Henrywood and Agarwal, Equation (12)

Average Accuracy: 58.2% → 71.4%

Time: 1.1min

Precision: binary64

Cost: 40596

\[ \begin{array}{c}[M, D] = \mathsf{sort}([M, D])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := \frac{t_0}{\sqrt{-\ell}}\\ t_4 := \left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\\ \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+157}:\\ \;\;\;\;\left(t_2 \cdot t_3\right) \cdot \left(1 - {\left(\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{h \cdot 0.5}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;\ell \leq -2.5 \cdot 10^{-93}:\\ \;\;\;\;\left(\frac{t_0}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_2 \cdot \left(t_3 \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\frac{D}{\ell} \cdot \left(M \cdot D\right)}{d} \cdot \frac{M}{\frac{d}{h}}\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{-12}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot t_1\right) \cdot \left(1 - {\left(\frac{\sqrt{h}}{\sqrt{\ell}} \cdot t_4\right)}^{2}\right)\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+198}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_2 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - {\left(t_4 \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \end{array} \]

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 (sqrt (/ d l)))
        (t_2 (sqrt (/ d h)))
        (t_3 (/ t_0 (sqrt (- l))))
        (t_4 (* (* 0.5 (/ (* M D) d)) (sqrt 0.5))))
   (if (<= l -1.8e+157)
     (*
      (* t_2 t_3)
      (- 1.0 (pow (* (* M (* 0.5 (/ D d))) (sqrt (/ (* h 0.5) l))) 2.0)))
     (if (<= l -2.5e-93)
       (*
        (* (/ t_0 (sqrt (- h))) (pow (/ d l) 0.5))
        (- 1.0 (* (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0)) (/ h l))))
       (if (<= l -5e-310)
         (*
          t_2
          (*
           t_3
           (-
            1.0
            (* 0.5 (* 0.25 (* (/ (* (/ D l) (* M D)) d) (/ M (/ d h))))))))
         (if (<= l 1.15e-12)
           (*
            (* (pow (/ d h) 0.5) t_1)
            (- 1.0 (pow (* (/ (sqrt h) (sqrt l)) t_4) 2.0)))
           (if (<= l 1.15e+198)
             (*
              (/ (sqrt d) (sqrt h))
              (*
               t_1
               (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D d) (/ M 2.0)) 2.0))))))
             (*
              (* t_2 (/ (sqrt d) (sqrt l)))
              (- 1.0 (pow (* t_4 (sqrt (/ h l))) 2.0))))))))))

