Henrywood and Agarwal, Equation (12)

Average Error: 25.9 → 18.1

Time: 14.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := 1 - {\left(\frac{\ell}{h \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)}\right)}^{-1}\\ \mathbf{if}\;h \leq -1.733687270610834 \cdot 10^{-182}:\\ \;\;\;\;\left(\sqrt{d \cdot \frac{1}{h}} \cdot \frac{t_0}{\sqrt{-\ell}}\right) \cdot t_1\\ \mathbf{elif}\;h \leq -5.96337956912148 \cdot 10^{-309}:\\ \;\;\;\;t_1 \cdot \left(\left(t_0 \cdot \sqrt{\frac{-1}{h}}\right) \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right)\\ \mathbf{elif}\;h \leq 1.2123265266183649 \cdot 10^{-139}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (*
  (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
  (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))

↓

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (- d)))
        (t_1
         (-
          1.0
          (pow (/ l (* h (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0)))) -1.0))))
   (if (<= h -1.733687270610834e-182)
     (* (* (sqrt (* d (/ 1.0 h))) (/ t_0 (sqrt (- l)))) t_1)
     (if (<= h -5.96337956912148e-309)
       (* t_1 (* (* t_0 (sqrt (/ -1.0 h))) (pow (/ d l) 0.5)))
       (if (<= h 1.2123265266183649e-139)
         (* d (sqrt (/ (/ 1.0 l) h)))
         (* t_1 (* (pow (/ d h) 0.5) (/ (sqrt d) (sqrt l)))))))))

C

double code(double d, double h, double l, double M, double D) {
	return (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

↓

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(-d);
	double t_1 = 1.0 - pow((l / (h * (0.5 * pow(((M * D) / (d * 2.0)), 2.0)))), -1.0);
	double tmp;
	if (h <= -1.733687270610834e-182) {
		tmp = (sqrt((d * (1.0 / h))) * (t_0 / sqrt(-l))) * t_1;
	} else if (h <= -5.96337956912148e-309) {
		tmp = t_1 * ((t_0 * sqrt((-1.0 / h))) * pow((d / l), 0.5));
	} else if (h <= 1.2123265266183649e-139) {
		tmp = d * sqrt(((1.0 / l) / h));
	} else {
		tmp = t_1 * (pow((d / h), 0.5) * (sqrt(d) / sqrt(l)));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
end function

↓

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(-d)
    t_1 = 1.0d0 - ((l / (h * (0.5d0 * (((m * d_1) / (d * 2.0d0)) ** 2.0d0)))) ** (-1.0d0))
    if (h <= (-1.733687270610834d-182)) then
        tmp = (sqrt((d * (1.0d0 / h))) * (t_0 / sqrt(-l))) * t_1
    else if (h <= (-5.96337956912148d-309)) then
        tmp = t_1 * ((t_0 * sqrt(((-1.0d0) / h))) * ((d / l) ** 0.5d0))
    else if (h <= 1.2123265266183649d-139) then
        tmp = d * sqrt(((1.0d0 / l) / h))
    else
        tmp = t_1 * (((d / h) ** 0.5d0) * (sqrt(d) / sqrt(l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

↓

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(-d);
	double t_1 = 1.0 - Math.pow((l / (h * (0.5 * Math.pow(((M * D) / (d * 2.0)), 2.0)))), -1.0);
	double tmp;
	if (h <= -1.733687270610834e-182) {
		tmp = (Math.sqrt((d * (1.0 / h))) * (t_0 / Math.sqrt(-l))) * t_1;
	} else if (h <= -5.96337956912148e-309) {
		tmp = t_1 * ((t_0 * Math.sqrt((-1.0 / h))) * Math.pow((d / l), 0.5));
	} else if (h <= 1.2123265266183649e-139) {
		tmp = d * Math.sqrt(((1.0 / l) / h));
	} else {
		tmp = t_1 * (Math.pow((d / h), 0.5) * (Math.sqrt(d) / Math.sqrt(l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	return (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))

↓

def code(d, h, l, M, D):
	t_0 = math.sqrt(-d)
	t_1 = 1.0 - math.pow((l / (h * (0.5 * math.pow(((M * D) / (d * 2.0)), 2.0)))), -1.0)
	tmp = 0
	if h <= -1.733687270610834e-182:
		tmp = (math.sqrt((d * (1.0 / h))) * (t_0 / math.sqrt(-l))) * t_1
	elif h <= -5.96337956912148e-309:
		tmp = t_1 * ((t_0 * math.sqrt((-1.0 / h))) * math.pow((d / l), 0.5))
	elif h <= 1.2123265266183649e-139:
		tmp = d * math.sqrt(((1.0 / l) / h))
	else:
		tmp = t_1 * (math.pow((d / h), 0.5) * (math.sqrt(d) / math.sqrt(l)))
	return tmp

