Average Error: 13.8 → 9.5
Time: 20.4s
Precision: binary64
Cost: 28296
\[ \begin{array}{c}[M, D] = \mathsf{sort}([M, D])\\ \end{array} \]
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
↓
\[\begin{array}{l} t_0 := {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+307}:\\ \;\;\;\;w0 \cdot \sqrt{1 + D \cdot \frac{D \cdot \left({\left(\frac{M}{d}\right)}^{2} \cdot \left(h \cdot 0.25\right)\right)}{-\ell}}\\ \mathbf{elif}\;t_0 \leq 10^{-42}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(M \cdot \left(M \cdot h\right)\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot -0.25\right)}{\ell}}\\ \end{array} \]
(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))
↓
(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))
   (if (<= t_0 -2e+307)
     (*
      w0
      (sqrt (+ 1.0 (* D (/ (* D (* (pow (/ M d) 2.0) (* h 0.25))) (- l))))))
     (if (<= t_0 1e-42)
       (* w0 (sqrt (- 1.0 t_0)))
       (*
        w0
        (sqrt
         (+ 1.0 (/ (* (* M (* M h)) (* (* (/ D d) (/ D d)) -0.25)) l))))))))
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}
↓
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = pow(((M * D) / (2.0 * d)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -2e+307) {
		tmp = w0 * sqrt((1.0 + (D * ((D * (pow((M / d), 2.0) * (h * 0.25))) / -l))));
	} else if (t_0 <= 1e-42) {
		tmp = w0 * sqrt((1.0 - t_0));
	} else {
		tmp = w0 * sqrt((1.0 + (((M * (M * h)) * (((D / d) * (D / d)) * -0.25)) / l)));
	}
	return tmp;
}

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))
end function
↓
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)
    if (t_0 <= (-2d+307)) then
        tmp = w0 * sqrt((1.0d0 + (d * ((d * (((m / d_1) ** 2.0d0) * (h * 0.25d0))) / -l))))
    else if (t_0 <= 1d-42) then
        tmp = w0 * sqrt((1.0d0 - t_0))
    else
        tmp = w0 * sqrt((1.0d0 + (((m * (m * h)) * (((d / d_1) * (d / d_1)) * (-0.25d0))) / l)))
    end if
    code = tmp
end function

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}
↓
public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -2e+307) {
		tmp = w0 * Math.sqrt((1.0 + (D * ((D * (Math.pow((M / d), 2.0) * (h * 0.25))) / -l))));
	} else if (t_0 <= 1e-42) {
		tmp = w0 * Math.sqrt((1.0 - t_0));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (((M * (M * h)) * (((D / d) * (D / d)) * -0.25)) / l)));
	}
	return tmp;
}

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))
↓
def code(w0, M, D, h, l, d):
	t_0 = math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)
	tmp = 0
	if t_0 <= -2e+307:
		tmp = w0 * math.sqrt((1.0 + (D * ((D * (math.pow((M / d), 2.0) * (h * 0.25))) / -l))))
	elif t_0 <= 1e-42:
		tmp = w0 * math.sqrt((1.0 - t_0))
	else:
		tmp = w0 * math.sqrt((1.0 + (((M * (M * h)) * (((D / d) * (D / d)) * -0.25)) / l)))
	return tmp

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))))
end
↓
function code(w0, M, D, h, l, d)
	t_0 = Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_0 <= -2e+307)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(D * Float64(Float64(D * Float64((Float64(M / d) ^ 2.0) * Float64(h * 0.25))) / Float64(-l))))));
	elseif (t_0 <= 1e-42)
		tmp = Float64(w0 * sqrt(Float64(1.0 - t_0)));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(M * Float64(M * h)) * Float64(Float64(Float64(D / d) * Float64(D / d)) * -0.25)) / l))));
	end
	return tmp
end