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 = sqrt((d / l));
	double t_2 = sqrt((d / h));
	double t_3 = t_0 / sqrt(-l);
	double t_4 = (0.5 * ((M * D) / d)) * sqrt(0.5);
	double tmp;
	if (l <= -1.8e+157) {
		tmp = (t_2 * t_3) * (1.0 - pow(((M * (0.5 * (D / d))) * sqrt(((h * 0.5) / l))), 2.0));
	} else if (l <= -2.5e-93) {
		tmp = ((t_0 / sqrt(-h)) * pow((d / l), 0.5)) * (1.0 - ((0.5 * pow(((M * D) / (d * 2.0)), 2.0)) * (h / l)));
	} else if (l <= -5e-310) {
		tmp = t_2 * (t_3 * (1.0 - (0.5 * (0.25 * ((((D / l) * (M * D)) / d) * (M / (d / h)))))));
	} else if (l <= 1.15e-12) {
		tmp = (pow((d / h), 0.5) * t_1) * (1.0 - pow(((sqrt(h) / sqrt(l)) * t_4), 2.0));
	} else if (l <= 1.15e+198) {
		tmp = (sqrt(d) / sqrt(h)) * (t_1 * (1.0 - (0.5 * ((h / l) * pow(((D / d) * (M / 2.0)), 2.0)))));
	} else {
		tmp = (t_2 * (sqrt(d) / sqrt(l))) * (1.0 - pow((t_4 * sqrt((h / l))), 2.0));
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = sqrt(-d)
    t_1 = sqrt((d / l))
    t_2 = sqrt((d / h))
    t_3 = t_0 / sqrt(-l)
    t_4 = (0.5d0 * ((m * d_1) / d)) * sqrt(0.5d0)
    if (l <= (-1.8d+157)) then
        tmp = (t_2 * t_3) * (1.0d0 - (((m * (0.5d0 * (d_1 / d))) * sqrt(((h * 0.5d0) / l))) ** 2.0d0))
    else if (l <= (-2.5d-93)) then
        tmp = ((t_0 / sqrt(-h)) * ((d / l) ** 0.5d0)) * (1.0d0 - ((0.5d0 * (((m * d_1) / (d * 2.0d0)) ** 2.0d0)) * (h / l)))
    else if (l <= (-5d-310)) then
        tmp = t_2 * (t_3 * (1.0d0 - (0.5d0 * (0.25d0 * ((((d_1 / l) * (m * d_1)) / d) * (m / (d / h)))))))
    else if (l <= 1.15d-12) then
        tmp = (((d / h) ** 0.5d0) * t_1) * (1.0d0 - (((sqrt(h) / sqrt(l)) * t_4) ** 2.0d0))
    else if (l <= 1.15d+198) then
        tmp = (sqrt(d) / sqrt(h)) * (t_1 * (1.0d0 - (0.5d0 * ((h / l) * (((d_1 / d) * (m / 2.0d0)) ** 2.0d0)))))
    else
        tmp = (t_2 * (sqrt(d) / sqrt(l))) * (1.0d0 - ((t_4 * sqrt((h / l))) ** 2.0d0))
    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 = Math.sqrt((d / l));
	double t_2 = Math.sqrt((d / h));
	double t_3 = t_0 / Math.sqrt(-l);
	double t_4 = (0.5 * ((M * D) / d)) * Math.sqrt(0.5);
	double tmp;
	if (l <= -1.8e+157) {
		tmp = (t_2 * t_3) * (1.0 - Math.pow(((M * (0.5 * (D / d))) * Math.sqrt(((h * 0.5) / l))), 2.0));
	} else if (l <= -2.5e-93) {
		tmp = ((t_0 / Math.sqrt(-h)) * Math.pow((d / l), 0.5)) * (1.0 - ((0.5 * Math.pow(((M * D) / (d * 2.0)), 2.0)) * (h / l)));
	} else if (l <= -5e-310) {
		tmp = t_2 * (t_3 * (1.0 - (0.5 * (0.25 * ((((D / l) * (M * D)) / d) * (M / (d / h)))))));
	} else if (l <= 1.15e-12) {
		tmp = (Math.pow((d / h), 0.5) * t_1) * (1.0 - Math.pow(((Math.sqrt(h) / Math.sqrt(l)) * t_4), 2.0));
	} else if (l <= 1.15e+198) {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * (t_1 * (1.0 - (0.5 * ((h / l) * Math.pow(((D / d) * (M / 2.0)), 2.0)))));
	} else {
		tmp = (t_2 * (Math.sqrt(d) / Math.sqrt(l))) * (1.0 - Math.pow((t_4 * Math.sqrt((h / l))), 2.0));
	}
	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 = math.sqrt((d / l))
	t_2 = math.sqrt((d / h))
	t_3 = t_0 / math.sqrt(-l)
	t_4 = (0.5 * ((M * D) / d)) * math.sqrt(0.5)
	tmp = 0
	if l <= -1.8e+157:
		tmp = (t_2 * t_3) * (1.0 - math.pow(((M * (0.5 * (D / d))) * math.sqrt(((h * 0.5) / l))), 2.0))
	elif l <= -2.5e-93:
		tmp = ((t_0 / math.sqrt(-h)) * math.pow((d / l), 0.5)) * (1.0 - ((0.5 * math.pow(((M * D) / (d * 2.0)), 2.0)) * (h / l)))
	elif l <= -5e-310:
		tmp = t_2 * (t_3 * (1.0 - (0.5 * (0.25 * ((((D / l) * (M * D)) / d) * (M / (d / h)))))))
	elif l <= 1.15e-12:
		tmp = (math.pow((d / h), 0.5) * t_1) * (1.0 - math.pow(((math.sqrt(h) / math.sqrt(l)) * t_4), 2.0))
	elif l <= 1.15e+198:
		tmp = (math.sqrt(d) / math.sqrt(h)) * (t_1 * (1.0 - (0.5 * ((h / l) * math.pow(((D / d) * (M / 2.0)), 2.0)))))
	else:
		tmp = (t_2 * (math.sqrt(d) / math.sqrt(l))) * (1.0 - math.pow((t_4 * math.sqrt((h / l))), 2.0))
	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 = sqrt(Float64(d / l))
	t_2 = sqrt(Float64(d / h))
	t_3 = Float64(t_0 / sqrt(Float64(-l)))
	t_4 = Float64(Float64(0.5 * Float64(Float64(M * D) / d)) * sqrt(0.5))
	tmp = 0.0
	if (l <= -1.8e+157)
		tmp = Float64(Float64(t_2 * t_3) * Float64(1.0 - (Float64(Float64(M * Float64(0.5 * Float64(D / d))) * sqrt(Float64(Float64(h * 0.5) / l))) ^ 2.0)));
	elseif (l <= -2.5e-93)
		tmp = Float64(Float64(Float64(t_0 / sqrt(Float64(-h))) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(0.5 * (Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0)) * Float64(h / l))));
	elseif (l <= -5e-310)
		tmp = Float64(t_2 * Float64(t_3 * Float64(1.0 - Float64(0.5 * Float64(0.25 * Float64(Float64(Float64(Float64(D / l) * Float64(M * D)) / d) * Float64(M / Float64(d / h))))))));
	elseif (l <= 1.15e-12)
		tmp = Float64(Float64((Float64(d / h) ^ 0.5) * t_1) * Float64(1.0 - (Float64(Float64(sqrt(h) / sqrt(l)) * t_4) ^ 2.0)));
	elseif (l <= 1.15e+198)
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(t_1 * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0))))));
	else
		tmp = Float64(Float64(t_2 * Float64(sqrt(d) / sqrt(l))) * Float64(1.0 - (Float64(t_4 * sqrt(Float64(h / l))) ^ 2.0)));
	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 = sqrt((d / l));
	t_2 = sqrt((d / h));
	t_3 = t_0 / sqrt(-l);
	t_4 = (0.5 * ((M * D) / d)) * sqrt(0.5);
	tmp = 0.0;
	if (l <= -1.8e+157)
		tmp = (t_2 * t_3) * (1.0 - (((M * (0.5 * (D / d))) * sqrt(((h * 0.5) / l))) ^ 2.0));
	elseif (l <= -2.5e-93)
		tmp = ((t_0 / sqrt(-h)) * ((d / l) ^ 0.5)) * (1.0 - ((0.5 * (((M * D) / (d * 2.0)) ^ 2.0)) * (h / l)));
	elseif (l <= -5e-310)
		tmp = t_2 * (t_3 * (1.0 - (0.5 * (0.25 * ((((D / l) * (M * D)) / d) * (M / (d / h)))))));
	elseif (l <= 1.15e-12)
		tmp = (((d / h) ^ 0.5) * t_1) * (1.0 - (((sqrt(h) / sqrt(l)) * t_4) ^ 2.0));
	elseif (l <= 1.15e+198)
		tmp = (sqrt(d) / sqrt(h)) * (t_1 * (1.0 - (0.5 * ((h / l) * (((D / d) * (M / 2.0)) ^ 2.0)))));
	else
		tmp = (t_2 * (sqrt(d) / sqrt(l))) * (1.0 - ((t_4 * sqrt((h / l))) ^ 2.0));
	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[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(0.5 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.8e+157], N[(N[(t$95$2 * t$95$3), $MachinePrecision] * N[(1.0 - N[Power[N[(N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(h * 0.5), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2.5e-93], N[(N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.5 * N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(t$95$2 * N[(t$95$3 * N[(1.0 - N[(0.5 * N[(0.25 * N[(N[(N[(N[(D / l), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(M / N[(d / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.15e-12], N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(1.0 - N[Power[N[(N[(N[Sqrt[h], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.15e+198], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Power[N[(t$95$4 * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if l < -1.80000000000000012e157