Julia

function code(d, h, l, M, D)
	return Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
end

↓

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(-d))
	t_1 = Float64(1.0 - (Float64(l / Float64(h * Float64(0.5 * (Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0)))) ^ -1.0))
	tmp = 0.0
	if (h <= -1.733687270610834e-182)
		tmp = Float64(Float64(sqrt(Float64(d * Float64(1.0 / h))) * Float64(t_0 / sqrt(Float64(-l)))) * t_1);
	elseif (h <= -5.96337956912148e-309)
		tmp = Float64(t_1 * Float64(Float64(t_0 * sqrt(Float64(-1.0 / h))) * (Float64(d / l) ^ 0.5)));
	elseif (h <= 1.2123265266183649e-139)
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / h)));
	else
		tmp = Float64(t_1 * Float64((Float64(d / h) ^ 0.5) * Float64(sqrt(d) / sqrt(l))));
	end
	return tmp
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
end

↓

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(-d);
	t_1 = 1.0 - ((l / (h * (0.5 * (((M * D) / (d * 2.0)) ^ 2.0)))) ^ -1.0);
	tmp = 0.0;
	if (h <= -1.733687270610834e-182)
		tmp = (sqrt((d * (1.0 / h))) * (t_0 / sqrt(-l))) * t_1;
	elseif (h <= -5.96337956912148e-309)
		tmp = t_1 * ((t_0 * sqrt((-1.0 / h))) * ((d / l) ^ 0.5));
	elseif (h <= 1.2123265266183649e-139)
		tmp = d * sqrt(((1.0 / l) / h));
	else
		tmp = t_1 * (((d / h) ^ 0.5) * (sqrt(d) / sqrt(l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Power[N[(l / N[(h * N[(0.5 * N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -1.733687270610834e-182], N[(N[(N[Sqrt[N[(d * N[(1.0 / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[h, -5.96337956912148e-309], N[(t$95$1 * N[(N[(t$95$0 * N[Sqrt[N[(-1.0 / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 1.2123265266183649e-139], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Derivation

1. Split input into 4 regimes

2. if h < -1.7336872706108341e-182

   1. Initial program 24.0

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

   2. Applied egg-rr 23.3

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

   3. Applied egg-rr 22.8

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

   4. Taylor expanded in h around -inf 22.7

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

   5. Simplified 22.9

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

   6. Applied egg-rr 16.9

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

   if -1.7336872706108341e-182 < h < -5.9633795691214818e-309

   1. Initial program 36.3

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

   2. Applied egg-rr 37.2

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

   3. Applied egg-rr 36.3

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

   4. Taylor expanded in h around -inf 23.6

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

   5. Simplified 36.3

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

   6. Applied egg-rr 19.6

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

   if -5.9633795691214818e-309 < h < 1.21232652661836487e-139

   1. Initial program 31.7

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

   2. Applied egg-rr 32.4

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

   3. Applied egg-rr 31.7

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

   4. Taylor expanded in d around inf 24.2

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   5. Simplified 24.2

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]

   if 1.21232652661836487e-139 < h

   1. Initial program 23.5

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

   2. Applied egg-rr 23.0

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

   3. Applied egg-rr 22.9

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

   4. Applied egg-rr 16.8

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - {\left(\frac{\ell}{h \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right)}^{-1}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 18.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.733687270610834 \cdot 10^{-182}:\\ \;\;\;\;\left(\sqrt{d \cdot \frac{1}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - {\left(\frac{\ell}{h \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)}\right)}^{-1}\right)\\ \mathbf{elif}\;h \leq -5.96337956912148 \cdot 10^{-309}:\\ \;\;\;\;\left(1 - {\left(\frac{\ell}{h \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)}\right)}^{-1}\right) \cdot \left(\left(\sqrt{-d} \cdot \sqrt{\frac{-1}{h}}\right) \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right)\\ \mathbf{elif}\;h \leq 1.2123265266183649 \cdot 10^{-139}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - {\left(\frac{\ell}{h \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)}\right)}^{-1}\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  :precision binary64
  (* (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))) (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))