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))));
end
↓
function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = (((M * D) / (2.0 * d)) ^ 2.0) * (h / l);
	tmp = 0.0;
	if (t_0 <= -2e+307)
		tmp = w0 * sqrt((1.0 + (D * ((D * (((M / d) ^ 2.0) * (h * 0.25))) / -l))));
	elseif (t_0 <= 1e-42)
		tmp = w0 * sqrt((1.0 - t_0));
	else
		tmp = w0 * sqrt((1.0 + (((M * (M * h)) * (((D / d) * (D / d)) * -0.25)) / l)));
	end
	tmp_2 = tmp;
end

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
↓
code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+307], N[(w0 * N[Sqrt[N[(1.0 + N[(D * N[(N[(D * N[(N[Power[N[(M / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-l)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-42], N[(w0 * N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
t_0 := {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\
\mathbf{if}\;t_0 \leq -2 \cdot 10^{+307}:\\
\;\;\;\;w0 \cdot \sqrt{1 + D \cdot \frac{D \cdot \left({\left(\frac{M}{d}\right)}^{2} \cdot \left(h \cdot 0.25\right)\right)}{-\ell}}\\

\mathbf{elif}\;t_0 \leq 10^{-42}:\\
\;\;\;\;w0 \cdot \sqrt{1 - t_0}\\

\mathbf{else}:\\
\;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(M \cdot \left(M \cdot h\right)\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot -0.25\right)}{\ell}}\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -1.99999999999999997e307

    1. Initial program 64.0

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

    2. Taylor expanded in M around 0 62.4

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}} \]

    3. Simplified61.0

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{D}{\ell} \cdot D\right) \cdot \left(\left(\frac{M \cdot M}{d \cdot d} \cdot h\right) \cdot 0.25\right)}} \]
      Proof
      (*.f64 (*.f64 (/.f64 D l) D) (*.f64 (*.f64 (/.f64 (*.f64 M M) (*.f64 d d)) h) 1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/r/_binary64 (/.f64 D (/.f64 l D))) (*.f64 (*.f64 (/.f64 (*.f64 M M) (*.f64 d d)) h) 1/4)): 3 points increase in error, 2 points decrease in error
      (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 D D) l)) (*.f64 (*.f64 (/.f64 (*.f64 M M) (*.f64 d d)) h) 1/4)): 17 points increase in error, 2 points decrease in error
      (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) l) (*.f64 (*.f64 (/.f64 (*.f64 M M) (*.f64 d d)) h) 1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 D 2) l) (*.f64 (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 M 2)) (*.f64 d d)) h) 1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 D 2) l) (*.f64 (*.f64 (/.f64 (pow.f64 M 2) (Rewrite<= unpow2_binary64 (pow.f64 d 2))) h) 1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 D 2) l) (*.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (pow.f64 M 2) (/.f64 (pow.f64 d 2) h))) 1/4)): 12 points increase in error, 3 points decrease in error
      (*.f64 (/.f64 (pow.f64 D 2) l) (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 M 2) h) (pow.f64 d 2))) 1/4)): 7 points increase in error, 12 points decrease in error
      (*.f64 (/.f64 (pow.f64 D 2) l) (*.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 h (pow.f64 M 2))) (pow.f64 d 2)) 1/4)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 D 2) l) (/.f64 (*.f64 h (pow.f64 M 2)) (pow.f64 d 2))) 1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 h (pow.f64 M 2))) (*.f64 l (pow.f64 d 2)))) 1/4): 9 points increase in error, 8 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (*.f64 (pow.f64 D 2) (*.f64 h (pow.f64 M 2))) 1/4) (*.f64 l (pow.f64 d 2)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/4 (*.f64 (pow.f64 D 2) (*.f64 h (pow.f64 M 2))))) (*.f64 l (pow.f64 d 2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 1/4 (*.f64 (pow.f64 D 2) (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 M 2) h)))) (*.f64 l (pow.f64 d 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (*.f64 l (pow.f64 d 2))))): 0 points increase in error, 0 points decrease in error

    4. Applied egg-rr56.4

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{D \cdot \left({\left(\frac{M}{d}\right)}^{2} \cdot \left(h \cdot 0.25\right)\right)}{\frac{\ell}{D}}}} \]

    5. Applied egg-rr56.8

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{D \cdot \left({\left(\frac{M}{d}\right)}^{2} \cdot \left(h \cdot 0.25\right)\right)}{-\ell} \cdot \left(-D\right)}} \]

    if -1.99999999999999997e307 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < 1.00000000000000004e-42