   1. Initial program 53.2%

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

   2. Simplified 53.2%

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

      [Start] 53.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 53.2	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 53.2	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 53.2	\[ \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 53.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      *-commutative [=>] 53.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)} \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 53.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)}\right) \]
      times-frac [=>] 53.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 53.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(\color{blue}{0.5} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied egg-rr 62.5%

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

      [Start] 53.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      frac-2neg [=>] 53.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{-d}{-\ell}}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      sqrt-div [=>] 62.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   4. Applied egg-rr 67.7%

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

      [Start] 62.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      add-sqr-sqrt [=>] 62.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - \color{blue}{\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)}}\right) \]
      pow2 [=>] 62.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)}\right)}^{2}}\right) \]
      sqrt-prod [=>] 62.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{0.5 \cdot \frac{h}{\ell}}\right)}}^{2}\right) \]
      unpow2 [=>] 62.5	\[ \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - {\left(\sqrt{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}} \cdot \sqrt{0.5 \cdot \frac{h}{\ell}}\right)}^{2}\right) \]
      sqrt-prod [=>] 43.2	\[ \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - {\left(\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)} \cdot \sqrt{0.5 \cdot \frac{h}{\ell}}\right)}^{2}\right) \]
      add-sqr-sqrt [<=] 67.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{0.5 \cdot \frac{h}{\ell}}\right)}^{2}\right) \]
      div-inv [=>] 67.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - {\left(\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right) \cdot \sqrt{0.5 \cdot \frac{h}{\ell}}\right)}^{2}\right) \]
      metadata-eval [=>] 67.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - {\left(\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5 \cdot \frac{h}{\ell}}\right)}^{2}\right) \]
      associate-*l* [=>] 67.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - {\left(\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)} \cdot \sqrt{0.5 \cdot \frac{h}{\ell}}\right)}^{2}\right) \]
      associate-*r/ [=>] 67.7	\[ \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - {\left(\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\color{blue}{\frac{0.5 \cdot h}{\ell}}}\right)}^{2}\right) \]

   if -1.80000000000000012e157 < l < -2.49999999999999997e-93

   1. Initial program 67.3%

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

   2. Applied egg-rr 78.0%

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

      [Start] 67.3	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      metadata-eval [=>] 67.3	\[ \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      unpow1/2 [=>] 67.3	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      frac-2neg [=>] 67.3	\[ \left(\sqrt{\color{blue}{\frac{-d}{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      sqrt-div [=>] 78.0	\[ \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   if -2.49999999999999997e-93 < l < -4.999999999999985e-310

   1. Initial program 49.8%

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

   2. Simplified 48.6%

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

      [Start] 49.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 49.1	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 49.1	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 49.1	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 49.1	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 49.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      associate-*l* [=>] 49.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right) \]
      metadata-eval [=>] 49.1	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      times-frac [=>] 48.6	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   3. Taylor expanded in M around 0 25.7%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)}\right)\right) \]

   4. Simplified 34.2%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(0.25 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)\right)}\right)\right) \]
      Proof

      [Start] 25.7	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}\right)\right)\right) \]
      *-commutative [=>] 25.7	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{\ell \cdot {d}^{2}}\right)\right)\right) \]
      times-frac [=>] 34.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)}\right)\right)\right) \]
      unpow2 [=>] 34.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{h \cdot {M}^{2}}{{d}^{2}}\right)\right)\right)\right) \]
      *-commutative [<=] 34.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2}}\right)\right)\right)\right) \]
      unpow2 [=>] 34.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)\right)\right)\right) \]
      unpow2 [=>] 34.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d \cdot d}}\right)\right)\right)\right) \]

