-/l* 50.1%

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

      4. associate-*r/ 76.6%

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

   9. Simplified 76.6%

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

   if 1.99998e-320 < (*.f64 V l)

   1. Initial program 79.6%

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

      1. associate-/r* 74.9%

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

      2. clear-num 74.6%

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

      3. sqrt-div 75.0%

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

      4. metadata-eval 75.0%

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

      5. div-inv 74.3%

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

      6. clear-num 74.3%

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

   4. Applied egg-rr 74.3%

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

   5. Taylor expanded in l around 0 79.8%

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

      1. frac-2neg 79.8%

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

      2. *-commutative 79.8%

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

      3. distribute-rgt-neg-out 79.8%

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

      4. sqrt-div 0.0%

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

      5. frac-2neg 0.0%

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

      6. clear-num 0.0%

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

      7. div-inv 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. sqr-neg 0.0%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 30.0%

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

      15. sqr-neg 30.0%

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

      16. sqrt-unprod 46.2%

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

      17. add-sqr-sqrt 92.0%

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

   7. Applied egg-rr 92.0%

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

      1. *-commutative 92.0%

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

      2. neg-mul-1 92.0%

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

      3. associate-/r* 92.0%

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

      4. metadata-eval 92.0%

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

      5. /-rgt-identity 92.0%

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

      6. distribute-neg-frac 92.0%

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

      7. times-frac 92.0%

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

      8. neg-mul-1 92.0%

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

      9. remove-double-neg 92.0%

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

      10. *-rgt-identity 92.0%

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

   9. Simplified 92.0%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{-222}:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-320}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l) :precision binary64 (* c0 (sqrt (/ A (* V l)))))

C

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

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this 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
    code = c0 * sqrt((a / (v * l)))
end function

Java

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

Python

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

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	return Float64(c0 * sqrt(Float64(A / Float64(V * l))))
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp = code(c0, A, V, l)
	tmp = c0 * sqrt((A / (V * l)));
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.3%

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

3. Final simplification 75.3%

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

Reproduce

herbie shell --seed 2024010 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))