    1. Initial program 0.1

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

    if 1.00000000000000004e-42 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

    1. Initial program 55.0

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

    2. Applied egg-rr22.1

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot h}{\ell}}} \]

    3. Taylor expanded in M around 0 34.5

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}} \]

    4. Simplified20.2

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(0.25 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right) \cdot \left(M \cdot \left(M \cdot h\right)\right)}{\ell}}} \]
      Proof
      (/.f64 (*.f64 (*.f64 1/4 (*.f64 (/.f64 D d) (/.f64 D d))) (*.f64 M (*.f64 M h))) l): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (*.f64 1/4 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 D D) (*.f64 d d)))) (*.f64 M (*.f64 M h))) l): 59 points increase in error, 9 points decrease in error
      (/.f64 (*.f64 (*.f64 1/4 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) (*.f64 d d))) (*.f64 M (*.f64 M h))) l): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (*.f64 1/4 (/.f64 (pow.f64 D 2) (Rewrite<= unpow2_binary64 (pow.f64 d 2)))) (*.f64 M (*.f64 M h))) l): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 1/4 (pow.f64 D 2)) (pow.f64 d 2))) (*.f64 M (*.f64 M h))) l): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 D 2) 1/4)) (pow.f64 d 2)) (*.f64 M (*.f64 M h))) l): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (*.f64 (pow.f64 D 2) 1/4) (pow.f64 d 2)) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 M M) h))) l): 9 points increase in error, 4 points decrease in error
      (/.f64 (*.f64 (/.f64 (*.f64 (pow.f64 D 2) 1/4) (pow.f64 d 2)) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 M 2)) h)) l): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (*.f64 (pow.f64 D 2) 1/4) (pow.f64 d 2)) (Rewrite<= *-commutative_binary64 (*.f64 h (pow.f64 M 2)))) l): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (*.f64 (pow.f64 D 2) 1/4) (/.f64 (pow.f64 d 2) (*.f64 h (pow.f64 M 2))))) l): 3 points increase in error, 6 points decrease in error
      (/.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (pow.f64 D 2) (/.f64 (pow.f64 d 2) (*.f64 h (pow.f64 M 2)))) 1/4)) l): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 h (pow.f64 M 2))) (pow.f64 d 2))) 1/4) l): 2 points increase in error, 4 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 h (pow.f64 M 2))) (pow.f64 d 2)))) l): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 1/4 (*.f64 (pow.f64 D 2) (*.f64 h (pow.f64 M 2)))) (pow.f64 d 2))) l): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-/l/_binary64 (/.f64 (*.f64 1/4 (*.f64 (pow.f64 D 2) (*.f64 h (pow.f64 M 2)))) (*.f64 l (pow.f64 d 2)))): 9 points increase in error, 3 points decrease in error
      (/.f64 (*.f64 1/4 (*.f64 (pow.f64 D 2) (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 M 2) h)))) (*.f64 l (pow.f64 d 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (*.f64 l (pow.f64 d 2))))): 0 points increase in error, 0 points decrease in error
  3. Recombined 3 regimes into one program.

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+307}:\\ \;\;\;\;w0 \cdot \sqrt{1 + D \cdot \frac{D \cdot \left({\left(\frac{M}{d}\right)}^{2} \cdot \left(h \cdot 0.25\right)\right)}{-\ell}}\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{-42}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(M \cdot \left(M \cdot h\right)\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot -0.25\right)}{\ell}}\\ \end{array} \]

Alternatives

Alternative 1
Error9.6
Cost21188
\[\begin{array}{l} t_0 := 1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{+307}:\\ \;\;\;\;w0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(M \cdot \left(M \cdot h\right)\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot -0.25\right)}{\ell}}\\ \end{array} \]
Alternative 2
Error10.8
Cost13824
\[w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}}{\ell}} \]
Alternative 3
Error13.6
Cost64
\[w0 \]

Error

Reproduce

herbie shell --seed 2022295 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  :precision binary64
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))