   5. Applied egg-rr 39.8%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\frac{\left(\frac{D}{\ell} \cdot D\right) \cdot \left(M \cdot M\right)}{\frac{d}{\frac{h}{d}}}}\right)\right)\right) \]
      Proof

      [Start] 34.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)\right)\right)\right) \]
      associate-/l* [=>] 33.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D \cdot D}{\ell} \cdot \color{blue}{\frac{M \cdot M}{\frac{d \cdot d}{h}}}\right)\right)\right)\right) \]
      associate-*r/ [=>] 33.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\frac{\frac{D \cdot D}{\ell} \cdot \left(M \cdot M\right)}{\frac{d \cdot d}{h}}}\right)\right)\right) \]
      associate-/l* [=>] 34.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{\color{blue}{\frac{D}{\frac{\ell}{D}}} \cdot \left(M \cdot M\right)}{\frac{d \cdot d}{h}}\right)\right)\right) \]
      associate-/r/ [=>] 34.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{\color{blue}{\left(\frac{D}{\ell} \cdot D\right)} \cdot \left(M \cdot M\right)}{\frac{d \cdot d}{h}}\right)\right)\right) \]
      associate-/l* [=>] 39.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{\left(\frac{D}{\ell} \cdot D\right) \cdot \left(M \cdot M\right)}{\color{blue}{\frac{d}{\frac{h}{d}}}}\right)\right)\right) \]

   6. Applied egg-rr 53.2%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{\frac{D}{\ell} \cdot \left(D \cdot M\right)}{d} \cdot \frac{M}{\frac{d}{h}}\right)}\right)\right)\right) \]
      Proof

      [Start] 39.8	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{\left(\frac{D}{\ell} \cdot D\right) \cdot \left(M \cdot M\right)}{\frac{d}{\frac{h}{d}}}\right)\right)\right) \]
      associate-*r* [=>] 42.7	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{\color{blue}{\left(\left(\frac{D}{\ell} \cdot D\right) \cdot M\right) \cdot M}}{\frac{d}{\frac{h}{d}}}\right)\right)\right) \]
      div-inv [=>] 42.7	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{\left(\left(\frac{D}{\ell} \cdot D\right) \cdot M\right) \cdot M}{\color{blue}{d \cdot \frac{1}{\frac{h}{d}}}}\right)\right)\right) \]
      times-frac [=>] 49.3	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \color{blue}{\left(\frac{\left(\frac{D}{\ell} \cdot D\right) \cdot M}{d} \cdot \frac{M}{\frac{1}{\frac{h}{d}}}\right)}\right)\right)\right) \]
      associate-*l* [=>] 52.9	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\color{blue}{\frac{D}{\ell} \cdot \left(D \cdot M\right)}}{d} \cdot \frac{M}{\frac{1}{\frac{h}{d}}}\right)\right)\right)\right) \]
      clear-num [<=] 53.2	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\frac{D}{\ell} \cdot \left(D \cdot M\right)}{d} \cdot \frac{M}{\color{blue}{\frac{d}{h}}}\right)\right)\right)\right) \]

   7. Applied egg-rr 73.8%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\frac{D}{\ell} \cdot \left(D \cdot M\right)}{d} \cdot \frac{M}{\frac{d}{h}}\right)\right)\right)\right) \]
      Proof

      [Start] 49.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      frac-2neg [=>] 49.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{-d}{-\ell}}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]
      sqrt-div [=>] 59.9	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}}\right) \cdot \left(1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \]

   if -4.999999999999985e-310 < l < 1.14999999999999995e-12

   1. Initial program 54.8%

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

   2. Applied egg-rr 55.9%

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

      [Start] 54.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      add-sqr-sqrt [=>] 54.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}} \cdot \sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}}\right) \]
      pow2 [=>] 54.8	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

   3. Applied egg-rr 40.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)} - 1\right)}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      Proof

      [Start] 55.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      expm1-log1p-u [=>] 53.1	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right)}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      expm1-udef [=>] 40.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} - 1\right)}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      metadata-eval [=>] 40.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(e^{\mathsf{log1p}\left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right)} - 1\right)\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      unpow1/2 [=>] 40.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{d}{\ell}}}\right)} - 1\right)\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]

   4. Simplified 55.9%

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

      [Start] 40.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)} - 1\right)\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      expm1-def [=>] 53.1	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      expm1-log1p [=>] 55.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]

   5. Applied egg-rr 65.2%

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

      [Start] 55.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      sqrt-div [=>] 65.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\color{blue}{\frac{\sqrt{h}}{\sqrt{\ell}}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      clear-num [=>] 65.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\color{blue}{\frac{1}{\frac{\sqrt{\ell}}{\sqrt{h}}}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]

   6. Simplified 65.2%

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

      [Start] 65.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{1}{\frac{\sqrt{\ell}}{\sqrt{h}}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      associate-/l* [<=] 65.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\color{blue}{\frac{1 \cdot \sqrt{h}}{\sqrt{\ell}}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      *-lft-identity [=>] 65.2	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{\color{blue}{\sqrt{h}}}{\sqrt{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]

   if 1.14999999999999995e-12 < l < 1.15e198

   1. Initial program 64.1%

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

   2. Simplified 62.5%

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

      [Start] 64.1	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      associate-*l* [=>] 64.0	\[ \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} \]
      metadata-eval [=>] 64.0	\[ {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 64.0	\[ \color{blue}{\sqrt{\frac{d}{h}}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      metadata-eval [=>] 64.0	\[ \sqrt{\frac{d}{h}} \cdot \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      unpow1/2 [=>] 64.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) \]
      associate-*l* [=>] 64.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right) \]
      metadata-eval [=>] 64.0	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \color{blue}{0.5} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      times-frac [=>] 62.5	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   3. Applied egg-rr 74.1%

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

      [Start] 62.5	\[ \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      sqrt-div [=>] 74.2	\[ \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      div-inv [=>] 74.1	\[ \color{blue}{\left(\sqrt{d} \cdot \frac{1}{\sqrt{h}}\right)} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   4. Simplified 74.2%

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

      [Start] 74.1	\[ \left(\sqrt{d} \cdot \frac{1}{\sqrt{h}}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      associate-*r/ [=>] 74.2	\[ \color{blue}{\frac{\sqrt{d} \cdot 1}{\sqrt{h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
      *-rgt-identity [=>] 74.2	\[ \frac{\color{blue}{\sqrt{d}}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   if 1.15e198 < l

   1. Initial program 50.9%

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

   2. Applied egg-rr 53.6%

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

      [Start] 50.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
      add-sqr-sqrt [=>] 50.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}} \cdot \sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}}\right) \]
      pow2 [=>] 50.9	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

   3. Applied egg-rr 17.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)} - 1\right)}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      Proof

      [Start] 53.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      expm1-log1p-u [=>] 53.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right)}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      expm1-udef [=>] 17.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} - 1\right)}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      metadata-eval [=>] 17.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(e^{\mathsf{log1p}\left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right)} - 1\right)\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      unpow1/2 [=>] 17.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{d}{\ell}}}\right)} - 1\right)\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]

   4. Simplified 53.6%

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

      [Start] 17.5	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)} - 1\right)\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      expm1-def [=>] 53.4	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      expm1-log1p [=>] 53.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]

   5. Applied egg-rr 38.5%

      \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h}}\right)} - 1\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      Proof

      [Start] 53.6	\[ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      expm1-log1p-u [=>] 51.7	\[ \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      expm1-udef [=>] 38.5	\[ \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} - 1\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      metadata-eval [=>] 38.5	\[ \left(\left(e^{\mathsf{log1p}\left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}}\right)} - 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      unpow1/2 [=>] 38.5	\[ \left(\left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{d}{h}}}\right)} - 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]

   6. Simplified 53.6%

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

      [Start] 38.5	\[ \left(\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h}}\right)} - 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      expm1-def [=>] 51.7	\[ \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h}}\right)\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      expm1-log1p [=>] 53.6	\[ \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]

   7. Applied egg-rr 66.3%

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

      [Start] 53.6	\[ \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      sqrt-div [=>] 66.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      div-inv [=>] 66.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left(\sqrt{d} \cdot \frac{1}{\sqrt{\ell}}\right)}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]

   8. Simplified 66.3%

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

      [Start] 66.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \left(\sqrt{d} \cdot \frac{1}{\sqrt{\ell}}\right)\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      associate-*r/ [=>] 66.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{d} \cdot 1}{\sqrt{\ell}}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]
      *-rgt-identity [=>] 66.3	\[ \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{\ell}}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \]

3. Recombined 6 regimes into one program.

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+157}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - {\left(\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{h \cdot 0.5}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;\ell \leq -2.5 \cdot 10^{-93}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\frac{D}{\ell} \cdot \left(M \cdot D\right)}{d} \cdot \frac{M}{\frac{d}{h}}\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{-12}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - {\left(\frac{\sqrt{h}}{\sqrt{\ell}} \cdot \left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right)\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+198}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - {\left(\left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	71.9%
Cost	40596

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \frac{\sqrt{d}}{\sqrt{\ell}}\\ t_2 := 1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\frac{D}{\ell} \cdot \left(M \cdot D\right)}{d} \cdot \frac{M}{\frac{d}{h}}\right)\right)\\ t_3 := \sqrt{\frac{d}{h}}\\ t_4 := \frac{t_0}{\sqrt{-\ell}}\\ \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+161}:\\ \;\;\;\;\left(t_3 \cdot t_4\right) \cdot \left(1 - {\left(\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \sqrt{\frac{h \cdot 0.5}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;\ell \leq -8 \cdot 10^{-93}:\\ \;\;\;\;\left(\frac{t_0}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_3 \cdot \left(t_4 \cdot t_2\right)\\ \mathbf{elif}\;\ell \leq 2.15 \cdot 10^{-181}:\\ \;\;\;\;t_3 \cdot \left(t_2 \cdot t_1\right)\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+198}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_3 \cdot t_1\right) \cdot \left(1 - {\left(\left(\left(0.5 \cdot \frac{M \cdot D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \end{array} \]

Alternative 2
Accuracy	68.9%
Cost	27928

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := 1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\frac{D}{\ell} \cdot \left(M \cdot D\right)}{d} \cdot \frac{M}{\frac{d}{h}}\right)\right)\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := \sqrt{\frac{d}{\ell}}\\ t_4 := \frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t_3 \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{if}\;h \leq -9.6 \cdot 10^{+219}:\\ \;\;\;\;\left(t_2 \cdot t_3\right) \cdot \left(1 - \frac{\left(h \cdot 0.5\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;h \leq -1.38 \cdot 10^{-151}:\\ \;\;\;\;t_2 \cdot \left(\frac{t_0}{\sqrt{-\ell}} \cdot t_1\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{t_3}{\frac{\sqrt{-h}}{t_0}}\\ \mathbf{elif}\;h \leq 3.5 \cdot 10^{-41}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;h \leq 1.36 \cdot 10^{+179}:\\ \;\;\;\;t_2 \cdot \left(t_1 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{elif}\;h \leq 7 \cdot 10^{+232}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(t_3 \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{M \cdot \frac{D}{\ell}}{\frac{d}{h} \cdot \frac{d}{M \cdot D}}\right)\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	68.9%
Cost	27928

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := 1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\frac{D}{\ell} \cdot \left(M \cdot D\right)}{d} \cdot \frac{M}{\frac{d}{h}}\right)\right)\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := \sqrt{\frac{d}{\ell}}\\ t_4 := t_3 \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{if}\;h \leq -9.6 \cdot 10^{+219}:\\ \;\;\;\;\left(t_2 \cdot t_3\right) \cdot \left(1 - \frac{\left(h \cdot 0.5\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;h \leq -1.7 \cdot 10^{-151}:\\ \;\;\;\;t_2 \cdot \left(\frac{t_0}{\sqrt{-\ell}} \cdot t_1\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{t_3}{\frac{\sqrt{-h}}{t_0}}\\ \mathbf{elif}\;h \leq 3.8 \cdot 10^{-41}:\\ \;\;\;\;t_4 \cdot \frac{1}{\frac{\sqrt{h}}{\sqrt{d}}}\\ \mathbf{elif}\;h \leq 1.5 \cdot 10^{+184}:\\ \;\;\;\;t_2 \cdot \left(t_1 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{elif}\;h \leq 8.5 \cdot 10^{+233}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot t_4\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(t_3 \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{M \cdot \frac{D}{\ell}}{\frac{d}{h} \cdot \frac{d}{M \cdot D}}\right)\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	71.2%
Cost	27924

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\\ t_3 := \sqrt{-d}\\ t_4 := 1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\frac{D}{\ell} \cdot \left(M \cdot D\right)}{d} \cdot \frac{M}{\frac{d}{h}}\right)\right)\\ t_5 := \frac{\sqrt[3]{h}}{\ell}\\ \mathbf{if}\;d \leq -5 \cdot 10^{+62}:\\ \;\;\;\;\frac{t_3}{\sqrt{-h}} \cdot \left(t_0 \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot t_2\right)\right)\right)\\ \mathbf{elif}\;d \leq -1.1 \cdot 10^{-197}:\\ \;\;\;\;t_1 \cdot \left(\frac{t_3}{\sqrt{-\ell}} \cdot t_4\right)\\ \mathbf{elif}\;d \leq -1.95 \cdot 10^{-308}:\\ \;\;\;\;\left(t_5 \cdot \sqrt{t_5}\right) \cdot \left(\left(M \cdot \frac{D \cdot D}{\frac{d}{M}}\right) \cdot 0.125\right) - d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;d \leq 8.2 \cdot 10^{-196}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;d \leq 8.2 \cdot 10^{-138}:\\ \;\;\;\;\left(t_0 \cdot \left(\sqrt{d} \cdot \sqrt{\frac{1}{h}}\right)\right) \cdot \left(1 - t_2 \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq 50:\\ \;\;\;\;t_1 \cdot \left(t_4 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{+24}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{h}{\ell} \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d \cdot d}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 5
Accuracy	64.2%
Cost	21980

\[\begin{array}{l} t_0 := 1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\frac{D}{\ell} \cdot \left(M \cdot D\right)}{d} \cdot \frac{M}{\frac{d}{h}}\right)\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \left(t_1 \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D}{\frac{d}{D}} \cdot \frac{h \cdot \frac{M}{\frac{\ell}{M}}}{d}\right)\right)\\ t_3 := \sqrt{\frac{d}{\ell}}\\ t_4 := t_1 \cdot t_3\\ t_5 := t_1 \cdot \left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{if}\;\ell \leq -1.5 \cdot 10^{+160}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -4.6 \cdot 10^{-90}:\\ \;\;\;\;\left(1 - {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \cdot t_4\\ \mathbf{elif}\;\ell \leq -2.15 \cdot 10^{-187}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{-192}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;\ell \leq 6800000000:\\ \;\;\;\;t_4 \cdot \left(1 - \frac{\left(h \cdot 0.5\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 4.3 \cdot 10^{+246}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t_0 \cdot t_3\right)\\ \mathbf{else}:\\ \;\;\;\;t_5\\ \end{array} \]

Alternative 6
Accuracy	67.2%
Cost	21848

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := 1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\frac{D}{\ell} \cdot \left(M \cdot D\right)}{d} \cdot \frac{M}{\frac{d}{h}}\right)\right)\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := t_2 \cdot \left(t_1 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ t_4 := \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{if}\;\ell \leq -1.32 \cdot 10^{+115}:\\ \;\;\;\;\left(t_2 \cdot t_4\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D}{\frac{d}{D}} \cdot \frac{h \cdot \frac{M}{\frac{\ell}{M}}}{d}\right)\right)\\ \mathbf{elif}\;\ell \leq -2.8 \cdot 10^{-27}:\\ \;\;\;\;\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(D \cdot \left(0.125 \cdot \left(M \cdot \left(M \cdot \frac{D}{d}\right)\right)\right)\right) - d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_2 \cdot \left(t_4 \cdot t_1\right)\\ \mathbf{elif}\;\ell \leq 2.15 \cdot 10^{-192}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 7500000000:\\ \;\;\;\;\left(t_2 \cdot t_0\right) \cdot \left(1 - \frac{\left(h \cdot 0.5\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+246}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 7
Accuracy	66.3%
Cost	21848

\[\begin{array}{l} t_0 := 1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\frac{D}{\ell} \cdot \left(M \cdot D\right)}{d} \cdot \frac{M}{\frac{d}{h}}\right)\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := t_1 \cdot \left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ t_3 := \sqrt{\frac{d}{\ell}}\\ t_4 := \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{if}\;\ell \leq -2.05 \cdot 10^{+153}:\\ \;\;\;\;\left(t_1 \cdot t_4\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D}{\frac{d}{D}} \cdot \frac{h \cdot \frac{M}{\frac{\ell}{M}}}{d}\right)\right)\\ \mathbf{elif}\;\ell \leq -4.7 \cdot 10^{-88}:\\ \;\;\;\;\left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot {\left(\frac{d}{h}\right)}^{0.5}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_1 \cdot \left(t_4 \cdot t_0\right)\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{-192}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 3050000000:\\ \;\;\;\;\left(t_1 \cdot t_3\right) \cdot \left(1 - \frac{\left(h \cdot 0.5\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+247}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t_0 \cdot t_3\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Accuracy	67.9%
Cost	21584

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -6.5 \cdot 10^{+100}:\\ \;\;\;\;\frac{t_1}{\frac{\sqrt{-h}}{\sqrt{-d}}}\\ \mathbf{elif}\;d \leq -2.1 \cdot 10^{-103}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{M \cdot \frac{D}{\ell}}{\frac{d}{h} \cdot \frac{d}{M \cdot D}}\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.14 \cdot 10^{-138}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \frac{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2}}{\frac{\ell}{h}}\right)\right)\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{+39}:\\ \;\;\;\;t_0 \cdot \left(\left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\frac{D}{\ell} \cdot \left(M \cdot D\right)}{d} \cdot \frac{M}{\frac{d}{h}}\right)\right)\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 9
Accuracy	68.9%
Cost	21452

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := t_1 \cdot \left(t_2 \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{M \cdot \frac{D}{\ell}}{\frac{d}{h} \cdot \frac{d}{M \cdot D}}\right)\right)\right)\\ \mathbf{if}\;d \leq -2.4 \cdot 10^{+97}:\\ \;\;\;\;\frac{t_2}{\frac{\sqrt{-h}}{t_0}}\\ \mathbf{elif}\;d \leq -1.05 \cdot 10^{-61}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(t_1 \cdot \frac{t_0}{\sqrt{-\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D}{\frac{d}{D}} \cdot \frac{h \cdot \frac{M}{\frac{\ell}{M}}}{d}\right)\right)\\ \mathbf{elif}\;d \leq 2.25 \cdot 10^{-194}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{+44}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 10
Accuracy	67.1%
Cost	21132

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -2.45 \cdot 10^{+97}:\\ \;\;\;\;\frac{t_1}{\frac{\sqrt{-h}}{\sqrt{-d}}}\\ \mathbf{elif}\;d \leq -8.8 \cdot 10^{-103}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{M \cdot \frac{D}{\ell}}{\frac{d}{h} \cdot \frac{d}{M \cdot D}}\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.62 \cdot 10^{-138}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \frac{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2}}{\frac{\ell}{h}}\right)\right)\\ \mathbf{elif}\;d \leq 160000:\\ \;\;\;\;t_0 \cdot \left(\left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\frac{D}{\ell} \cdot \left(M \cdot D\right)}{d} \cdot \frac{M}{\frac{d}{h}}\right)\right)\right) \cdot t_1\right)\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{+25}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{h}{\ell} \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d \cdot d}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 11
Accuracy	68.5%
Cost	21068

\[\begin{array}{l} t_0 := d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{d}{h}} \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{M \cdot \frac{D}{\ell}}{\frac{d}{h} \cdot \frac{d}{M \cdot D}}\right)\right)\right)\\ \mathbf{if}\;d \leq -1.36 \cdot 10^{+106}:\\ \;\;\;\;\frac{t_1}{\frac{\sqrt{-h}}{\sqrt{-d}}}\\ \mathbf{elif}\;d \leq -6 \cdot 10^{-154}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -1.2 \cdot 10^{-308}:\\ \;\;\;\;\left(\left(M \cdot \frac{D \cdot D}{\frac{d}{M}}\right) \cdot 0.125\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} - t_0\\ \mathbf{elif}\;d \leq 10^{-195}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{+40}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 12
Accuracy	67.1%
Cost	19908

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{M \cdot \frac{D}{\ell}}{\frac{d}{h} \cdot \frac{d}{M \cdot D}}\right)\right)\right)\\ \mathbf{if}\;d \leq -1.02 \cdot 10^{+104}:\\ \;\;\;\;\frac{t_1}{\frac{\sqrt{-h}}{\sqrt{-d}}}\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-99}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -4.4 \cdot 10^{-213}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D}{\frac{d}{D}} \cdot \frac{h \cdot \frac{M}{\frac{\ell}{M}}}{d}\right)\right)\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-272}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{+46}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 13
Accuracy	64.5%
Cost	15316

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -1.9 \cdot 10^{+146}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;d \leq -5.5 \cdot 10^{-101}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\ell} \cdot \left(D \cdot \left(M \cdot \left(h \cdot \frac{M}{d \cdot d}\right)\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 3 \cdot 10^{-309}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D}{\frac{d}{D}} \cdot \frac{h \cdot \frac{M}{\frac{\ell}{M}}}{d}\right)\right)\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-14}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{+23}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{h}{\ell} \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d \cdot d}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 14
Accuracy	66.6%
Cost	15316

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \frac{M \cdot \frac{D}{\ell}}{\frac{d}{h} \cdot \frac{d}{M \cdot D}}\right)\right)\right)\\ \mathbf{if}\;d \leq -2.05 \cdot 10^{+153}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;d \leq -3.8 \cdot 10^{-98}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -1.8 \cdot 10^{-211}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D}{\frac{d}{D}} \cdot \frac{h \cdot \frac{M}{\frac{\ell}{M}}}{d}\right)\right)\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-272}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+46}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 15
Accuracy	63.0%
Cost	15188

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := t_0 \cdot t_1\\ t_3 := t_2 \cdot \left(1 - 0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{M \cdot M}{\frac{\ell}{h}}\right)\right)\\ \mathbf{if}\;d \leq -5.1 \cdot 10^{+94}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;d \leq -1.25 \cdot 10^{+15}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq -8.5 \cdot 10^{-46}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -2.55 \cdot 10^{-73}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(-0.125 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{M \cdot \left(h \cdot M\right)}{d \cdot d}\right)\right)\right)\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-174}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq 2.45 \cdot 10^{-290}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 16
Accuracy	65.3%
Cost	15184

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}} \cdot \left(t_0 \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\frac{D}{\ell} \cdot \left(M \cdot D\right)}{\frac{d}{h}} \cdot \frac{M}{d}\right)\right)\right)\right)\\ \mathbf{if}\;d \leq -5.7 \cdot 10^{+150}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;d \leq -1.75 \cdot 10^{-168}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 5.6 \cdot 10^{-199}:\\ \;\;\;\;\frac{t_0}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 17
Accuracy	64.9%
Cost	14924

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -3.5 \cdot 10^{+150}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-100}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{D}{\ell} \cdot \left(D \cdot \left(M \cdot \left(h \cdot \frac{M}{d \cdot d}\right)\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-310}:\\ \;\;\;\;\left(t_0 \cdot t_1\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D}{\frac{d}{D}} \cdot \frac{h \cdot \frac{M}{\frac{\ell}{M}}}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 18
Accuracy	66.9%
Cost	14920

\[\begin{array}{l} \mathbf{if}\;d \leq -3.5 \cdot 10^{+153}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(0.25 \cdot \left(\frac{\frac{D}{\ell} \cdot \left(M \cdot D\right)}{d} \cdot \left(M \cdot \frac{h}{d}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-290}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 19
Accuracy	63.9%
Cost	14792

\[\begin{array}{l} \mathbf{if}\;d \leq -8.5 \cdot 10^{+94}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{-308}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.125 \cdot \left(\frac{D}{\frac{d}{D}} \cdot \frac{h \cdot \frac{M}{\frac{\ell}{M}}}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 20
Accuracy	64.6%
Cost	13644

\[\begin{array}{l} \mathbf{if}\;d \leq -9.8 \cdot 10^{+147}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;d \leq -2.3 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.92 \cdot 10^{-290}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 21
Accuracy	64.8%
Cost	13644

\[\begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{+145}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-149}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-290}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 22
Accuracy	64.8%
Cost	13644

\[\begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{+147}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;d \leq -8.8 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{h}{d}\right)}^{-0.5}\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-290}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]

Alternative 23
Accuracy	63.3%
Cost	13516

\[\begin{array}{l} \mathbf{if}\;d \leq -2.15 \cdot 10^{+145}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{elif}\;d \leq -3.6 \cdot 10^{-149}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-290}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 24
Accuracy	62.8%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;d \leq 7.2 \cdot 10^{-282}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]

Alternative 25
Accuracy	58.1%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-304}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternative 26
Accuracy	58.3%
Cost	7044

\[\begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{-304}:\\ \;\;\;\;d \cdot \left(-t_0\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot t_0\\ \end{array} \]

Alternative 27
Accuracy	48.6%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;h \leq -1.02 \cdot 10^{-283}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]

Alternative 28
Accuracy	48.4%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;h \leq -1.02 \cdot 10^{-283}:\\ \;\;\;\;\sqrt{d \cdot \frac{\frac{d}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]

Alternative 29
Accuracy	48.5%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;h \leq -1.02 \cdot 10^{-283}:\\ \;\;\;\;\sqrt{\frac{d}{\ell \cdot \frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]

Alternative 30
Accuracy	48.6%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;h \leq -1.02 \cdot 10^{-283}:\\ \;\;\;\;\sqrt{\frac{d}{\ell \cdot \frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternative 31
Accuracy	31.6%
Cost	6720

\[\frac{d}{\sqrt{\ell \cdot h}} \]

Reproduce

herbie shell --seed 2023151 
(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)))))